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Created January 18, 2026 03:50
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Claude session: e2ec0911-f06f-40f3-8a73-4cc719ea9e7c
Raw
import.sh
#!/bin/bash
# Import this Claude session: curl -sL <raw_url> | bash
if ! command -v uvx &> /dev/null; then
echo "Installing uv..."
curl -LsSf https://astral.sh/uv/install.sh | sh
export PATH="$HOME/.local/bin:$PATH"
fi
uvx one_claude gist import 0623b9a9a2b1a5b5b949c97e5a8066a5
Raw
session.json
This file has been truncated, but you can view the full file.
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"title": "I kind of want to chase down the engram grokking thing, seems doable on one 5090?",
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{
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"content": "Web search results for query: \"Pedro Domingos tensor logic machine learning\"\n\nLinks: [{\"title\":\"[2510.12269] Tensor Logic: The Language of AI\",\"url\":\"https://arxiv.org/abs/2510.12269\"},{\"title\":\"Pedro Domingos\",\"url\":\"https://homes.cs.washington.edu/~pedrod/\"},{\"title\":\"Tensor Logic Tensor Logic: The Language of AI Pedro Domingos\",\"url\":\"https://arxiv.org/pdf/2510.12269\"},{\"title\":\"Paper page - Tensor Logic: The Language of AI\",\"url\":\"https://huggingface.co/papers/2510.12269\"},{\"title\":\"Tensor Logic The Language of AI Pedro Domingos University of Washington\",\"url\":\"https://homes.cs.washington.edu/~pedrod/tls.pdf\"},{\"title\":\"Pedro Domingos (@pmddomingos) / X\",\"url\":\"https://x.com/pmddomingos?lang=en\"},{\"title\":\"Tensor Logic: A New Language for AI by Pedro Domingos\",\"url\":\"https://www.linkedin.com/posts/machine-learning-street-talk_tensor-logic-is-new-language-for-ai-which-activity-7396269442768195584-qq3A\"},{\"title\":\"Pedro Domingos: Tensor Logic Unifies AI Paradigms by Machine Learning Street Talk (MLST)\",\"url\":\"https://creators.spotify.com/pod/profile/machinelearningstreettalk/episodes/Pedro-Domingos-Tensor-Logic-Unifies-AI-Paradigms-e3c16cv\"},{\"title\":\"FEP - DaSSWeb | Tensor Logic: A Simple Unification of Neural and Symbolic AI\",\"url\":\"https://sigarra.up.pt/fep/en/noticias_geral.ver_noticia?p_nr=51570\"},{\"title\":\"Pedro Domingos\",\"url\":\"https://scholar.google.com/citations?user=KOrhfVMAAAAJ&hl=en\"}]\n\nBased on the search results, here's what I found about Pedro Domingos and Tensor Logic:\n\n## Pedro Domingos - Tensor Logic\n\nPedro Domingos is a Professor of Computer Science at the University of Washington and author of 'The Master Algorithm' and '2040'.(no content)He is a winner of the SIGKDD Innovation Award and the IJCAI John McCarthy Award, two of the highest honors in data science and AI.\n\n### Tensor Logic: The Language of AI\n\nIn October 2025, Domingos published a paper on arXiv titled \"Tensor Logic: The Language of AI.\"\n\n**The Problem It Addresses:**\nProgress in AI is hindered by the lack of a programming language with all the requisite features. Libraries like PyTorch and TensorFlow provide automatic differentiation and efficient GPU implementation, but are additions to Python, which was never intended for AI. Their lack of support for automated reasoning and knowledge acquisition has led to a long and costly series of hacky attempts to tack them on.(no content)On the other hand, AI languages like LISP and Prolog lack scalability and support for learning.\n\n**The Solution:**\nThis paper proposes tensor logic, a language that solves these problems by unifying neural and symbolic AI at a fundamental level. The sole construct in tensor logic is the tensor equation, based on the observation that logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them.\n\n**Key Capabilities:**\nDomingos shows how to elegantly implement key forms of neural, symbolic and statistical AI in tensor logic, including transformers, formal reasoning, kernel machines and graphical models. Most importantly, tensor logic makes new directions possible, such as sound reasoning in embedding space. This combines the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning.\n\n**Technical Details:**\nTensor logic is based on the observation that predicates are compactly represented Boolean tensors, and can be straightforwardly extended to compactly represent numeric ones. The two key constructs in tensor logic are tensor join and project, numeric operations that generalize database join and project.(no content)Pedro argues that just as physics found its language in calculus and circuit design found its language in Boolean logic, AI has been missing its language - until now.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
}
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"content": "Web search results for query: \"NVIDIA cuTile 2026\"\n\nLinks: [{\"title\":\"Siemens unveils industrial AI innovations at CES 2026 | Siemens\",\"url\":\"https://news.siemens.com/en-us/siemens-unveils-technologies-to-accelerate-the-industrial-ai-revolution-at-ces-2026/\"},{\"title\":\"Nvidia plans heavy cuts to GPU supply in early 2026 - OC3D\",\"url\":\"https://overclock3d.net/news/gpu-displays/nvidia-plans-heavy-cuts-to-gpu-supply-in-early-2026/\"},{\"title\":\"CUDA Tile | NVIDIA Developer\",\"url\":\"https://developer.nvidia.com/cuda/tile\"},{\"title\":\"NVIDIA cuTile Python Guide Shows 90% cuBLAS Performance for Matrix Ops\",\"url\":\"https://bitcoinethereumnews.com/tech/nvidia-cutile-python-guide-shows-90-cublas-performance-for-matrix-ops/\"},{\"title\":\"How to Write High-Performance Matrix Multiply in NVIDIA CUDA Tile | NVIDIA Technical Blog\",\"url\":\"https://developer.nvidia.com/blog/how-to-write-high-performance-matrix-multiply-in-nvidia-cuda-tile/\"},{\"title\":\"GitHub - NVIDIA/cutile-python: cuTile is a programming model for writing parallel kernels for NVIDIA GPUs\",\"url\":\"https://github.com/NVIDIA/cutile-python\"},{\"title\":\"cuTile Python — cuTile Python\",\"url\":\"https://docs.nvidia.com/cuda/cutile-python/\"},{\"title\":\"NVIDIA at CES 2026 | January 5–9 | Las Vegas, NV\",\"url\":\"https://www.nvidia.com/en-us/events/ces/\"},{\"title\":\"NVIDIA CUDA Tile IR Open-Sourced - Phoronix\",\"url\":\"https://www.phoronix.com/news/NVIDIA-CUDA-Tile-IR-Open-Source\"},{\"title\":\"NVIDIA Makes GPU Acceleration Easier, More Portable with CUDA Tiles, cuTile Python - Hackster.io\",\"url\":\"https://www.hackster.io/news/nvidia-makes-gpu-acceleration-easier-more-portable-with-cuda-tiles-cutile-python-b808c1ad7e4d\"}]\n\nBased on the search results, here's information about **NVIDIA cuTile**:\n\n## Overview\n\nNVIDIA CUDA Tile is a tile-based GPU programming model that targets portability for NVIDIA Tensor Cores.(no content)cuTile is a parallel programming model for NVIDIA GPUs and a Python-based DSL. It automatically leverages advanced hardware capabilities, such as tensor cores and tensor memory accelerators, while providing portability across different NVIDIA GPU architectures.\n\n## Recent Developments (2025-2026)\n\nEarlier this month with CUDA 13.1 NVIDIA announced the CUDA Tile programming model as \"the largest and most comprehensive update to the CUDA platform since it was invented two decades ago.\"(no content)As a wonderful Christmas gift to open-source fans, NVIDIA dropped their proprietary license on the CUDA Tile intermediate representation and has now made the IR open-source software.(no content)The open-source code is under an Apache 2.0 license.\n\n## Performance\n\nNVIDIA has published a comprehensive developer guide for its cuTile Python framework, demonstrating how the new tile-based programming model can achieve over 90% of cuBLAS performance for matrix multiplication operations on Blackwell architecture GPUs.(no content)A complete matrix multiplication kernel in cuTile requires roughly 30 lines of Python code. The key operations: load tiles from matrices A and B, call ct.mma() for matrix multiply-accumulate (which auto-invokes tensor cores), and store results. The framework handles thread synchronization and memory access patterns internally.\n\n## Requirements\n\nCurrent requirements limit adoption: CUDA 13.1 minimum, Blackwell architecture only (RTX 50 series, compute capability 10.x and 12.x), and Python 3.10+.(no content)NVIDIA indicates broader architecture support will come in future CUDA releases.\n\n## Key Benefits\n\n\"cuTile automates block-level parallelism and asynchrony, memory movement, and other low-level details of GPU programming. It will leverage the advanced capabilities of NVIDIA hardware (such as tensor cores, shared memory, and tensor memory accelerators) without requiring explicit programming. cuTile is portable across different NVIDIA GPU architectures, enabling you to use the latest hardware features without rewriting your code.\"(no content)cuTile Python is a programming language for NVIDIA GPUs. The official documentation can be found on docs.nvidia.com.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "# Tensor Logic: Key Technical Overview\n\nBased on the arXiv abstract provided, here's what we can extract about tensor logic:\n\n## Core Concept\nThe fundamental building block is the **tensor equation**. According to the paper, \"logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them.\"\n\n## Unifying Neural and Symbolic AI\nTensor logic aims to bridge two traditionally separate domains by recognizing that mathematical operations underlying both approaches share common foundations. This unified framework allows implementation of diverse AI paradigms within a single language.\n\n## Supported AI Implementations\nThe paper demonstrates that tensor logic can elegantly implement:\n- Transformers (neural approach)\n- Formal reasoning (symbolic approach)\n- Kernel machines (statistical approach)\n- Graphical models (probabilistic approach)\n\n## Novel Direction: Sound Reasoning in Embedding Space\nThe most distinctive contribution is enabling \"sound reasoning in embedding space,\" which the author describes as combining \"the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning.\"\n\n## Motivation\nThe proposal addresses limitations in current tools: PyTorch and TensorFlow lack automated reasoning support, while traditional AI languages like LISP and Prolog don't scale or support learning effectively.\n\n**Note:** The abstract doesn't provide specific details about tensor join/project operations or implementation examples—only the high-level conceptual framework."
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"result": "# Tensor Logic: Key Technical Overview\n\nBased on the arXiv abstract provided, here's what we can extract about tensor logic:\n\n## Core Concept\nThe fundamental building block is the **tensor equation**. According to the paper, \"logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them.\"\n\n## Unifying Neural and Symbolic AI\nTensor logic aims to bridge two traditionally separate domains by recognizing that mathematical operations underlying both approaches share common foundations. This unified framework allows implementation of diverse AI paradigms within a single language.\n\n## Supported AI Implementations\nThe paper demonstrates that tensor logic can elegantly implement:\n- Transformers (neural approach)\n- Formal reasoning (symbolic approach)\n- Kernel machines (statistical approach)\n- Graphical models (probabilistic approach)\n\n## Novel Direction: Sound Reasoning in Embedding Space\nThe most distinctive contribution is enabling \"sound reasoning in embedding space,\" which the author describes as combining \"the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning.\"\n\n## Motivation\nThe proposal addresses limitations in current tools: PyTorch and TensorFlow lack automated reasoning support, while traditional AI languages like LISP and Prolog don't scale or support learning effectively.\n\n**Note:** The abstract doesn't provide specific details about tensor join/project operations or implementation examples—only the high-level conceptual framework.",
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"content": "# cuTile: Key Technical Details\n\n## Programming Model\n\ncuTile shifts from **thread-level to block-level thinking**. Each CUDA block processes a single tile of the output matrix, with the framework handling memory access and synchronization automatically.\n\n## Kernel Structure\n\nKernels use the `@ct.kernel` decorator and follow this pattern:\n1. **Determine work scope** via block ID mapping\n2. **Load tiles** from input matrices\n3. **Compute** using `ct.mma()` (matrix multiply-accumulate)\n4. **Store results** back to global memory\n\n## The Tile Abstraction\n\n\"Each Block processes a (tm × tn) tile of the output matrix C. Loop over the K dimension, loading corresponding tiles of A and B one by one.\"\n\nTile sizes use **compile-time constants** (annotated as `ct.Constant[int]`), enabling compiler optimizations like loop unrolling and specialized Tensor Core instruction generation.\n\n## Tensor Core Integration\n\n`ct.mma(a, b, accumulator)` automatically invokes Tensor Cores when tile shapes meet hardware requirements, eliminating manual coordination.\n\n## Code Example\n\n```python\n@ct.kernel\ndef matmul_kernel(A, B, C, tm: ConstInt, tn: ConstInt, tk: ConstInt):\n bidx, bidy = swizzle_2d(M, N, tm, tn, GROUP_SIZE_M)\n accumulator = ct.full((tm, tn), 0, dtype=ct.float32)\n for k in range(ct.num_tiles(A, axis=1, shape=(tm, tk))):\n a = ct.load(A, index=(bidx, k), shape=(tm, tk))\n b = ct.load(B, index=(k, bidy), shape=(tk, tn))\n accumulator = ct.mma(a, b, accumulator)\n ct.store(C, index=(bidx, bidy), tile=accumulator)\n```\n\n## Performance Optimization: Swizzling\n\nBlock ID remapping improves cache locality by reorganizing computation into grouped patterns, reducing memory traffic by ~20% versus linear access."
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"result": "# cuTile: Key Technical Details\n\n## Programming Model\n\ncuTile shifts from **thread-level to block-level thinking**. Each CUDA block processes a single tile of the output matrix, with the framework handling memory access and synchronization automatically.\n\n## Kernel Structure\n\nKernels use the `@ct.kernel` decorator and follow this pattern:\n1. **Determine work scope** via block ID mapping\n2. **Load tiles** from input matrices\n3. **Compute** using `ct.mma()` (matrix multiply-accumulate)\n4. **Store results** back to global memory\n\n## The Tile Abstraction\n\n\"Each Block processes a (tm × tn) tile of the output matrix C. Loop over the K dimension, loading corresponding tiles of A and B one by one.\"\n\nTile sizes use **compile-time constants** (annotated as `ct.Constant[int]`), enabling compiler optimizations like loop unrolling and specialized Tensor Core instruction generation.\n\n## Tensor Core Integration\n\n`ct.mma(a, b, accumulator)` automatically invokes Tensor Cores when tile shapes meet hardware requirements, eliminating manual coordination.\n\n## Code Example\n\n```python\n@ct.kernel\ndef matmul_kernel(A, B, C, tm: ConstInt, tn: ConstInt, tk: ConstInt):\n bidx, bidy = swizzle_2d(M, N, tm, tn, GROUP_SIZE_M)\n accumulator = ct.full((tm, tn), 0, dtype=ct.float32)\n for k in range(ct.num_tiles(A, axis=1, shape=(tm, tk))):\n a = ct.load(A, index=(bidx, k), shape=(tm, tk))\n b = ct.load(B, index=(k, bidy), shape=(tk, tn))\n accumulator = ct.mma(a, b, accumulator)\n ct.store(C, index=(bidx, bidy), tile=accumulator)\n```\n\n## Performance Optimization: Swizzling\n\nBlock ID remapping improves cache locality by reorganizing computation into grouped patterns, reducing memory traffic by ~20% versus linear access.",
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"thinking": "Now I have a good overview of both topics. Let me summarize what I've found for the user. They're in a directory called \"tensorlogic\" which suggests they might be interested in implementing something related to tensor logic, possibly using cuTile. Let me present the findings clearly.",
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"text": "## Pedro Domingos' Tensor Logic\n\n**Core Idea:** Tensor logic proposes that *logical rules and Einstein summation are essentially the same operation*. The sole construct is the **tensor equation**.\n\n**Key insight:** Predicates can be represented as compact Boolean tensors, which naturally extend to numeric tensors. The two fundamental operations are:\n- **Tensor join** - generalizes database join\n- **Tensor project** - generalizes database project\n\n**What it unifies:** The paper shows elegant implementations of:\n- Transformers (neural)\n- Formal reasoning (symbolic)\n- Kernel machines (statistical)\n- Graphical models (probabilistic)\n\n**Novel capability:** Sound reasoning in embedding space — combining neural network scalability/learnability with symbolic reasoning reliability/transparency.\n\n**Motivation:** PyTorch/TensorFlow lack automated reasoning; LISP/Prolog lack scalability and learning support. Tensor logic aims to be \"the language AI has been missing.\"\n\n---\n\n## NVIDIA cuTile\n\n**Core Idea:** A tile-based GPU programming model that shifts from thread-level to **block-level thinking**. Each CUDA block processes one tile of output.\n\n**Key features:**\n- ~30 lines of Python for a complete matrix multiply kernel\n- Achieves **>90% cuBLAS performance** on Blackwell\n- Automatic Tensor Core invocation via `ct.mma()`\n- Portable across NVIDIA architectures (no rewrite needed)\n\n**Requirements:** CUDA 13.1+, Blackwell GPUs only (RTX 50 series), Python 3.10+\n\n**Recent news:** NVIDIA open-sourced the CUDA Tile IR under Apache 2.0 in December 2025.\n\n---\n\n**Sources:**\n- [Tensor Logic: The Language of AI (arXiv)](https://arxiv.org/abs/2510.12269)\n- [Pedro Domingos' Homepage](https://homes.cs.washington.edu/~pedrod/)\n- [NVIDIA cuTile Developer Page](https://developer.nvidia.com/cuda/tile)\n- [cuTile Python Documentation](https://docs.nvidia.com/cuda/cutile-python/)\n- [High-Performance Matrix Multiply in cuTile (NVIDIA Blog)](https://developer.nvidia.com/blog/how-to-write-high-performance-matrix-multiply-in-nvidia-cuda-tile/)\n- [cuTile Python GitHub](https://github.com/NVIDIA/cutile-python)\n\nWould you like me to dig deeper into either topic, or are you thinking about implementing tensor logic operations using cuTile?"
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"content": "yeah the idea is to do an implementation of tensor logic using cutile. Here's an early take :\n\n\"\"\"\nTensor Logic Compiler for cuTile\n\nCompiles Datalog-style tensor rules to GPU kernels.\n\nExamples:\n # Standard matmul\n C[i,k] <- A[i,j], B[j,k]\n \n # Attention scores \n Score[b,p,q] <- Q[b,p,d], K[b,q,d]\n \n # CTM Neuron-Level Model\n H[b,d,h] <- A[b,d,m], W[m,h,d]\n \n # Transitive closure\n Path[x,z] <- Path[x,y], Edge[y,z]\n\nUsage:\n rule = parse_rule(\"C[i,k] <- A[i,j], B[j,k]\")\n shapes = {\"A\": {\"i\": 1024, \"j\": 512}, \"B\": {\"j\": 512, \"k\": 256}, \"C\": {\"i\": 1024, \"k\": 256}}\n kernel_code = compile_rule(rule, shapes, \"matmul_kernel\")\n\"\"\"\n\nfrom __future__ import annotations\nfrom dataclasses import dataclass, field\nfrom typing import Dict, List, Set, Tuple, Optional\nfrom enum import Enum, auto\n\n\n# ============================================================================\n# Data Structures\n# ============================================================================\n\n@dataclass(frozen=True)\nclass TensorRef:\n \"\"\"Reference to a tensor with named indices.\"\"\"\n name: str\n indices: Tuple[str, ...]\n \n def __repr__(self) -> str:\n return f\"{self.name}[{','.join(self.indices)}]\"\n \n @staticmethod\n def parse(s: str) -> TensorRef:\n \"\"\"Parse 'Name[i,j,k]' format.\"\"\"\n s = s.strip()\n name, rest = s.split(\"[\")\n indices = tuple(idx.strip() for idx in rest.rstrip(\"]\").split(\",\"))\n return TensorRef(name, indices)\n\n\n@dataclass(frozen=True)\nclass Rule:\n \"\"\"A tensor logic rule: head <- body[0], body[1], ...\"\"\"\n head: TensorRef\n body: Tuple[TensorRef, ...]\n \n def __repr__(self) -> str:\n body_str = \", \".join(str(t) for t in self.body)\n return f\"{self.head} <- {body_str}\"\n \n def to_einsum(self) -> str:\n \"\"\"Convert to einsum notation.\"\"\"\n inputs = \",\".join(\"\".join(t.indices) for t in self.body)\n output = \"\".join(self.head.indices)\n return f\"{inputs}->{output}\"\n \n @staticmethod\n def parse(s: str) -> Rule:\n \"\"\"Parse 'Head[i,j] <- A[i,k], B[k,j]' format.\"\"\"\n head_str, body_str = s.split(\"<-\")\n head = TensorRef.parse(head_str)\n body = tuple(\n TensorRef.parse(t) \n for t in body_str.split(\",\") \n if \"[\" in t\n )\n return Rule(head, body)\n\n\nclass IndexRole(Enum):\n \"\"\"Role of an index in a contraction.\"\"\"\n CONTRACTED = auto() # Summed over (in inputs, not output)\n BATCH = auto() # In all inputs and output\n FREE_LEFT = auto() # In output and first input only\n FREE_RIGHT = auto() # In output and second input only \n FREE_SHARED = auto() # In output and multiple inputs\n\n\n@dataclass\nclass IndexInfo:\n \"\"\"Analysis of a single index.\"\"\"\n name: str\n role: IndexRole\n size: int\n tensors: List[str] # Which tensors contain this index\n\n\n@dataclass\nclass MatmulPattern:\n \"\"\"Describes how a contraction maps to matrix multiplication.\"\"\"\n is_matmul: bool\n m_indices: Tuple[str, ...] # Row dimensions (in A and output)\n n_indices: Tuple[str, ...] # Column dimensions (in B and output)\n k_indices: Tuple[str, ...] # Contracted dimensions\n batch_indices: Tuple[str, ...] # Batch dimensions (in all)\n a_transpose: bool\n b_transpose: bool\n reason: str # Why it is/isn't matmul\n\n\n@dataclass\nclass TilingPlan:\n \"\"\"How to tile the computation.\"\"\"\n grid_dims: List[str] # Indices to parallelize over\n tile_sizes: Dict[str, int]\n reduction_loops: List[str] # Contracted indices\n free_loops: List[str] # Non-grid free indices\n\n\n# ============================================================================\n# Analysis\n# ============================================================================\n\ndef analyze_indices(rule: Rule, shapes: Dict[str, Dict[str, int]]) -> Dict[str, IndexInfo]:\n \"\"\"Determine the role and size of each index in a rule.\"\"\"\n \n output_indices = set(rule.head.indices)\n \n # Track which tensors each index appears in\n index_to_tensors: Dict[str, List[str]] = {}\n index_to_size: Dict[str, int] = {}\n \n # Scan inputs\n for tensor in rule.body:\n tensor_shapes = shapes.get(tensor.name, {})\n for idx in tensor.indices:\n if idx not in index_to_tensors:\n index_to_tensors[idx] = []\n index_to_tensors[idx].append(tensor.name)\n if idx in tensor_shapes:\n index_to_size[idx] = tensor_shapes[idx]\n \n # Scan output\n output_shapes = shapes.get(rule.head.name, {})\n for idx in rule.head.indices:\n if idx not in index_to_tensors:\n index_to_tensors[idx] = []\n index_to_tensors[idx].append(rule.head.name)\n if idx in output_shapes:\n index_to_size[idx] = output_shapes[idx]\n \n # Classify each index\n result = {}\n n_inputs = len(rule.body)\n \n for idx in set(index_to_tensors.keys()):\n in_output = idx in output_indices\n input_tensors = [t for t in index_to_tensors[idx] if t != rule.head.name]\n n_in_inputs = len(input_tensors)\n \n if not in_output:\n role = IndexRole.CONTRACTED\n elif n_in_inputs == n_inputs:\n role = IndexRole.BATCH\n elif n_in_inputs > 1:\n role = IndexRole.FREE_SHARED\n elif n_inputs == 2 and input_tensors[0] == rule.body[0].name:\n role = IndexRole.FREE_LEFT\n elif n_inputs == 2 and input_tensors[0] == rule.body[1].name:\n role = IndexRole.FREE_RIGHT\n else:\n role = IndexRole.FREE_SHARED\n \n result[idx] = IndexInfo(\n name=idx,\n role=role,\n size=index_to_size.get(idx, 0),\n tensors=index_to_tensors[idx]\n )\n \n return result\n\n\ndef detect_matmul(rule: Rule, index_info: Dict[str, IndexInfo]) -> MatmulPattern:\n \"\"\"Detect if a rule can be computed as matrix multiplication.\"\"\"\n \n # Must be binary\n if len(rule.body) != 2:\n return MatmulPattern(\n False, (), (), (), (), False, False,\n f\"Not binary: {len(rule.body)} inputs\"\n )\n \n t1, t2 = rule.body\n t1_set = set(t1.indices)\n t2_set = set(t2.indices)\n out_set = set(rule.head.indices)\n \n # Classify indices\n contracted = tuple(idx for idx, info in index_info.items() \n if info.role == IndexRole.CONTRACTED)\n batch = tuple(idx for idx, info in index_info.items()\n if info.role == IndexRole.BATCH)\n m_indices = tuple(idx for idx, info in index_info.items()\n if info.role == IndexRole.FREE_LEFT)\n n_indices = tuple(idx for idx, info in index_info.items()\n if info.role == IndexRole.FREE_RIGHT)\n shared = tuple(idx for idx, info in index_info.items()\n if info.role == IndexRole.FREE_SHARED)\n \n # Shared indices act like batch for matmul purposes\n effective_batch = batch + shared\n \n if not contracted:\n return MatmulPattern(\n False, (), (), (), (), False, False,\n \"No contracted indices\"\n )\n \n if not m_indices and not n_indices:\n # Both inputs contribute to same output dimensions\n # Still valid if we treat shared as providing both M and N\n pass\n \n # Determine transpose requirements\n # Standard: A[..., M, K] @ B[..., K, N] -> C[..., M, N]\n \n def needs_transpose(tensor_indices: Tuple[str, ...], \n row_indices: Tuple[str, ...],\n col_indices: Tuple[str, ...]) -> bool:\n \"\"\"Check if tensor needs transpose for matmul.\"\"\"\n # Get positions of row and col indices (excluding batch)\n non_batch = [idx for idx in tensor_indices if idx not in effective_batch]\n if len(non_batch) < 2:\n return False\n \n # Find last positions\n row_positions = [non_batch.index(idx) for idx in row_indices if idx in non_batch]\n col_positions = [non_batch.index(idx) for idx in col_indices if idx in non_batch]\n \n if not row_positions or not col_positions:\n return False\n \n # For A: rows (M) should come before cols (K) -> no transpose\n # For B: rows (K) should come before cols (N) -> no transpose\n return max(row_positions) > min(col_positions)\n \n a_transpose = needs_transpose(t1.indices, m_indices, contracted)\n b_transpose = needs_transpose(t2.indices, contracted, n_indices)\n \n return MatmulPattern(\n is_matmul=True,\n m_indices=m_indices if m_indices else tuple(shared[:1]) if shared else (),\n n_indices=n_indices if n_indices else tuple(shared[1:2]) if len(shared) > 1 else (),\n k_indices=contracted,\n batch_indices=effective_batch,\n a_transpose=a_transpose,\n b_transpose=b_transpose,\n reason=f\"M={m_indices}, N={n_indices}, K={contracted}, batch={effective_batch}\"\n )\n\n\ndef plan_tiling(\n rule: Rule,\n index_info: Dict[str, IndexInfo],\n min_parallelism: int = 1024,\n default_tile: int = 64\n) -> TilingPlan:\n \"\"\"Create a tiling plan for the computation.\"\"\"\n \n # Separate by role\n contracted = [idx for idx, info in index_info.items() \n if info.role == IndexRole.CONTRACTED]\n free = [idx for idx, info in index_info.items()\n if info.role != IndexRole.CONTRACTED]\n \n # Sort free indices by size (largest first for grid)\n free_sorted = sorted(\n free, \n key=lambda x: -(index_info[x].size or default_tile)\n )\n \n # Select grid dimensions\n grid_dims = []\n parallelism = 1\n for idx in free_sorted:\n size = index_info[idx].size or default_tile\n grid_dims.append(idx)\n parallelism *= size\n if parallelism >= min_parallelism and len(grid_dims) >= 2:\n break\n \n # Compute tile sizes\n tile_sizes = {}\n for idx, info in index_info.items():\n size = info.size or default_tile\n if info.role == IndexRole.CONTRACTED:\n tile_sizes[idx] = min(32, size) # Smaller tiles for reduction\n elif idx in grid_dims:\n tile_sizes[idx] = min(default_tile, size)\n else:\n tile_sizes[idx] = min(32, size)\n \n # Remaining free indices become loops\n free_loops = [idx for idx in free if idx not in grid_dims]\n \n return TilingPlan(\n grid_dims=grid_dims,\n tile_sizes=tile_sizes,\n reduction_loops=contracted,\n free_loops=free_loops\n )\n\n\n# ============================================================================\n# Code Generation\n# ============================================================================\n\nclass CodeBuilder:\n \"\"\"Helper for building indented code.\"\"\"\n \n def __init__(self):\n self.lines: List[str] = []\n self.indent_level: int = 0\n self.indent_str: str = \" \"\n \n def line(self, text: str = \"\") -> None:\n if text:\n self.lines.append(self.indent_str * self.indent_level + text)\n else:\n self.lines.append(\"\")\n \n def indent(self) -> None:\n self.indent_level += 1\n \n def dedent(self) -> None:\n self.indent_level = max(0, self.indent_level - 1)\n \n def build(self) -> str:\n return \"\\n\".join(self.lines)\n\n\ndef generate_kernel(\n rule: Rule,\n index_info: Dict[str, IndexInfo],\n tiling: TilingPlan,\n pattern: MatmulPattern,\n kernel_name: str\n) -> str:\n \"\"\"Generate a complete cuTile kernel.\"\"\"\n \n cb = CodeBuilder()\n \n # Collect unique names\n all_tensors = [rule.head.name] + [t.name for t in rule.body]\n unique_tensors = list(dict.fromkeys(all_tensors))\n \n # Build parameter list (avoiding duplicates)\n tensor_params = unique_tensors\n size_params = []\n seen_sizes = set()\n for idx in index_info:\n param_name = f\"{idx.upper()}\"\n if param_name not in seen_sizes:\n size_params.append(f\"{param_name}: ct.Constant[int]\")\n seen_sizes.add(param_name)\n \n tile_params = []\n seen_tiles = set()\n for idx in tiling.tile_sizes:\n param_name = f\"TILE_{idx.upper()}\"\n if param_name not in seen_tiles:\n tile_params.append(f\"{param_name}: ct.Constant[int]\")\n seen_tiles.add(param_name)\n \n all_params = tensor_params + size_params + tile_params\n \n # Kernel signature\n cb.line(\"@ct.kernel\")\n cb.line(f\"def {kernel_name}({', '.join(all_params)}):\")\n cb.indent()\n \n # Docstring\n cb.line('\"\"\"')\n cb.line(f\"Compiled from: {rule}\")\n cb.line(f\"Einsum: {rule.to_einsum()}\")\n if pattern.is_matmul:\n cb.line(f\"Pattern: matmul with {pattern.reason}\")\n cb.line('\"\"\"')\n cb.line()\n \n # Grid index setup\n for i, idx in enumerate(tiling.grid_dims):\n cb.line(f\"# Grid dimension {i}: {idx}\")\n cb.line(f\"{idx}_tile_idx = ct.program_id({i})\")\n cb.line(f\"{idx}_offset = {idx}_tile_idx * TILE_{idx.upper()}\")\n cb.line()\n \n # Accumulator\n acc_shape = \", \".join(f\"TILE_{idx.upper()}\" for idx in rule.head.indices)\n cb.line(f\"# Output accumulator: [{', '.join(rule.head.indices)}]\")\n cb.line(f\"acc = ct.zeros(({acc_shape},), dtype=ct.float32)\")\n cb.line()\n \n # Generate loops for non-grid free indices\n for idx in tiling.free_loops:\n cb.line(f\"for {idx}_offset in range(0, {idx.upper()}, TILE_{idx.upper()}):\")\n cb.indent()\n \n # Generate reduction loops\n for idx in tiling.reduction_loops:\n cb.line(f\"# Reduction over {idx}\")\n cb.line(f\"for {idx}_offset in range(0, {idx.upper()}, TILE_{idx.upper()}):\")\n cb.indent()\n \n # Load tiles\n cb.line(\"# Load input tiles\")\n tile_vars = []\n for tensor in rule.body:\n var_name = f\"tile_{tensor.name.lower()}\"\n tile_vars.append(var_name)\n \n coords = []\n shape_parts = []\n for idx in tensor.indices:\n if idx in tiling.grid_dims:\n coords.append(f\"{idx}_offset\")\n elif idx in tiling.free_loops or idx in tiling.reduction_loops:\n coords.append(f\"{idx}_offset\")\n else:\n coords.append(\"0\")\n shape_parts.append(f\"TILE_{idx.upper()}\")\n \n coord_str = \", \".join(coords)\n shape_str = \", \".join(shape_parts)\n cb.line(f\"{var_name} = ct.load({tensor.name}, ({coord_str},), shape=({shape_str},))\")\n cb.line()\n \n # Generate contraction\n if pattern.is_matmul:\n generate_matmul_op(cb, rule, pattern, tile_vars, tiling)\n else:\n generate_broadcast_op(cb, rule, index_info, tile_vars)\n \n # Close loops\n for _ in tiling.reduction_loops:\n cb.dedent()\n for _ in tiling.free_loops:\n cb.dedent()\n \n cb.line()\n \n # Store result\n cb.line(\"# Store result\")\n out_coords = []\n for idx in rule.head.indices:\n if idx in tiling.grid_dims:\n out_coords.append(f\"{idx}_offset\")\n elif idx in tiling.free_loops:\n out_coords.append(f\"{idx}_offset\")\n else:\n out_coords.append(\"0\")\n \n cb.line(f\"ct.store({rule.head.name}, ({', '.join(out_coords)},), tile=acc)\")\n \n cb.dedent()\n return cb.build()\n\n\ndef generate_matmul_op(\n cb: CodeBuilder,\n rule: Rule,\n pattern: MatmulPattern,\n tile_vars: List[str],\n tiling: TilingPlan\n) -> None:\n \"\"\"Generate matmul-based contraction.\"\"\"\n \n a_var, b_var = tile_vars\n has_batch = len(pattern.batch_indices) > 0\n \n cb.line(f\"# Matmul: M={pattern.m_indices}, N={pattern.n_indices}, K={pattern.k_indices}\")\n \n if not has_batch and len(pattern.m_indices) <= 1 and len(pattern.n_indices) <= 1 and len(pattern.k_indices) == 1:\n # Simple 2D matmul\n a_expr = a_var\n b_expr = b_var\n \n if pattern.a_transpose:\n cb.line(f\"a = ct.transpose({a_var})\")\n a_expr = \"a\"\n \n if pattern.b_transpose:\n cb.line(f\"b = ct.transpose({b_var})\")\n b_expr = \"b\"\n \n cb.line(f\"acc += ct.dot({a_expr}, {b_expr})\")\n \n else:\n # Batched or multi-dimensional - need reshape\n cb.line(\"# Reshape for batched matmul\")\n \n batch_size = \" * \".join(f\"TILE_{idx.upper()}\" for idx in pattern.batch_indices) if pattern.batch_indices else \"1\"\n m_size = \" * \".join(f\"TILE_{idx.upper()}\" for idx in pattern.m_indices) if pattern.m_indices else \"1\"\n n_size = \" * \".join(f\"TILE_{idx.upper()}\" for idx in pattern.n_indices) if pattern.n_indices else \"1\"\n k_size = \" * \".join(f\"TILE_{idx.upper()}\" for idx in pattern.k_indices)\n \n # Determine permutation for A: want [batch..., m..., k...]\n t1 = rule.body[0]\n a_target_order = list(pattern.batch_indices) + list(pattern.m_indices) + list(pattern.k_indices)\n a_current_order = [idx for idx in t1.indices if idx in a_target_order]\n \n if a_current_order != a_target_order:\n perm = [a_current_order.index(idx) if idx in a_current_order else 0 for idx in a_target_order]\n perm_str = \", \".join(str(p) for p in perm)\n cb.line(f\"a_perm = ct.permute({a_var}, ({perm_str},))\")\n a_reshape_var = \"a_perm\"\n else:\n a_reshape_var = a_var\n \n if pattern.batch_indices:\n cb.line(f\"a_mat = ct.reshape({a_reshape_var}, ({batch_size}, {m_size}, {k_size}))\")\n else:\n cb.line(f\"a_mat = ct.reshape({a_reshape_var}, ({m_size}, {k_size}))\")\n \n if pattern.a_transpose:\n if pattern.batch_indices:\n cb.line(\"a_mat = ct.permute(a_mat, (0, 2, 1))\")\n else:\n cb.line(\"a_mat = ct.transpose(a_mat)\")\n \n # Similar for B: want [batch..., k..., n...]\n t2 = rule.body[1]\n b_target_order = list(pattern.batch_indices) + list(pattern.k_indices) + list(pattern.n_indices)\n b_current_order = [idx for idx in t2.indices if idx in b_target_order]\n \n if b_current_order != b_target_order:\n perm = [b_current_order.index(idx) if idx in b_current_order else 0 for idx in b_target_order]\n perm_str = \", \".join(str(p) for p in perm)\n cb.line(f\"b_perm = ct.permute({b_var}, ({perm_str},))\")\n b_reshape_var = \"b_perm\"\n else:\n b_reshape_var = b_var\n \n if pattern.batch_indices:\n cb.line(f\"b_mat = ct.reshape({b_reshape_var}, ({batch_size}, {k_size}, {n_size}))\")\n else:\n cb.line(f\"b_mat = ct.reshape({b_reshape_var}, ({k_size}, {n_size}))\")\n \n if pattern.b_transpose:\n if pattern.batch_indices:\n cb.line(\"b_mat = ct.permute(b_mat, (0, 2, 1))\")\n else:\n cb.line(\"b_mat = ct.transpose(b_mat)\")\n \n # Matmul\n if pattern.batch_indices:\n cb.line(\"result = ct.bmm(a_mat, b_mat)\")\n else:\n cb.line(\"result = ct.dot(a_mat, b_mat)\")\n \n # Reshape back\n out_shape = \", \".join(f\"TILE_{idx.upper()}\" for idx in rule.head.indices)\n cb.line(f\"acc += ct.reshape(result, ({out_shape},))\")\n\n\ndef generate_broadcast_op(\n cb: CodeBuilder,\n rule: Rule,\n index_info: Dict[str, IndexInfo],\n tile_vars: List[str]\n) -> None:\n \"\"\"Generate broadcast + elementwise + reduce contraction.\"\"\"\n \n cb.line(\"# Broadcast and reduce (fallback)\")\n \n # Canonical order: output indices then contracted\n contracted = [idx for idx, info in index_info.items() \n if info.role == IndexRole.CONTRACTED]\n canonical = list(rule.head.indices) + contracted\n \n aligned_vars = []\n for i, tensor in enumerate(rule.body):\n var = tile_vars[i]\n present = [idx for idx in canonical if idx in tensor.indices]\n missing = [idx for idx in canonical if idx not in tensor.indices]\n \n current = var\n \n # Permute to canonical order\n if present != [idx for idx in tensor.indices]:\n perm = [tensor.indices.index(idx) for idx in present if idx in tensor.indices]\n cb.line(f\"{var}_p = ct.permute({current}, ({', '.join(map(str, perm))},))\")\n current = f\"{var}_p\"\n \n # Broadcast missing dimensions\n for j, idx in enumerate(missing):\n pos = canonical.index(idx)\n cb.line(f\"{var}_b{j} = ct.unsqueeze({current}, {pos}) # {idx}\")\n current = f\"{var}_b{j}\"\n \n aligned_vars.append(current)\n \n # Multiply\n cb.line(f\"product = {aligned_vars[0]} * {aligned_vars[1]}\")\n \n # Reduce contracted dimensions (from end to preserve positions)\n current = \"product\"\n contracted_positions = sorted([canonical.index(idx) for idx in contracted], reverse=True)\n \n for i, pos in enumerate(contracted_positions):\n next_var = \"result\" if i == len(contracted_positions) - 1 else f\"r{i}\"\n cb.line(f\"{next_var} = ct.sum({current}, axis={pos})\")\n current = next_var\n \n cb.line(f\"acc += {current}\")\n\n\n# ============================================================================\n# Main API\n# ============================================================================\n\ndef compile_rule(\n rule: Rule,\n shapes: Dict[str, Dict[str, int]],\n kernel_name: str = \"kernel\"\n) -> str:\n \"\"\"\n Compile a tensor logic rule to a cuTile kernel.\n \n Args:\n rule: The rule to compile (use Rule.parse() to create)\n shapes: Tensor shapes as {tensor_name: {index_name: size}}\n kernel_name: Name for the generated kernel function\n \n Returns:\n Generated cuTile kernel source code\n \"\"\"\n index_info = analyze_indices(rule, shapes)\n pattern = detect_matmul(rule, index_info)\n tiling = plan_tiling(rule, index_info)\n return generate_kernel(rule, index_info, tiling, pattern, kernel_name)\n\n\ndef compile_rules(\n rules: List[Rule],\n shapes: Dict[str, Dict[str, int]],\n prefix: str = \"kernel\"\n) -> str:\n \"\"\"Compile multiple rules to kernels.\"\"\"\n kernels = []\n for i, rule in enumerate(rules):\n kernels.append(compile_rule(rule, shapes, f\"{prefix}_{i}\"))\n return \"\\n\\n\".join(kernels)\n\n\n# Convenience aliases\nparse_rule = Rule.parse\nparse_tensor = TensorRef.parse\n\n\n# ============================================================================\n# Tests\n# ============================================================================\n\ndef run_tests():\n \"\"\"Run comprehensive tests.\"\"\"\n \n test_cases = [\n {\n \"name\": \"Standard Matmul\",\n \"rule\": \"C[i,k] <- A[i,j], B[j,k]\",\n \"shapes\": {\n \"A\": {\"i\": 1024, \"j\": 512},\n \"B\": {\"j\": 512, \"k\": 256},\n \"C\": {\"i\": 1024, \"k\": 256}\n }\n },\n {\n \"name\": \"Transposed Matmul (A.T @ B)\",\n \"rule\": \"C[i,k] <- A[j,i], B[j,k]\",\n \"shapes\": {\n \"A\": {\"j\": 512, \"i\": 1024},\n \"B\": {\"j\": 512, \"k\": 256},\n \"C\": {\"i\": 1024, \"k\": 256}\n }\n },\n {\n \"name\": \"Batched Attention Scores\",\n \"rule\": \"Score[b,p,q] <- Q[b,p,d], K[b,q,d]\",\n \"shapes\": {\n \"Q\": {\"b\": 32, \"p\": 512, \"d\": 64},\n \"K\": {\"b\": 32, \"q\": 512, \"d\": 64},\n \"Score\": {\"b\": 32, \"p\": 512, \"q\": 512}\n }\n },\n {\n \"name\": \"CTM NLM Layer\",\n \"rule\": \"H[b,d,h] <- A[b,d,m], W[m,h,d]\",\n \"shapes\": {\n \"A\": {\"b\": 64, \"d\": 2048, \"m\": 25},\n \"W\": {\"m\": 25, \"h\": 32, \"d\": 2048},\n \"H\": {\"b\": 64, \"d\": 2048, \"h\": 32}\n }\n },\n {\n \"name\": \"CTM Synchronization\",\n \"rule\": \"S[b,i,j] <- Z1[b,i,t], Z2[b,j,t]\",\n \"shapes\": {\n \"Z1\": {\"b\": 64, \"i\": 2048, \"t\": 50},\n \"Z2\": {\"b\": 64, \"j\": 2048, \"t\": 50},\n \"S\": {\"b\": 64, \"i\": 2048, \"j\": 2048}\n }\n },\n {\n \"name\": \"Transitive Closure\",\n \"rule\": \"Path[x,z] <- Path[x,y], Edge[y,z]\",\n \"shapes\": {\n \"Path\": {\"x\": 1000, \"y\": 1000, \"z\": 1000},\n \"Edge\": {\"y\": 1000, \"z\": 1000}\n }\n },\n ]\n \n print(\"=\" * 80)\n print(\"TENSOR LOGIC COMPILER - TEST SUITE\")\n print(\"=\" * 80)\n \n for tc in test_cases:\n print(f\"\\n{'─' * 80}\")\n print(f\"TEST: {tc['name']}\")\n print(f\"{'─' * 80}\")\n \n rule = Rule.parse(tc[\"rule\"])\n shapes = tc[\"shapes\"]\n \n # Analysis\n index_info = analyze_indices(rule, shapes)\n pattern = detect_matmul(rule, index_info)\n tiling = plan_tiling(rule, index_info)\n \n print(f\"\\nRule: {rule}\")\n print(f\"Einsum: {rule.to_einsum()}\")\n print(f\"\\nIndex Analysis:\")\n for idx, info in index_info.items():\n print(f\" {idx}: {info.role.name}, size={info.size}\")\n \n print(f\"\\nMatmul Pattern: {'✓' if pattern.is_matmul else '✗'}\")\n if pattern.is_matmul:\n print(f\" M={pattern.m_indices}, N={pattern.n_indices}, K={pattern.k_indices}\")\n print(f\" Batch={pattern.batch_indices}\")\n print(f\" A_transpose={pattern.a_transpose}, B_transpose={pattern.b_transpose}\")\n \n print(f\"\\nTiling Plan:\")\n print(f\" Grid: {tiling.grid_dims}\")\n print(f\" Reduction loops: {tiling.reduction_loops}\")\n print(f\" Tile sizes: {tiling.tile_sizes}\")\n \n print(f\"\\nGenerated Kernel:\")\n print(\"─\" * 40)\n kernel = compile_rule(rule, shapes, tc[\"name\"].lower().replace(\" \", \"_\"))\n print(kernel)\n \n print(\"\\n\" + \"=\" * 80)\n print(\"ALL TESTS COMPLETE\")\n print(\"=\" * 80)\n\n\nif __name__ == \"__main__\":\n run_tests()\n```\n\n## Sample Output\n```\n================================================================================\nTENSOR LOGIC COMPILER - TEST SUITE\n================================================================================\n\n────────────────────────────────────────────────────────────────────────────────\nTEST: Standard Matmul\n────────────────────────────────────────────────────────────────────────────────\n\nRule: C[i,k] <- A[i,j], B[j,k]\nEinsum: ij,jk->ik\n\nIndex Analysis:\n i: FREE_LEFT, size=1024\n j: CONTRACTED, size=512\n k: FREE_RIGHT, size=256\n\nMatmul Pattern: ✓\n M=('i',), N=('k',), K=('j',)\n Batch=()\n A_transpose=False, B_transpose=False\n\nTiling Plan:\n Grid: ['i', 'k']\n Reduction loops: ['j']\n Tile sizes: {'i': 64, 'j': 32, 'k': 64}\n\nGenerated Kernel:\n────────────────────────────────────────\n@ct.kernel\ndef standard_matmul(C, A, B, I: ct.Constant[int], J: ct.Constant[int], K: ct.Constant[int], TILE_I: ct.Constant[int], TILE_J: ct.Constant[int], TILE_K: ct.Constant[int]):\n \"\"\"\n Compiled from: C[i,k] <- A[i,j], B[j,k]\n Einsum: ij,jk->ik\n Pattern: matmul with M=('i',), N=('k',), K=('j',), batch=()\n \"\"\"\n\n # Grid dimension 0: i\n i_tile_idx = ct.program_id(0)\n i_offset = i_tile_idx * TILE_I\n # Grid dimension 1: k\n k_tile_idx = ct.program_id(1)\n k_offset = k_tile_idx * TILE_K\n\n # Output accumulator: [i, k]\n acc = ct.zeros((TILE_I, TILE_K,), dtype=ct.float32)\n\n # Reduction over j\n for j_offset in range(0, J, TILE_J):\n # Load input tiles\n tile_a = ct.load(A, (i_offset, j_offset,), shape=(TILE_I, TILE_J,))\n tile_b = ct.load(B, (j_offset, k_offset,), shape=(TILE_J, TILE_K,))\n\n # Matmul: M=('i',), N=('k',), K=('j',)\n acc += ct.dot(tile_a, tile_b)\n\n # Store result\n ct.store(C, (i_offset, k_offset,), tile=acc)\n\n────────────────────────────────────────────────────────────────────────────────\nTEST: CTM NLM Layer\n────────────────────────────────────────────────────────────────────────────────\n\nRule: H[b,d,h] <- A[b,d,m], W[m,h,d]\nEinsum: bdm,mhd->bdh\n\nIndex Analysis:\n b: FREE_LEFT, size=64\n d: FREE_SHARED, size=2048\n m: CONTRACTED, size=25\n h: FREE_RIGHT, size=32\n\nMatmul Pattern: ✓\n M=('b',), N=('h',), K=('m',)\n Batch=('d',)\n A_transpose=False, B_transpose=False\n\nTiling Plan:\n Grid: ['d', 'b']\n Reduction loops: ['m']\n Tile sizes: {'b': 64, 'd': 64, 'm': 25, 'h': 32}\n\nGenerated Kernel:\n────────────────────────────────────────\n@ct.kernel\ndef ctm_nlm_layer(H, A, W, B: ct.Constant[int], D: ct.Constant[int], M: ct.Constant[int], H: ct.Constant[int], TILE_B: ct.Constant[int], TILE_D: ct.Constant[int], TILE_M: ct.Constant[int], TILE_H: ct.Constant[int]):\n \"\"\"\n Compiled from: H[b,d,h] <- A[b,d,m], W[m,h,d]\n Einsum: bdm,mhd->bdh\n Pattern: matmul with M=('b',), N=('h',), K=('m',), batch=('d',)\n \"\"\"\n\n # Grid dimension 0: d\n d_tile_idx = ct.program_id(0)\n d_offset = d_tile_idx * TILE_D\n # Grid dimension 1: b\n b_tile_idx = ct.program_id(1)\n b_offset = b_tile_idx * TILE_B\n\n # Output accumulator: [b, d, h]\n acc = ct.zeros((TILE_B, TILE_D, TILE_H,), dtype=ct.float32)\n\n for h_offset in range(0, H, TILE_H):\n # Reduction over m\n for m_offset in range(0, M, TILE_M):\n # Load input tiles\n tile_a = ct.load(A, (b_offset, d_offset, m_offset,), shape=(TILE_B, TILE_D, TILE_M,))\n tile_w = ct.load(W, (m_offset, h_offset, d_offset,), shape=(TILE_M, TILE_H, TILE_D,))\n\n # Matmul: M=('b',), N=('h',), K=('m',)\n # Reshape for batched matmul\n a_perm = ct.permute(tile_a, (1, 0, 2,))\n a_mat = ct.reshape(a_perm, (TILE_D, TILE_B, TILE_M))\n b_perm = ct.permute(tile_w, (2, 0, 1,))\n b_mat = ct.reshape(b_perm, (TILE_D, TILE_M, TILE_H))\n result = ct.bmm(a_mat, b_mat)\n acc += ct.reshape(result, (TILE_B, TILE_D, TILE_H,))\n\n # Store result\n ct.store(H, (b_offset, d_offset, h_offset,), tile=acc)"
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"thinking": "This is a fascinating project - implementing Pedro Domingos' tensor logic using NVIDIA's cuTile. The user has shared an early implementation of a compiler that translates Datalog-style tensor rules into cuTile GPU kernels.\n\nLet me analyze what they have:\n\n1. **Data Structures**: `TensorRef`, `Rule`, `IndexRole`, `IndexInfo`, `MatmulPattern`, `TilingPlan`\n2. **Analysis**: `analyze_indices`, `detect_matmul`, `plan_tiling`\n3. **Code Generation**: `CodeBuilder`, `generate_kernel`, `generate_matmul_op`, `generate_broadcast_op`\n4. **Main API**: `compile_rule`, `compile_rules`\n\nLooking at the sample output, I can see some issues:\n\n1. **CTM NLM Layer bug**: The generated kernel has `H` as both a tensor parameter and a size constant parameter - name collision!\n ```python\n def ctm_nlm_layer(H, A, W, B: ct.Constant[int], D: ct.Constant[int], M: ct.Constant[int], H: ct.Constant[int], ...\n ```\n\n2. **The store at the end of CTM NLM Layer** references `h_offset` but that's inside the loop - so the store would only save the last tile of h.\n\n3. **Index classification issues**: In the CTM NLM Layer case, `d` is classified as `FREE_SHARED` but it appears in all three tensors (A, W, and H), so it should be `BATCH`. Let me trace through:\n - `d` is in output H[b,d,h]\n - `d` is in A[b,d,m] \n - `d` is in W[m,h,d]\n - So `d` is in all inputs AND output → should be BATCH\n\n Looking at the code:\n ```python\n if not in_output:\n role = IndexRole.CONTRACTED\n elif n_in_inputs == n_inputs:\n role = IndexRole.BATCH\n ```\n For `d`, it appears in both A and W (n_in_inputs = 2) and n_inputs = 2, so the condition `n_in_inputs == n_inputs` should be true, making it BATCH. But the output shows `d: FREE_SHARED`, which suggests the index classification logic isn't correctly identifying when an index spans all inputs.\n\nThere's likely an issue with how `input_tensors` gets constructed. Additionally, the `h` index is treated as a free loop dimension while simultaneously being part of the output shape—the accumulator gets initialized with TILE_H included, but then h_offset shifts during iteration, which doesn't align with how the accumulator should only track grid dimensions rather than loop dimensions.",
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"text": "This is an ambitious and interesting project. Let me review your implementation and identify some issues.\n\n## Issues I Found\n\n### 1. Parameter Name Collision (Critical)\n\nIn the CTM NLM Layer output, you have:\n```python\ndef ctm_nlm_layer(H, A, W, B: ct.Constant[int], ..., H: ct.Constant[int], ...)\n```\n`H` is both the output tensor and a size constant. This needs disambiguation.\n\n### 2. Store Outside Loop Problem\n\nIn the CTM NLM kernel, `h_offset` is used in the store but defined inside a loop:\n```python\nfor h_offset in range(0, H, TILE_H):\n ...\n# Store result\nct.store(H, (b_offset, d_offset, h_offset,), tile=acc) # h_offset is undefined here\n```\n\nThe accumulator logic is also wrong - you're accumulating across `h` iterations but should be storing each tile separately.\n\n### 3. Index Classification Bug\n\nFor the CTM NLM rule `H[b,d,h] <- A[b,d,m], W[m,h,d]`:\n- Index `d` appears in **all** tensors (A, W, and H)\n- Should be `BATCH`, but classified as `FREE_SHARED`\n\nThe issue is in `analyze_indices` - looks like the `b` index is incorrectly classified as `FREE_LEFT` when it should also be `BATCH` (it's in A and H but not W... wait, that's actually correct for `b`).\n\nLet me trace `d` more carefully. In your code:\n```python\ninput_tensors = [t for t in index_to_tensors[idx] if t != rule.head.name]\nn_in_inputs = len(input_tensors)\n```\nFor `d`: it's in A, W, and H. So `input_tensors` = ['A', 'W'], `n_in_inputs` = 2, `n_inputs` = 2. This should trigger `role = IndexRole.BATCH`. Let me check the output again... The output shows `d: FREE_SHARED`. There's likely a bug where tensor names are being double-counted or the shapes dict keys don't match.\n\n### 4. Tiling Plan vs Accumulator Mismatch\n\nThe accumulator shape includes all output indices:\n```python\nacc_shape = \", \".join(f\"TILE_{idx.upper()}\" for idx in rule.head.indices)\n```\n\nBut if some output indices (`h` in CTM NLM) are looped over separately, the accumulator should only cover the grid dimensions, and you need a separate store per loop iteration.\n\n---\n\nWant me to explore the codebase further and propose fixes, or would you prefer to discuss the architectural approach first? Specifically:\n\n1. Should non-grid free indices (`h` in CTM NLM) be parallelized differently?\n2. How should we handle the tensor/size parameter naming collision?\n3. Do you want me to verify the analysis logic with more test cases?"
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"content": "first talk me through how this owuld compare w/ just pytorch"
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"thinking": "The user wants me to compare the tensor logic + cuTile approach with just using PyTorch. This is a conceptual/research discussion, not an implementation task, so I don't need to enter plan mode or use tools extensively. Let me think through the comparison.\n\n## PyTorch approach\n\nIn PyTorch, you'd express these operations using:\n1. `torch.einsum()` - directly supports Einstein summation notation\n2. `torch.matmul()` / `@` operator - for matrix multiplication\n3. `torch.bmm()` - for batched matrix multiply\n4. Manual reshaping + matmul for complex patterns\n\nFor example:\n- `C[i,k] <- A[i,j], B[j,k]` → `C = torch.einsum('ij,jk->ik', A, B)` or `C = A @ B`\n- `Score[b,p,q] <- Q[b,p,d], K[b,q,d]` → `Score = torch.einsum('bpd,bqd->bpq', Q, K)` or `Score = Q @ K.transpose(-1,-2)`\n\n## Comparison dimensions\n\n### 1. Abstraction level\n- **PyTorch**: High-level, einsum or matmul calls, relies on cuBLAS/cuDNN under the hood\n- **Tensor Logic + cuTile**: You're compiling to explicit GPU kernels with control over tiling, memory access patterns\n\n### 2. Performance\n- **PyTorch einsum**: Actually quite good, but can have overhead for complex patterns, may not always fuse operations optimally\n- **cuTile**: Direct tensor core access, can achieve 90%+ of cuBLAS, potential for custom fusion\n\n### 3. Expressiveness\n- **PyTorch**: einsum is essentially the same as tensor logic's notation for contractions\n- **Tensor Logic**: The vision is broader - sound reasoning in embedding space, not just numerical computation\n\n\n- **Compilation and control**: PyTorch's JIT can optimize, but you're constrained by what the framework decides to do. With tensor logic compiled to cuTile, you generate the actual kernel and dictate tiling strategies and memory patterns.\n\n### 4. Where this distinction matters\n- Custom attention patterns that don't fit standard operations\n- Sparse tensor operations\n- Novel architectures like CTM that lack existing optimized kernels\n- Research exploring neuro-symbolic computation where you need both the math and the reasoning framework",
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"text": "Great question. Let me walk through the comparison.\n\n## PyTorch Equivalent\n\nYour tensor logic rules map directly to `torch.einsum`:\n\n```python\n# C[i,k] <- A[i,j], B[j,k]\nC = torch.einsum('ij,jk->ik', A, B) # or just A @ B\n\n# Score[b,p,q] <- Q[b,p,d], K[b,q,d]\nScore = torch.einsum('bpd,bqd->bpq', Q, K) # or Q @ K.transpose(-1,-2)\n\n# H[b,d,h] <- A[b,d,m], W[m,h,d]\nH = torch.einsum('bdm,mhd->bdh', A, W)\n```\n\nSo at the **notation level**, you're not gaining much — einsum already *is* tensor logic for contractions.\n\n## Where This Approach Differs\n\n| Aspect | PyTorch | Tensor Logic → cuTile |\n|--------|---------|----------------------|\n| **Abstraction** | High-level, opaque kernel selection | You control the generated kernel |\n| **Backend** | cuBLAS/cuDNN (black box) | Direct tensor core access via cuTile |\n| **Fusion** | Relies on `torch.compile` heuristics | You decide what fuses |\n| **Custom patterns** | May fall back to slow paths | You generate exactly what you need |\n| **Tiling strategy** | Library chooses | You specify tile sizes |\n\n## When Does This Matter?\n\n**1. Novel architectures (CTM, custom attention)**\n```python\n# PyTorch einsum for CTM NLM layer\nH = torch.einsum('bdm,mhd->bdh', A, W)\n```\nPyTorch will likely decompose this into reshapes + batched matmul. It might not be optimal. With your compiler, you control exactly how it maps to tensor cores.\n\n**2. Fused operations**\nIf you want `Softmax(Q @ K.T) @ V` as a single kernel (FlashAttention-style), PyTorch won't do that automatically. Your compiler could potentially fuse multiple rules.\n\n**3. Sparse/structured patterns**\nTransitive closure `Path[x,z] <- Path[x,y], Edge[y,z]` with sparse tensors — PyTorch einsum doesn't handle sparsity well. Custom kernels can.\n\n**4. The bigger vision (Domingos' point)**\nTensor logic isn't just about efficient contractions. It's about:\n- Logical inference as tensor ops\n- Differentiable reasoning\n- Unifying neural + symbolic in one substrate\n\nPyTorch gives you the *numerical* part but not the *logical reasoning* part.\n\n## Honest Assessment\n\nFor **pure numerical contractions** like matmul and attention, PyTorch + `torch.compile` is already very good. You'd need to demonstrate:\n\n1. **Comparable performance** to cuBLAS (cuTile claims 90%+)\n2. **Something PyTorch can't do** — custom fusion, symbolic reasoning, novel patterns\n\nThe real value proposition isn't \"faster matmul\" — it's **extensibility to operations that don't exist as library calls** and potentially the symbolic AI integration Domingos envisions.\n\n---\n\nWhat's your primary motivation here — performance for custom patterns like CTM, or exploring the neuro-symbolic direction?"
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"content": "the problem IMO w/ the neurosymbolic stuff is superposition"
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"thinking": "The user is making a really interesting point about superposition being a problem for neuro-symbolic approaches. Let me think about what they mean.\n\nSuperposition in the context of neural networks (particularly from the Anthropic interpretability research) refers to the phenomenon where neural networks represent more features than they have dimensions - essentially packing multiple features into the same neurons/dimensions in a way that they can be recovered but creates interference.\n\nThe connection to tensor logic / neuro-symbolic:\n\n1. **Domingos' vision**: Predicates as tensors, logical reasoning in embedding space. The idea is you can do \"sound reasoning\" with neural representations.\n\n2. **The superposition problem**: If neural representations are in superposition, then:\n - A single dimension doesn't correspond to a single concept/predicate\n - Multiple features are entangled in the same space\n - Logical operations (AND, OR, implication) become ill-defined because you can't cleanly separate the things you're reasoning about\n\n3. **Why this matters for tensor logic**:\n - If `P[x]` is a predicate represented as a tensor, but that representation is superposed with other predicates, then `P[x] AND Q[x]` doesn't have clean semantics\n - The \"soundness\" of reasoning breaks down because you're not operating on clean logical entities\n - You get interference/crosstalk between concepts\n\n4. **The fundamental tension**:\n - Neural networks use superposition because it's efficient (more concepts than dimensions)\n - Symbolic reasoning requires clean separation of concepts\n - These seem fundamentally at odds\n\nThis is actually a deep critique of the neuro-symbolic dream. The user seems to be saying that tensor logic is fine for the numerical/computational side, but the promise of \"sound reasoning in embedding space\" runs into the wall of superposition.\n\nPossible responses to this critique:\n1. Use sparse/disentangled representations (SAEs, dictionary learning)\n2. Accept approximate reasoning (but then you lose \"soundness\")\n\n\n3. Work in a regime where superposition is minimal (high-dimensional, low feature count)\n4. The symbolic structure might help *reduce* superposition by providing constraints\n\nLet me engage with this thoughtfully.",
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"text": "That's a sharp observation.\n\n## The Problem\n\nDomingos' pitch: predicates are tensors, do sound logical inference in embedding space.\n\nBut neural representations are **superposed** — multiple features packed into fewer dimensions. From the interpretability work (Anthropic et al.):\n\n- A single neuron/dimension ≠ a single concept\n- Features interfere with each other\n- The \"clean\" predicate `IsBlue[x]` doesn't exist — it's smeared across dimensions, entangled with `IsCold[x]` and `IsSad[x]`\n\n## Why This Breaks \"Sound Reasoning\"\n\n```\n# What tensor logic wants:\nCat[x] AND Mammal[x] → Animal[x]\n\n# What you actually have:\nentangled_mess[x] ⊗ different_entangled_mess[x] → ???\n```\n\nLogical operations assume you can cleanly pick out the things you're operating on. Superposition means you can't. You get:\n\n- **Crosstalk**: Reasoning about cats accidentally affects reasoning about cars\n- **Interference**: `A AND B` picks up ghost contributions from `C`\n- **No soundness**: The guarantees of logic evaporate\n\n## The Fundamental Tension\n\n| Neural Networks | Symbolic Logic |\n|-----------------|----------------|\n| Superposition is a feature (efficiency) | Requires clean separation |\n| Distributed representations | Discrete symbols |\n| Graceful degradation | Brittle but sound |\n\nThese seem **architecturally incompatible** at the representation level.\n\n## Possible Outs?\n\n1. **Sparse autoencoders / dictionary learning** — decompose superposition into monosemantic features, *then* do logic. But this is expensive and imperfect.\n\n2. **Work in the \"sparse regime\"** — high dimensions, few active features. But then you lose the efficiency that makes neural nets attractive.\n\n3. **Accept approximate reasoning** — but then you've lost the main selling point over just training a transformer.\n\n4. **Hybrid**: Use neural for perception/embedding, switch to explicit symbolic for reasoning. But that's not \"unified\" anymore.\n\n---\n\nSo where does that leave the cuTile project? More as a **kernel compiler for custom tensor operations** (CTM, attention variants, graph algorithms) rather than a path to neuro-symbolic AI?"
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"thinking": "The user is wondering if anyone has asked Pedro Domingos about the superposition problem and how it affects his tensor logic vision for neuro-symbolic AI. Let me search for any discussions, interviews, or Q&A where this might have come up.",
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"content": "Web search results for query: \"Pedro Domingos tensor logic superposition neural representations interpretability\"\n\nLinks: [{\"title\":\"[2510.12269] Tensor Logic: The Language of AI\",\"url\":\"https://arxiv.org/abs/2510.12269\"},{\"title\":\"Tensor Logic Tensor Logic: The Language of AI Pedro Domingos\",\"url\":\"https://arxiv.org/pdf/2510.12269\"},{\"title\":\"Tensor Logic for Bridging Neural and Symbolic AI\",\"url\":\"https://bengoertzel.substack.com/p/tensor-logic-for-bridging-neural\"},{\"title\":\"Tensor Logic: Unified Language for AI\",\"url\":\"https://www.emergentmind.com/papers/2510.12269\"},{\"title\":\"Ecmlpkdd\",\"url\":\"https://ecmlpkdd.org/2025/keynotes/\"},{\"title\":\"Tensor Logic The Language of AI Pedro Domingos University of Washington\",\"url\":\"https://homes.cs.washington.edu/~pedrod/tls.pdf\"},{\"title\":\"Tensor Logic: The Language of AI\",\"url\":\"https://arxiv.org/html/2510.12269v3\"},{\"title\":\"[2510.12269v3] Tensor Logic: The Language of AI\",\"url\":\"https://arxiv.org/abs/2510.12269v3\"},{\"title\":\"Paper page - Tensor Logic: The Language of AI\",\"url\":\"https://huggingface.co/papers/2510.12269\"},{\"title\":\"FEP - DaSSWeb | Tensor Logic: A Simple Unification of Neural and Symbolic AI\",\"url\":\"https://sigarra.up.pt/fep/en/noticias_geral.ver_noticia?p_nr=51570\"}]\n\nHere are the search results on Pedro Domingos's work on tensor logic and its relation to superposition, neural representations, and interpretability:\n\n## Pedro Domingos's Tensor Logic\n\nPedro Domingos proposes tensor logic as a language that solves problems in AI by unifying neural and symbolic AI at a fundamental level.(no content)The sole construct in tensor logic is the tensor equation, based on the observation that logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them.\n\n### Core Motivation\n\n\"Deep learning has achieved remarkable successes in language generation and other tasks, but is extremely opaque and notoriously unreliable. Both of these problems can be overcome by combining it with the sound reasoning\"(no content)capabilities of symbolic AI. Tensor logic accomplishes this by unifying tensor algebra and logic programming, the formal languages underlying respectively deep learning and symbolic AI.\n\n### Superposition and Neural Representations\n\nA novel contribution is the formalization of reasoning in embedding space. Objects and relations are embedded as tensors, enabling superposition and retrieval via tensor products.(no content)The embedding of a relation \"is the superposition of the embeddings of the tuples in the relation, where the embedding of a tuple is the tensor product of the embeddings of its arguments. This is a type of tensor product representation (Smolensky, 1990).\"(no content)Reasoning over embedded rules is performed by chaining tensor equations, with the error probability controlled by embedding dimension. This approach supports analogical reasoning, compositionality, and transparency. The temperature parameter in sigmoid nonlinearities modulates the deductive vs. analogical nature of inference.\n\n### Interpretability and Transparency\n\nTensor logic makes new directions possible, such as sound reasoning in embedding space. This combines the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning.(no content)Tensor logic provides a unified formalism for AI, bridging neural, symbolic, and statistical paradigms. Its minimal design enables direct translation of mathematical models to code, facilitating transparency, reliability, and learnability. The framework supports new directions in reasoning, particularly in embedding space, with potential applications in knowledge representation, scientific computing, and interpretable AI.\n\n### Practical Applications\n\nA system based on tensor logic achieves state-of-the-art results on a suite of language and reasoning tasks. How tensor logic will fare on trillion-token corpora and associated tasks remains an open question.(no content)Ben Goertzel's team is exploring adopting and extending Pedro Domingos's tensor logic as an intermediate representation—\"a mathematical lingua franca that lets symbolic and tensorial computation meet on common ground.\"\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"text": "Interesting — he actually *leans into* superposition rather than avoiding it. Let me dig deeper into the paper and that podcast."
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"content": "# Embedding Space Reasoning in Tensor Logic\n\n## Core Mechanism\n\nThe framework uses random unit vector embeddings to represent objects. When retrieving relations, it computes: **\"the superposition of the embeddings of the tuples in the relation, where the embedding of a tuple is the tensor product of the embeddings of its arguments.\"**\n\n## Error Probability & Interference\n\nThe critical theoretical guarantee involves controlled interference. For set membership testing, **\"the dot product of two random unit vectors is approximately 0.\"** This orthogonality reduces crosstalk between different concepts.\n\nError probability is quantified as: **\"standard deviation N/D√, where N is the cardinality of the set and D is the embedding dimension.\"** Errors decrease predictably with higher-dimensional embeddings.\n\n## Theoretical Guarantees\n\nThe framework provides bounded error through mathematical properties:\n\n- Query results are approximately correct because **\"each inferred tensor can be expressed as a sum of projections of joins of embedded relations\"**\n- Thresholding at 0.5 enables binary decisions with controlled false positive/negative rates\n- The approach is **\"immune to hallucinations at sufficiently low temperature\"** when temperature parameter approaches zero\n\n## Denoising Strategy\n\nTo further reduce crosstalk errors, the system can **\"extract, threshold and re-embed the inferred tensors at regular intervals (in the limit, after each rule application).\"**\n\nThis periodic denoising prevents accumulation of interference effects through inference chains, though this introduces computational overhead."
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"result": "# Embedding Space Reasoning in Tensor Logic\n\n## Core Mechanism\n\nThe framework uses random unit vector embeddings to represent objects. When retrieving relations, it computes: **\"the superposition of the embeddings of the tuples in the relation, where the embedding of a tuple is the tensor product of the embeddings of its arguments.\"**\n\n## Error Probability & Interference\n\nThe critical theoretical guarantee involves controlled interference. For set membership testing, **\"the dot product of two random unit vectors is approximately 0.\"** This orthogonality reduces crosstalk between different concepts.\n\nError probability is quantified as: **\"standard deviation N/D√, where N is the cardinality of the set and D is the embedding dimension.\"** Errors decrease predictably with higher-dimensional embeddings.\n\n## Theoretical Guarantees\n\nThe framework provides bounded error through mathematical properties:\n\n- Query results are approximately correct because **\"each inferred tensor can be expressed as a sum of projections of joins of embedded relations\"**\n- Thresholding at 0.5 enables binary decisions with controlled false positive/negative rates\n- The approach is **\"immune to hallucinations at sufficiently low temperature\"** when temperature parameter approaches zero\n\n## Denoising Strategy\n\nTo further reduce crosstalk errors, the system can **\"extract, threshold and re-embed the inferred tensors at regular intervals (in the limit, after each rule application).\"**\n\nThis periodic denoising prevents accumulation of interference effects through inference chains, though this introduces computational overhead.",
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"content": "# Criticisms and Limitations of Tensor Logic in the Article\n\nBased on my review of Ben Goertzel's article, **no explicit criticisms or limitations of tensor logic are discussed**. The piece is entirely promotional.\n\n## What the Article Does Acknowledge:\n\nThe author frames tensor logic as a **solution to integration challenges** rather than discussing its shortcomings. The only cautionary note is:\n\n> \"This is stuff we're working on right now and it looks very promising but is not yet proven.\"\n\nThis disclaimer indicates the framework remains unvalidated, but Goertzel doesn't elaborate on potential failure modes.\n\n## Notable Absences:\n\nThe article contains **no discussion of**:\n- Superposition or distributed representations\n- Scalability limitations\n- Computational complexity tradeoffs\n- Cases where tensor logic might perform poorly\n- Comparison with competing neural-symbolic approaches\n\n## The Author's Position:\n\nGoertzel positions tensor logic modestly as **\"interfacing magic\"** between symbolic and neural systems rather than a universal solution (unlike Domingos's claims). However, he presents no empirical evidence, benchmarks, or failure case analysis.\n\nThe rough-draft status of accompanying documents suggests this remains early-stage work without peer-reviewed validation of either its theoretical soundness or practical effectiveness."
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"result": "# Criticisms and Limitations of Tensor Logic in the Article\n\nBased on my review of Ben Goertzel's article, **no explicit criticisms or limitations of tensor logic are discussed**. The piece is entirely promotional.\n\n## What the Article Does Acknowledge:\n\nThe author frames tensor logic as a **solution to integration challenges** rather than discussing its shortcomings. The only cautionary note is:\n\n> \"This is stuff we're working on right now and it looks very promising but is not yet proven.\"\n\nThis disclaimer indicates the framework remains unvalidated, but Goertzel doesn't elaborate on potential failure modes.\n\n## Notable Absences:\n\nThe article contains **no discussion of**:\n- Superposition or distributed representations\n- Scalability limitations\n- Computational complexity tradeoffs\n- Cases where tensor logic might perform poorly\n- Comparison with competing neural-symbolic approaches\n\n## The Author's Position:\n\nGoertzel positions tensor logic modestly as **\"interfacing magic\"** between symbolic and neural systems rather than a universal solution (unlike Domingos's claims). However, he presents no empirical evidence, benchmarks, or failure case analysis.\n\nThe rough-draft status of accompanying documents suggests this remains early-stage work without peer-reviewed validation of either its theoretical soundness or practical effectiveness.",
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"thinking": "This is really interesting. So Domingos' approach to superposition is:\n\n1. **Use random unit vectors** as embeddings - these are approximately orthogonal in high dimensions\n2. **Error probability scales as N/√D** where N is the number of items and D is the embedding dimension\n3. **Denoising step**: Periodically extract, threshold, and re-embed to prevent error accumulation\n4. **Temperature parameter**: Low temperature = more logic-like (immune to hallucinations), high temperature = more analogical\n\nSo he's aware of the interference problem and has a theoretical framework for controlling it. The question is whether this actually works in practice with real learned embeddings (not random unit vectors).\n\nThe key tension the user identified is still valid though:\n- Domingos assumes you can use random/controlled embeddings\n- Real neural network representations are NOT random unit vectors - they're learned and heavily superposed\n- The N/√D bound assumes orthogonality that may not hold in practice\n\nLet me present this to the user.",
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"text": "Okay, so Domingos **does** address it — but his answer might not satisfy you.\n\n## His Approach to Superposition\n\nHe explicitly uses [Smolensky's tensor product representations](https://arxiv.org/html/2510.12269v3) — superposition is **intentional**:\n\n> \"The embedding of a relation is the superposition of the embeddings of the tuples in the relation, where the embedding of a tuple is the tensor product of the embeddings of its arguments.\"\n\n## His Theoretical Guarantee\n\nThe interference is **bounded** under specific conditions:\n\n- Use **random unit vectors** as embeddings\n- Dot product of two random unit vectors ≈ 0 (approximate orthogonality)\n- Error probability: **N/√D** (N = number of items, D = embedding dimension)\n\nSo with D = 10,000 and N = 100 items, error ≈ 1%. Crank up dimensions, push down error.\n\n## His Escape Hatches\n\n1. **Low temperature** — at T→0, system is \"immune to hallucinations\", behaves like pure logic\n2. **Periodic denoising** — \"extract, threshold and re-embed the inferred tensors at regular intervals\" to prevent error accumulation\n\n## The Gap You're Pointing At\n\nHis analysis assumes **random unit vector embeddings**. But:\n\n| His Framework | Real Neural Nets |\n|---------------|------------------|\n| Random orthogonal vectors | Learned representations |\n| Controlled superposition | Emergent superposition |\n| N/√D error bound | Unknown interference |\n| You choose D | D determined by architecture |\n\nThe question nobody seems to have pressed him on:\n\n**Do his guarantees hold when you plug in embeddings from a trained transformer instead of random vectors?**\n\nMy guess: no. Learned embeddings violate the orthogonality assumption. The features are correlated, clustered, and packed in ways that serve gradient descent, not symbolic retrieval.\n\n---\n\nThe [Goertzel piece](https://bengoertzel.substack.com/p/tensor-logic-for-bridging-neural) is notably silent on limitations — just \"looks promising but not yet proven.\"\n\nWould be interesting to see someone test the actual error rates with GPT-style embeddings vs random vectors."
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"content": "so his temperature thing means you only accept actually orthogonal vectors as opposed to almost orthogonal?"
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"thinking": "The user is asking about what the temperature parameter actually does in Domingos' tensor logic framework. They're asking if low temperature means you only accept truly orthogonal vectors (exact matches) vs almost orthogonal (fuzzy matches).\n\nLet me think about this. In the context of tensor logic:\n\n- The temperature appears in sigmoid nonlinearities\n- At low T, sigmoid becomes a step function (hard threshold)\n- At high T, sigmoid is smooth (soft/fuzzy)\n\nSo when you do a query/retrieval:\n- Low T: Only exact matches pass the threshold (like a hard 0.5 cutoff that becomes very sharp)\n- High T: Near-matches also contribute (analogical reasoning, similar concepts bleed into each other)\n\nThe user's interpretation is roughly correct - low temperature means you're being strict about what counts as a match. But it's not about accepting \"orthogonal\" vectors per se - it's about thresholding the similarity scores.\n\nIf you have:\n- Query vector q\n- Stored vectors v1, v2, ... in superposition\n- You compute dot products: q·v1, q·v2, etc.\n\nWith random unit vectors:\n- q·q ≈ 1 (exact match)\n- q·vi ≈ 0 for i≠target (orthogonal, noise)\n\nThe sigmoid with temperature:\n- σ(x/T) where T is temperature\n- As T→0, σ becomes a step function at 0\n- So you threshold at 0.5, which means you need score > 0 to be \"true\"\n\nActually, thinking about it more carefully - the temperature controls how \"crisp\" the logic is:\n- T→0: Hard thresholding, only clear matches, no fuzzy reasoning\n- T→∞: Soft, everything bleeds together, analogical\n\nSo the user's intuition is right - low temperature means you're more strict. But I should clarify:\n- It's not about accepting \"orthogonal\" vectors (orthogonal would be the noise/non-matches)\n\n\n- It's about creating a sharp cutoff that filters out noise from near-orthogonal vectors, keeping only the strong signal from true matches.",
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"text": "Basically yes. The temperature controls the sigmoid sharpness:\n\n```\nσ(x/T)\n\nT → 0: step function (hard threshold)\nT → ∞: smooth curve (everything bleeds together)\n```\n\n## What This Means Practically\n\nWhen you query for \"Is X a cat?\":\n\n```\nscore = query · superposed_embedding\n```\n\nWith random vectors:\n- Correct match: score ≈ 1\n- Wrong items (noise): score ≈ 0 (but not exactly 0)\n\n**Low temperature**: Sharp threshold at 0.5\n- Score of 0.8 → YES\n- Score of 0.1 → NO\n- The noise gets rejected\n\n**High temperature**: Soft threshold\n- Score of 0.3 might partially activate\n- Similar concepts bleed in (\"cat\" query partially retrieves \"dog\")\n- This is his \"analogical reasoning\" mode\n\n## The Problem\n\nLow temperature gives you \"sound\" logic... **but only if the noise stays near zero**.\n\nHis N/√D bound says noise ≈ 0.01 with big D and small N. Fine.\n\nBut with learned embeddings:\n- \"Cat\" and \"dog\" aren't orthogonal — they're **close** (both animals, both pets)\n- The noise floor isn't 0.01, it's maybe 0.6\n- Now your threshold can't separate signal from noise\n\nSo low temperature doesn't save you — it just gives you **confident wrong answers** instead of fuzzy wrong answers.\n\nThe guarantee only holds if your embedding space is structured the way he assumes (random/orthogonal), not the way transformers actually learn it (clustered by semantic similarity)."
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"thinking": "The user is suggesting we could empirically test Domingos' theoretical claims by using actual pretrained embeddings (like BERT or NVIDIA's embedding model) and measuring whether the error bounds hold.\n\nThis would be a great experiment:\n1. Take a pretrained embedding model\n2. Embed a bunch of concepts/entities\n3. Measure the actual dot product distributions (how orthogonal are unrelated concepts?)\n4. Compare to Domingos' N/√D theoretical bound\n5. Test the superposition/retrieval accuracy\n\nThis shifts the project from \"implement tensor logic in cuTile\" to \"empirically test tensor logic's assumptions with real embeddings.\"\n\nLet me engage with this idea - it's a good one and would be valuable research.",
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"text": "Yeah, that would be a good empirical test of whether his theory holds with real embeddings.\n\n## The Experiment\n\n**Setup:**\n1. Pick an embedding model (BERT, Nemotron, etc.)\n2. Embed N entities from some domain (e.g., 1000 animals, or WordNet synsets)\n3. Create superposed representations using his tensor product method\n4. Query and measure actual retrieval accuracy\n\n**What to measure:**\n\n| Metric | Domingos' Assumption | What We'd Expect |\n|--------|---------------------|------------------|\n| Dot product of unrelated items | ≈ 0 | Probably 0.3-0.7 (semantic similarity) |\n| Std dev of noise | N/√D | Likely much higher |\n| Retrieval accuracy vs D | Improves predictably | May plateau or fail |\n\n**Concrete test:**\n```python\n# Embed 100 animals\nembeddings = model.encode([\"cat\", \"dog\", \"eagle\", ...])\n\n# Check \"orthogonality\" assumption\nfor i, j in pairs:\n dot = embeddings[i] @ embeddings[j]\n # Domingos assumes this ≈ 0\n # Reality: \"cat\" · \"dog\" ≈ 0.8?\n\n# Build superposed relation: \"has_fur\"\nhas_fur = sum(embed(x) for x in [\"cat\", \"dog\", \"bear\", ...])\n\n# Query: does \"cat\" have fur?\nscore = embed(\"cat\") @ has_fur # Should be high\nscore = embed(\"eagle\") @ has_fur # Should be low, but is it?\n```\n\n**The killer question:** What's the actual false positive rate when querying semantically similar but incorrect items?\n\n---\n\nWant to set this up? Could be a quick script with `sentence-transformers` or the NVIDIA API, measure the distributions, and see if tensor logic's assumptions survive contact with real embeddings."
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"thinking": "The user is asking a really good question - even if we validate that tensor logic works with real embeddings, what's the actual path to a \"better LLM\"? \n\nIf you took nanoGPT and rewrote it in tensor logic notation, you'd just have... nanoGPT. The transformer architecture is already expressible as tensor contractions (attention is just einsum operations). So what would you actually change?\n\nLet me think about what Domingos might claim:\n\n1. **Explicit reasoning chains** - Instead of hoping the transformer learns to reason in its forward pass, you could have explicit logical rules that get applied. But... transformers already do multi-step reasoning via layers.\n\n2. **Grounded symbols** - You could have explicit predicates that are interpretable. But this requires knowing what predicates you want ahead of time.\n\n3. **Guaranteed inference** - For queries that match your rules, you get sound answers. But LLMs are supposed to be general, not restricted to predefined rules.\n\n4. **Hybrid architecture** - Neural perception + symbolic reasoning. But where's the boundary? And who defines the symbols?\n\nThe real problem: \n- LLMs work precisely BECAUSE they don't commit to explicit symbolic structure\n- They learn distributed representations that generalize\n- Forcing symbolic structure might reduce capability, not increase it\n\nThe \"fidelity\" gain would be:\n- For tasks that match your predefined rules: perfect accuracy\n- For everything else: probably worse than a pure neural approach\n\nThe user is right to be skeptical. Tensor logic as a language for expressing computations is fine. Tensor logic as a path to better LLMs is unclear.\n\nPossible angles:\n1. **Constrained decoding** - Use tensor logic to enforce constraints on outputs\n2. **Retrieval augmentation** - Use tensor logic for structured knowledge retrieval\n3. **Interpretability** - Use tensor logic to analyze what the LLM learned (post-hoc)\n4. **Curriculum** - Train on tensor logic-structured data to encourage systematic reasoning\n\n\n5. **Auxiliary objectives** - Add tensor logic constraints as training signals\n\nNone of these fundamentally reshape the core model though—they're more like auxiliary tools that work alongside neural components rather than replacing the transformer itself. I should be direct about this uncertainty with the user.",
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"text": "This is the right question, and I don't think Domingos has a clear answer.\n\n## The Problem\n\nIf you rewrite nanoGPT in tensor logic notation:\n\n```\n# Attention in tensor logic\nScore[b,h,q,k] <- Q[b,h,q,d], K[b,h,k,d]\nAttn[b,h,q,k] <- Softmax(Score[b,h,q,k])\nOut[b,h,q,d] <- Attn[b,h,q,k], V[b,h,k,d]\n```\n\n...you just have nanoGPT. The transformer is already tensor contractions. Nothing changes.\n\n## What Would \"Tensor Logic LLM\" Even Mean?\n\n| Possible Interpretation | Problem |\n|------------------------|---------|\n| Add explicit logical rules alongside learned weights | Who writes the rules? How do they interact with learned representations? |\n| Force interpretable predicates in the embedding space | Constrains what the model can learn, probably hurts performance |\n| Use low-temperature inference for \"sound\" outputs | Only works if your rules cover the query (they won't for open-ended generation) |\n| Hybrid: neural perception → symbolic reasoning | Where's the boundary? The whole point of LLMs is end-to-end learning |\n\n## The Core Tension\n\nLLMs work **because** they don't commit to explicit structure:\n- Distributed representations generalize\n- No predefined ontology limits what they can express\n- Learning finds structure humans wouldn't design\n\nTensor logic wants:\n- Explicit predicates\n- Sound inference\n- Interpretable structure\n\nThese seem opposed. You'd gain \"fidelity\" on narrow tasks that match your rules, and lose generality everywhere else.\n\n## Maybe Useful For...\n\n- **Constrained decoding**: Enforce logical consistency on outputs\n- **Tool use**: Structured interface between LLM and external systems\n- **Verification**: Check if LLM outputs satisfy logical constraints\n- **Synthetic data**: Generate logically consistent training examples\n\nBut these are \"tensor logic alongside LLMs\" not \"tensor logic inside LLMs.\"\n\n---\n\nHonestly, I don't see the path from tensor logic to better next-token prediction. Do you have a hypothesis?"
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"content": "my thing was mostly that tensor logic and cutile seemed like a good match, idk if tensor logic has legs tbh"
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"thinking": "The user is being pragmatic here. They're saying:\n1. Tensor logic notation + cuTile kernel generation = good technical match (syntax → efficient GPU kernels)\n2. But the broader neuro-symbolic vision of tensor logic might not actually pan out\n\nSo the project becomes more about having a nice DSL for expressing tensor contractions that compiles to efficient cuTile kernels, rather than chasing the \"unified AI\" dream.\n\nThis is actually a reasonable and useful project:\n- Datalog-style syntax is clean for expressing einsum-like operations\n- Compiling to cuTile gives you tensor core performance\n- Useful for custom operations (CTM, novel attention, graph algorithms)\n- Don't need to buy into the neuro-symbolic stuff\n\nLet me respond to this pragmatically and maybe refocus on what the project could actually be useful for.",
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"text": "That's a fair assessment. The two things are somewhat separable:\n\n**Tensor logic as notation + cuTile as backend** — solid match:\n- Datalog-style rules are a clean DSL for contractions\n- Compiles to explicit GPU kernels with tensor core access\n- Useful for custom ops that don't have library calls (CTM, sparse patterns, novel attention)\n- You control tiling, fusion, memory layout\n\n**Tensor logic as path to neuro-symbolic AI** — unclear:\n- Superposition problem with real embeddings\n- No clear path to \"better LLM\"\n- The guarantees assume things that don't hold in practice\n- Mostly vibes and math that works on paper\n\n## What the Project Could Actually Be\n\nA **kernel compiler for custom tensor operations**:\n\n```python\n# Input: Datalog-ish rule\n\"H[b,d,h] <- A[b,d,m], W[m,h,d]\"\n\n# Output: cuTile kernel that hits tensor cores\n```\n\nUseful for:\n- Research architectures (CTM, state space models, whatever's next)\n- Attention variants that don't exist in FlashAttention\n- Graph neural nets with weird aggregation patterns\n- Anything where you'd otherwise write painful CUDA\n\nLess useful for:\n- Replacing PyTorch einsum for standard ops\n- The neuro-symbolic dream\n\n---\n\nWant to focus on making the compiler solid for the kernel-generation use case, and drop the grander claims? The code you have is a reasonable start for that narrower goal."
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"content": "Web search results for query: \"Continuous Thought Machine CTM NeurIPS 2025 spotlight paper\"\n\nLinks: [{\"title\":\"[2505.05522] Continuous Thought Machines\",\"url\":\"https://arxiv.org/abs/2505.05522\"},{\"title\":\"NeurIPS Poster Continuous Thought Machines\",\"url\":\"https://neurips.cc/virtual/2025/loc/san-diego/poster/115192\"},{\"title\":\"Continuous Thought Machine, Explained | The Neuron\",\"url\":\"https://www.theneuron.ai/explainer-articles/continuous-thought-machine-explained\"},{\"title\":\"Continuous Thought Machines | OpenReview\",\"url\":\"https://openreview.net/forum?id=y0wDflmpLk\"},{\"title\":\"Continuous Thought Machines\",\"url\":\"https://arxiv.org/html/2505.05522v4\"},{\"title\":\"Continuous Thought Machines Luke Darlow1 Ciaran Regan1,2 Sebastian Risi1,3\",\"url\":\"https://openreview.net/pdf/e74ece26bd36c4d9fece081ebf9d7cd8e3c6d1f5.pdf\"},{\"title\":\"Continuous Thought Machines - ADS\",\"url\":\"https://ui.adsabs.harvard.edu/abs/2025arXiv250505522D/abstract\"},{\"title\":\"Sakana AI on X: \\\"2/ “Continuous Thought Machines” will be presented as a Spotlight at #NeurIPS2025🧠 Paper: https://t.co/JKhZA0xYcd https://t.co/UNhaXDHfts Most neural networks abstract away time, but CTMs use neuron-level dynamics and synchronization as their core representation. This\\\" / X\",\"url\":\"https://x.com/SakanaAILabs/status/1992909033800716667\"},{\"title\":\"12 May 2025 Continuous Thought Machines\",\"url\":\"https://pub.sakana.ai/ctm/paper/ctm.pdf\"},{\"title\":\"Introducing Continuous Thought Machines\",\"url\":\"https://sakana.ai/ctm/\"}]\n\nBased on my search, here's what I found about the Continuous Thought Machine (CTM) NeurIPS 2025 spotlight paper:\n\n## Overview\n\n\"Continuous Thought Machines\" was accepted as a spotlight paper at NeurIPS 2025.(no content)The paper was developed by Llion Jones (a co-inventor of the Transformer) and researcher Luke Darlow.\n\n## What is the CTM?\n\nThe Continuous Thought Machine (CTM) is a model designed to leverage neural dynamics as its core representation.(no content)By incorporating neuron-level processing and synchronization, it reintroduces neural timing as a foundational element.(no content)The CTM has two core innovations: (1) neuron-level temporal processing, where each neuron uses unique weight parameters to process a history of incoming signals; and (2) neural synchronization employed as a latent representation.\n\n## Key Capabilities\n\nThe CTM demonstrates performance and versatility across a range of tasks, including solving 2D mazes, ImageNet-1K classification, parity computation, and more.(no content)Beyond displaying rich internal representations and offering a natural avenue for interpretation owing to its internal process, the CTM is able to perform tasks that require complex sequential reasoning. The CTM can also leverage adaptive compute, where it can stop earlier for simpler tasks, or keep computing when faced with more challenging instances.\n\n## Emergent Behaviors\n\nWhen trained on maze-solving, the CTM actually backtracks when it makes wrong turns. When it doesn't have enough time to solve a maze, it learns a leapfrogging algorithm to approximate solutions.(no content)When observing images, the CTM carefully moves its gaze around the scene, choosing to focus on the most salient features present.\n\n## Research Goal\n\nThe goal of this work is to share the CTM and its associated innovations, rather than pushing for new state-of-the-art results. The authors believe the CTM represents a significant step toward developing more biologically plausible and powerful artificial intelligence systems.\n\n## Resources\n\nAn interactive online demonstration is available at https://pub.sakana.ai/ctm/ and an extended technical report at https://pub.sakana.ai/ctm/paper.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "# CTM Technical Architecture Extraction\n\n## Neuron-Level Models (NLMs)\n\nNLMs are privately-parameterized MLPs where each neuron processes its own pre-activation history independently. Mathematically:\n\n```\nz_d^{t+1} = g_{θ_d}(A_d^t)\n```\n\nWhere `g_{θ_d}` is a depth-1 MLP with hidden dimension `d_hidden`, and `A_d^t` represents the M most recent pre-activations for neuron d. The implementation uses Einstein summation for efficiency:\n\n\"The first einsum computes a h-dimension latent for each neuron from the (truncated to M most recent) incoming history\" before projecting to final activations per neuron.\n\n## Synchronization Mechanism\n\nSynchronization forms the core latent representation through temporal correlations:\n\n```\nS^t = Z^t · (Z^t)^⊤ ∈ ℝ^{D×D}\n```\n\nWhere `Z^t` stacks all post-activation histories. Rather than using the full matrix, the CTM samples neuron pairs with learnable exponential decay:\n\n```\nS_{ij}^t = (Z_i^t)^⊤ · diag(R_{ij}^t) · (Z_j^t) / √(Σ[R_{ij}^t])\n```\n\nHigher decay rates `r_{ij}` bias toward recent internal ticks.\n\n## Internal State & Tensor Operations\n\nThe CTM maintains rolling histories:\n- **Pre-activations** `A^t ∈ ℝ^{D×M}`: Recent M pre-activation vectors concatenated\n- **Post-activations history** `Z^t ∈ ℝ^{D×t}`: All post-activations accumulated over time\n\nThe synapse model interconnects neurons:\n\n```\na^t = f_{θ_syn}(concat(z^t, o^t)) ∈ ℝ^D\n```\n\nUsing a U-Net architecture with skip connections and layer-norm for flexible synaptic computation.\n\n## Attention & Weight Mechanism\n\nAttention queries derive from action synchronization through linear projection:\n\n```\nq^t = W_in · S_{action}^t\n```\n\nStandard cross-attention then retrieves information: \"Attention(Q=q^t, KV=FeatureExtractor(data))\" generates attention output `o^t ∈ ℝ^{d_input}`.\n\nOutput predictions similarly project output synchronization:\n\n```\ny^t = W_out · S_{out}^t\n```\n\n## Forward Pass Overview\n\nThe iterative process unfolds across T internal ticks:\n\n1. Project query from action synchronization\n2. Attend to data via cross-attention\n3. Concatenate attention output with previous post-activations\n4. Pass through synapse model to generate pre-activations\n5. Maintain rolling pre-activation history (FIFO)\n6. Apply NLMs to generate post-activations\n7. Compute updated synchronization matrices\n8. Generate outputs and repeat\n\n## Loss Function\n\n\"The CTM produces outputs y^t at each internal tick t. We compute a loss and certainty per tick,\" selecting two critical points:\n\n```\nL = (L^{t_1} + L^{t_2}) / 2\n```\n\nWhere `t_1` minimizes loss and `t_2` maximizes certainty (1 - normalized entropy), enabling native adaptive computation without explicit halting mechanisms."
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"result": "# CTM Technical Architecture Extraction\n\n## Neuron-Level Models (NLMs)\n\nNLMs are privately-parameterized MLPs where each neuron processes its own pre-activation history independently. Mathematically:\n\n```\nz_d^{t+1} = g_{θ_d}(A_d^t)\n```\n\nWhere `g_{θ_d}` is a depth-1 MLP with hidden dimension `d_hidden`, and `A_d^t` represents the M most recent pre-activations for neuron d. The implementation uses Einstein summation for efficiency:\n\n\"The first einsum computes a h-dimension latent for each neuron from the (truncated to M most recent) incoming history\" before projecting to final activations per neuron.\n\n## Synchronization Mechanism\n\nSynchronization forms the core latent representation through temporal correlations:\n\n```\nS^t = Z^t · (Z^t)^⊤ ∈ ℝ^{D×D}\n```\n\nWhere `Z^t` stacks all post-activation histories. Rather than using the full matrix, the CTM samples neuron pairs with learnable exponential decay:\n\n```\nS_{ij}^t = (Z_i^t)^⊤ · diag(R_{ij}^t) · (Z_j^t) / √(Σ[R_{ij}^t])\n```\n\nHigher decay rates `r_{ij}` bias toward recent internal ticks.\n\n## Internal State & Tensor Operations\n\nThe CTM maintains rolling histories:\n- **Pre-activations** `A^t ∈ ℝ^{D×M}`: Recent M pre-activation vectors concatenated\n- **Post-activations history** `Z^t ∈ ℝ^{D×t}`: All post-activations accumulated over time\n\nThe synapse model interconnects neurons:\n\n```\na^t = f_{θ_syn}(concat(z^t, o^t)) ∈ ℝ^D\n```\n\nUsing a U-Net architecture with skip connections and layer-norm for flexible synaptic computation.\n\n## Attention & Weight Mechanism\n\nAttention queries derive from action synchronization through linear projection:\n\n```\nq^t = W_in · S_{action}^t\n```\n\nStandard cross-attention then retrieves information: \"Attention(Q=q^t, KV=FeatureExtractor(data))\" generates attention output `o^t ∈ ℝ^{d_input}`.\n\nOutput predictions similarly project output synchronization:\n\n```\ny^t = W_out · S_{out}^t\n```\n\n## Forward Pass Overview\n\nThe iterative process unfolds across T internal ticks:\n\n1. Project query from action synchronization\n2. Attend to data via cross-attention\n3. Concatenate attention output with previous post-activations\n4. Pass through synapse model to generate pre-activations\n5. Maintain rolling pre-activation history (FIFO)\n6. Apply NLMs to generate post-activations\n7. Compute updated synchronization matrices\n8. Generate outputs and repeat\n\n## Loss Function\n\n\"The CTM produces outputs y^t at each internal tick t. We compute a loss and certainty per tick,\" selecting two critical points:\n\n```\nL = (L^{t_1} + L^{t_2}) / 2\n```\n\nWhere `t_1` minimizes loss and `t_2` maximizes certainty (1 - normalized entropy), enabling native adaptive computation without explicit halting mechanisms.",
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"thinking": "This is really interesting. The CTM architecture has some unusual tensor operations that would be good candidates for custom kernels:\n\n1. **NLM layer**: Each neuron has its own MLP processing its pre-activation history. This is an einsum operation `H[b,d,h] <- A[b,d,m], W[m,h,d]` - exactly what the user had in their tensor logic compiler example.\n\n2. **Synchronization**: `S = Z @ Z.T` but with learned decay weights per pair. This is like a batched attention score computation.\n\n3. **Rolling histories**: FIFO buffer management across time steps.\n\nThese are non-standard operations that don't map cleanly to existing CUDA libraries. This is exactly the use case where a tensor logic → cuTile compiler would be valuable.\n\nLet me summarize this for the user and connect it back to their project.",
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"text": "This is from Sakana AI (Llion Jones, co-inventor of transformers). Interesting architecture.\n\n## CTM Core Ideas\n\n**1. Neuron-Level Models (NLMs)** — each neuron has its own MLP:\n```\nz_d^{t+1} = g_{θ_d}(A_d^t)\n```\nWhere `A_d^t` is that neuron's history of M recent pre-activations. This is exactly your tensor logic rule:\n```\nH[b,d,h] <- A[b,d,m], W[m,h,d]\n```\nEach of the D neurons has its own weights — not a shared matrix.\n\n**2. Synchronization as representation** — the latent state is temporal correlation between neurons:\n```\nS^t = Z^t · (Z^t)^⊤ ∈ ℝ^{D×D}\n```\nWith learned per-pair decay weights. Your other rule:\n```\nS[b,i,j] <- Z1[b,i,t], Z2[b,j,t]\n```\n\n**3. Iterative refinement** — runs for T \"internal ticks\", can stop early when confident (adaptive compute).\n\n## Why This Fits Your Project\n\nThese ops are **weird** from a library perspective:\n\n| Operation | Standard Library? | Notes |\n|-----------|------------------|-------|\n| Per-neuron MLPs | No | Einsum with broadcast, not batched matmul |\n| Pairwise sync with learned decay | No | Custom weighted inner product |\n| Rolling FIFO history | Sort of | But fused with the above |\n\nPyTorch will do reshapes + batched matmul + reshapes. Probably not optimal. A tensor logic → cuTile path could generate kernels that fuse these patterns directly.\n\n## The Interesting Bit\n\nCTM is explicitly about **temporal dynamics** and **interpretability** (you can watch it \"think\"). It's not trying to be neuro-symbolic, but it does have more structure than a standard transformer.\n\nThe synchronization matrix `S` is kind of like an explicit \"what's related to what\" representation at each timestep — more inspectable than attention weights.\n\n---\n\nSo: tensor logic compiler for CTM-style architectures might be a better framing than \"neuro-symbolic AI\". You're providing efficient custom kernels for research architectures that don't fit the standard matmul/attention mold.\n\n**Sources:**\n- [CTM Paper (arXiv)](https://arxiv.org/abs/2505.05522)\n- [Sakana AI Blog](https://sakana.ai/ctm/)\n- [NeurIPS 2025 Poster](https://neurips.cc/virtual/2025/loc/san-diego/poster/115192)"
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"content": "did they test ctm w/ pytorch? do you really think there are wins here?"
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"thinking": "The user is asking a practical question - did the CTM authors just use PyTorch, and is there actually performance to be gained from custom kernels?\n\nLet me think about this honestly:\n\n1. The CTM paper almost certainly uses PyTorch (Sakana AI typically does)\n2. PyTorch's einsum and torch.compile are pretty good these days\n3. The question is whether the specific patterns in CTM (per-neuron MLPs, pairwise sync) leave performance on the table\n\nThe honest answer is probably:\n- For standard-ish ops, PyTorch is fine\n- The NLM einsum `bdm,mhd->bdh` might not be optimally handled\n- But torch.compile with Triton backend might already do a decent job\n- The real wins would come from fusion across multiple ops, not single kernels\n\nLet me search for any performance details from the CTM paper or implementation.",
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"text": "Good question. Let me check what they actually used and whether there's real headroom."
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"content": "# Key PyTorch Operations in Continuous Thought Machine\n\n## NLM (Neuron-Level Models) Forward Pass\n\nThe NLM applies per-neuron MLPs to memory traces:\n\n```python\nSuperLinear(in_dims=memory_length, out_dims=2*memory_hidden_dims, N=d_model)\nnn.GLU()\nSuperLinear(in_dims=memory_hidden_dims, out_dims=2, N=d_model)\nnn.GLU()\n```\n\n**Key characteristic**: Uses `SuperLinear` with grouped convolutions (group size = 1), applying separate linear layers per neuron across the `d_model` dimension.\n\n## Synchronization Computation\n\nThe core synchronization calculation leverages recurrence rather than full trace dot products:\n\n```python\ndecay_alpha = r * decay_alpha + pairwise_product\ndecay_beta = r * decay_beta + 1\nsynchronisation = decay_alpha / torch.sqrt(decay_beta)\n```\n\n**Three neuron selection strategies**:\n- `first-last`: Upper triangular outer products from first/last neurons\n- `random`: Random neuron pairs with outer products\n- `random-pairing`: Direct element-wise multiplication of paired neurons\n\n## Attention Integration\n\n```python\nq = self.q_proj(synchronisation_action).unsqueeze(1)\nattn_out, attn_weights = self.attention(q, kv, kv)\n```\n\nUses standard `MultiheadAttention` with synchronization as query.\n\n## Performance Considerations\n\n**No custom CUDA or torch.compile detected** in provided code. Optimizations include:\n\n- Lazy module initialization via `nn.LazyLinear` and `nn.LazyConv2d`\n- Incremental synchronization updates avoiding redundant computations\n- Clamping decay parameters: `torch.clamp(decay_params, 0, 15)`\n\n**No einsum operations** present; uses explicit outer products and element-wise operations instead."
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"result": "# Key PyTorch Operations in Continuous Thought Machine\n\n## NLM (Neuron-Level Models) Forward Pass\n\nThe NLM applies per-neuron MLPs to memory traces:\n\n```python\nSuperLinear(in_dims=memory_length, out_dims=2*memory_hidden_dims, N=d_model)\nnn.GLU()\nSuperLinear(in_dims=memory_hidden_dims, out_dims=2, N=d_model)\nnn.GLU()\n```\n\n**Key characteristic**: Uses `SuperLinear` with grouped convolutions (group size = 1), applying separate linear layers per neuron across the `d_model` dimension.\n\n## Synchronization Computation\n\nThe core synchronization calculation leverages recurrence rather than full trace dot products:\n\n```python\ndecay_alpha = r * decay_alpha + pairwise_product\ndecay_beta = r * decay_beta + 1\nsynchronisation = decay_alpha / torch.sqrt(decay_beta)\n```\n\n**Three neuron selection strategies**:\n- `first-last`: Upper triangular outer products from first/last neurons\n- `random`: Random neuron pairs with outer products\n- `random-pairing`: Direct element-wise multiplication of paired neurons\n\n## Attention Integration\n\n```python\nq = self.q_proj(synchronisation_action).unsqueeze(1)\nattn_out, attn_weights = self.attention(q, kv, kv)\n```\n\nUses standard `MultiheadAttention` with synchronization as query.\n\n## Performance Considerations\n\n**No custom CUDA or torch.compile detected** in provided code. Optimizations include:\n\n- Lazy module initialization via `nn.LazyLinear` and `nn.LazyConv2d`\n- Incremental synchronization updates avoiding redundant computations\n- Clamping decay parameters: `torch.clamp(decay_params, 0, 15)`\n\n**No einsum operations** present; uses explicit outer products and element-wise operations instead.",
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"thinking": "Interesting. So the CTM implementation:1. Uses `SuperLinear` with grouped convolutions (group size = 1) for per-neuron MLPs - this is a workaround for the \"each neuron has its own weights\" pattern\n2. No einsum - they use explicit outer products and element-wise ops\n3. No torch.compile or custom CUDA\n4. Incremental synchronization (avoids recomputing full history each step)So the question is: is there room for optimization?\n\nThe `SuperLinear` with grouped conv is a PyTorch workaround for what tensor logic would express as `H[b,d,h] <- A[b,d,m], W[m,h,d]`. It might not be optimal.\n\nThe synchronization is done incrementally (good!) but the outer products / element-wise ops might have room for fusion.\n\nBut honestly... modern PyTorch with torch.compile might already do a decent job. The real question is whether the tensor logic → cuTile path would actually be faster.\n\nLet me be honest with the user about this.",
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"text": "Okay, here's the honest assessment:\n\n## What They Actually Do\n\n**NLM Layer:**\n```python\nSuperLinear(in_dims=memory_length, out_dims=..., N=d_model)\n```\nThey use grouped convolutions with `groups=d_model` as a PyTorch workaround for \"each neuron has its own weights.\" Not einsum.\n\n**Synchronization:**\n```python\ndecay_alpha = r * decay_alpha + pairwise_product\nsynchronisation = decay_alpha / torch.sqrt(decay_beta)\n```\nIncremental update, not full history recomputation. Already reasonably optimized.\n\n**No custom CUDA, no torch.compile** in the repo.\n\n## Is There Headroom?\n\n| Component | Their Approach | Potential Win? |\n|-----------|---------------|----------------|\n| NLM (per-neuron MLP) | Grouped conv hack | Maybe — einsum might be cleaner but not necessarily faster |\n| Sync outer products | Explicit loops over neuron pairs | Possibly — could fuse with decay |\n| Attention | Standard nn.MultiheadAttention | No — use FlashAttention |\n| Overall fusion | None | Modest — could fuse sync + attention query |\n\n## My Honest Take\n\n**Probably not huge wins.** Here's why:\n\n1. **torch.compile exists** — if they added `@torch.compile` to their forward pass, Triton would likely generate decent fused kernels automatically\n\n2. **Memory-bound, not compute-bound** — CTM's bottleneck is probably memory bandwidth (rolling histories, sync matrices), not raw FLOPS\n\n3. **Grouped conv is already fast** — PyTorch's grouped conv is well-optimized for this pattern\n\n4. **The interesting ops are small** — NLM has `memory_length ≈ 50`, not 4096. Tensor cores don't shine at small tile sizes.\n\n## Where There Might Be Wins\n\n- **Fusing the full tick** — sync update + attention query + synapse forward as one kernel\n- **Custom memory layout** — keep histories in a format that avoids reshapes\n- **Sparse sync** — they sample neuron pairs, could exploit sparsity\n\nBut these are \"maybe 20-30% faster\" wins, not \"10x\" wins. And you'd be competing against torch.compile which gets better every release.\n\n---\n\nThe tensor logic → cuTile compiler makes more sense as a **research tool** (express weird ops cleanly, get decent kernels) than a **performance play** (beat PyTorch on standard hardware).\n\nDoes that change your interest level?"
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"text": "## The Problem\n\nYou want each of D neurons to have its **own** MLP weights:\n\n```\n# For neuron d, with its own weights W_d:\nh_d = MLP_d(input_d)\n\n# Conceptually:\nfor d in range(D):\n output[d] = W[d] @ input[d] # Each d has different W\n```\n\nIn tensor logic notation:\n```\nH[b,d,h] <- A[b,d,m], W[d,m,h]\n# ↑ W is indexed by d — different weights per neuron\n```\n\nThis is **not** a standard batched matmul. Standard matmul shares weights across the batch dimension.\n\n## The Naive Approach (Slow)\n\n```python\n# Loop over neurons — terrible performance\noutputs = []\nfor d in range(D):\n outputs.append(self.mlps[d](input[:, d, :])) # D separate forward passes\noutput = torch.stack(outputs, dim=1)\n```\n\nD separate kernel launches. Awful.\n\n## The Grouped Conv Hack\n\nA 1D convolution with `groups=D` does exactly this:\n\n```python\n# input: [batch, D, M] — D neurons, M history length\n# Want: [batch, D, H] — D neurons, H hidden dim\n\nconv = nn.Conv1d(\n in_channels=D * M, # Treat each (neuron, feature) as a channel\n out_channels=D * H,\n kernel_size=1,\n groups=D # ← The magic: D independent convolutions\n)\n```\n\nWith `groups=D`:\n- Input channels are split into D groups of M\n- Output channels are split into D groups of H \n- Each group has its **own** weights\n- No cross-talk between groups\n\n```\nGroup 0: input[:, 0:M] → W_0 → output[:, 0:H]\nGroup 1: input[:, M:2M] → W_1 → output[:, H:2H]\n...\nGroup D: input[:, (D-1)*M:D*M] → W_D → output[:, (D-1)*H:D*H]\n```\n\n**One kernel launch, D independent linear transforms.**\n\n## Visual\n\n```\nStandard Conv (groups=1):\n┌─────────────────┐\n│ All inputs │ ──→ [shared W] ──→ All outputs\n└─────────────────┘\n\nGrouped Conv (groups=D):\n┌─────┐\n│ n_0 │ ──→ [W_0] ──→ out_0\n├─────┤\n│ n_1 │ ──→ [W_1] ──→ out_1\n├─────┤\n│ ... │ ...\n├─────┤\n│ n_D │ ──→ [W_D] ──→ out_D\n└─────┘\n```\n\n## Why It's a \"Hack\"\n\nIt works but it's awkward:\n- Have to reshape input from `[B, D, M]` to `[B, D*M, 1]`\n- Conv1d with kernel_size=1 is really just a linear layer in disguise\n- The semantics are obscured — you're not really doing convolution\n\n## The Tensor Logic Way\n\nYour einsum approach is more direct:\n\n```python\n# W: [D, M, H] — each neuron d has its own M×H matrix\n# A: [B, D, M] — input\n# H: [B, D, H] — output\n\nH = torch.einsum('bdm,dmh->bdh', A, W)\n```\n\nSame computation, clearer semantics. The question is whether PyTorch/cuTile can make this as fast as the grouped conv path (which has years of optimization behind it)."
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"text": "Not an idiot question — convolutions are weirdly named and overloaded.\n\n## Convolution: The Core Idea\n\nYou have a **sliding window** that moves across your input, doing the same operation at each position.\n\n```\nInput: [1, 2, 3, 4, 5, 6, 7, 8]\nKernel: [a, b, c] (size 3)\n\nSlide it across:\n\nPosition 0: [1, 2, 3] · [a, b, c] = 1a + 2b + 3c\nPosition 1: [2, 3, 4] · [a, b, c] = 2a + 3b + 4c\nPosition 2: [3, 4, 5] · [a, b, c] = 3a + 4b + 5c\n...\n\nOutput: [result_0, result_1, result_2, ...]\n```\n\nThat's it. Slide, dot product, repeat.\n\n## Why \"Convolution\"?\n\nMathematically, it's the integral/sum of one function multiplied by a shifted version of another. The name comes from signal processing. In practice, just think \"sliding dot product.\"\n\n## 1D vs 2D\n\n**1D Conv** — slides along a sequence:\n```\nAudio: [sample_0, sample_1, sample_2, ...]\n ←────kernel slides────→\n```\n\n**2D Conv** — slides across an image:\n```\nImage:\n┌─────────────┐\n│ □ □ □ □ □ □ │\n│ □ ■ ■ ■ □ □ │ ← 3×3 kernel at this position\n│ □ ■ ■ ■ □ □ │\n│ □ ■ ■ ■ □ □ │\n│ □ □ □ □ □ □ │\n└─────────────┘\n ↓\nOne output pixel = sum of (9 input pixels × 9 kernel weights)\n```\n\n## Multiple Channels\n\nImages have RGB (3 channels). So you convolve each channel and sum:\n\n```\nInput: [batch, channels_in, height, width]\nKernel: [channels_out, channels_in, kH, kW]\nOutput: [batch, channels_out, height', width']\n```\n\nEach output channel is a different \"filter\" looking for different patterns.\n\n## kernel_size=1 (The \"Hack\" Case)\n\nWhen kernel size is 1, there's no sliding — it's just a linear transform at each position:\n\n```\nConv1d with kernel_size=1:\n\nInput: [batch, C_in, length]\nKernel: [C_out, C_in, 1]\n\nAt each position:\n output[:, :, pos] = W @ input[:, :, pos]\n```\n\nThis is literally just `nn.Linear` applied independently to each position. People use it to mix channels without mixing positions.\n\n## Groups: The Key to the Hack\n\nNormal conv: **all** input channels contribute to **all** output channels.\n\n```\ngroups=1 (default):\n┌──────────────┐ ┌──────────────┐\n│ in_channel_0 │──┬───→│ out_channel_0│\n│ in_channel_1 │──┼───→│ out_channel_1│\n│ in_channel_2 │──┼───→│ out_channel_2│\n│ in_channel_3 │──┴───→│ out_channel_3│\n└──────────────┘ └──────────────┘\n All connected (one big weight matrix)\n```\n\nGrouped conv: split into independent groups, **no cross-talk**:\n\n```\ngroups=2:\n┌──────────────┐ ┌──────────────┐\n│ in_channel_0 │──────→│ out_channel_0│ Group 0\n│ in_channel_1 │──────→│ out_channel_1│ (own weights)\n├──────────────┤ ├──────────────┤\n│ in_channel_2 │──────→│ out_channel_2│ Group 1\n│ in_channel_3 │──────→│ out_channel_3│ (own weights)\n└──────────────┘ └──────────────┘\n```\n\n**groups=D with D neurons**: each neuron is its own group with its own weights. That's the CTM trick.\n\n## TL;DR\n\n| Concept | What It Means |\n|---------|--------------|\n| Convolution | Sliding window dot product |\n| kernel_size=1 | No sliding, just linear transform per position |\n| groups=D | D independent transforms, no mixing between groups |\n| Grouped conv1d, k=1 | D separate linear layers in one kernel launch |"
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"text": "Sort of — \"kernel\" is annoyingly overloaded. Here it means **the small array of weights** that slides across your input.\n\n## Different \"Kernels\" in CS/Math\n\n| Context | What \"kernel\" means |\n|---------|-------------------|\n| Convolution | The small weight array that slides (e.g., 3×3 = 9 numbers) |\n| OS/CUDA | The core program (Linux kernel, GPU kernel) |\n| Kernel methods (SVM) | A similarity function k(x, y) |\n| Linear algebra | The null space of a matrix |\n\nTotally different things, same word.\n\n## In Convolution\n\nThe \"kernel\" (also called \"filter\") is just the weights:\n\n```\nkernel = [0.5, 1.0, 0.5] # 3 learnable numbers\n\ninput: [1, 2, 3, 4, 5]\n ↑─────↑\n [0.5, 1.0, 0.5] ← this little array is \"the kernel\"\n \noutput[0] = 1×0.5 + 2×1.0 + 3×0.5 = 4.0\n```\n\nFor 2D (images):\n```\nkernel = [[1, 0, -1],\n [2, 0, -2], # 3×3 = 9 learnable numbers\n [1, 0, -1]]\n```\n\n## Why \"Kernel\"?\n\nComes from integral transforms in math — the function inside the integral that \"operates\" on your input. Convolution in continuous form:\n\n```\n(f * g)(t) = ∫ f(τ) · g(t - τ) dτ\n ↑\n \"kernel\" function\n```\n\nThe neural network version discretizes this: instead of a continuous function g, you have a small array of weights.\n\n## So in CTM Context\n\nWhen I said \"kernel_size=1\" — that means the sliding window has size 1 (no actual sliding). Just one weight per input position. Which makes it equivalent to a linear layer."
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"text": "Let me be super concrete with actual shapes.\n\n## Setup\n\n- D = 4 neurons\n- M = 3 (history length per neuron)\n- H = 2 (hidden dim of each neuron's MLP)\n\nInput shape: `[batch, D, M]` = `[batch, 4, 3]`\n\nEach neuron has 3 numbers (its history). We want each neuron to have its **own** MLP that transforms 3 → 2.\n\n## What We Want (Conceptually)\n\n```\nNeuron 0: [h0, h1, h2] → MLP_0 → [out0, out1]\nNeuron 1: [h0, h1, h2] → MLP_1 → [out0, out1]\nNeuron 2: [h0, h1, h2] → MLP_2 → [out0, out1]\nNeuron 3: [h0, h1, h2] → MLP_3 → [out0, out1]\n\nFour DIFFERENT MLPs, not shared weights.\n```\n\n## Step 1: Reshape\n\nReshape `[batch, D, M]` → `[batch, D*M, 1]`\n\n```\n[batch, 4, 3] → [batch, 12, 1]\n\nBefore (4 neurons × 3 history):\n neuron0: [a, b, c]\n neuron1: [d, e, f]\n neuron2: [g, h, i]\n neuron3: [j, k, l]\n\nAfter (12 channels × 1 position):\n channels: [a, b, c, d, e, f, g, h, i, j, k, l]\n ├──n0──┤ ├──n1──┤ ├──n2──┤ ├──n3──┤\n```\n\nNow each neuron's history is a **group of 3 channels**.\n\n## Step 2: Grouped Conv\n\n```python\nconv = nn.Conv1d(\n in_channels=12, # D*M = 4*3\n out_channels=8, # D*H = 4*2\n kernel_size=1,\n groups=4 # D groups\n)\n```\n\nWith `groups=4`, the conv creates **4 independent linear transforms**:\n\n```\nGroup 0: channels [0,1,2] → W_0 (3×2) → channels [0,1]\nGroup 1: channels [3,4,5] → W_1 (3×2) → channels [2,3]\nGroup 2: channels [6,7,8] → W_2 (3×2) → channels [4,5]\nGroup 3: channels [9,10,11] → W_3 (3×2) → channels [6,7]\n```\n\nEach W is different! That's the key.\n\n```\nWeight tensor shape: [out_channels, in_channels/groups, kernel_size]\n = [8, 3, 1]\n\nBut it's NOT one 8×3 matrix. It's:\n W_0: [2, 3] for group 0\n W_1: [2, 3] for group 1\n W_2: [2, 3] for group 2\n W_3: [2, 3] for group 3\n```\n\n## Step 3: Reshape Back\n\n`[batch, 8, 1]` → `[batch, 4, 2]` = `[batch, D, H]`\n\nEach neuron now has its 2-dim output, computed by its own weights.\n\n## Full MLP = Two Convs + Activation\n\n```python\n# Per-neuron MLP: M → hidden → output\nself.mlp = nn.Sequential(\n nn.Conv1d(D * M, D * hidden, kernel_size=1, groups=D),\n nn.ReLU(),\n nn.Conv1d(D * hidden, D * out, kernel_size=1, groups=D),\n)\n\n# Forward:\nx = x.view(batch, D * M, 1) # [B, D, M] → [B, D*M, 1]\nx = self.mlp(x) # [B, D*M, 1] → [B, D*out, 1]\nx = x.view(batch, D, out) # [B, D*out, 1] → [B, D, out]\n```\n\nTwo grouped convs with activation between = D independent MLPs.\n\n## The Trick Summarized\n\n| What you want | How grouped conv does it |\n|---------------|-------------------------|\n| D separate weight matrices | `groups=D` splits weights into D chunks |\n| No cross-talk between neurons | Groups don't see each other's channels |\n| One kernel launch | Conv is a single fused CUDA op |"
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"thinking": "The user is making a good point - PyTorch practitioners have figured out clever tricks (like grouped convolutions) to express unusual operations efficiently. This undermines the case for a tensor logic compiler - if people can already map weird ops to PyTorch idioms that are well-optimized, what's the point?\n\nLet me acknowledge this and think about what value tensor logic -> cuTile might actually provide:\n\n1. Maybe the value isn't performance but expressiveness/clarity?\n2. Maybe it's for truly novel ops where no one has figured out the trick yet?\n3. Maybe torch.compile is eating this space anyway?\n\nI should be honest that the case for this project is getting weaker.",
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"text": "Yeah, that's the thing. The PyTorch ecosystem has had years of people figuring out these mappings:\n\n- Grouped conv for per-element weights\n- `einsum` for arbitrary contractions \n- `torch.vmap` for batching weird ops\n- Clever reshapes to hit fast paths\n\nAnd now `torch.compile` + Triton generates fused kernels automatically for a lot of patterns.\n\n## Where Does That Leave Tensor Logic → cuTile?\n\n**The case is weaker than I initially thought:**\n\n| Argument | Counter |\n|----------|---------|\n| \"Express weird ops cleanly\" | Einsum already does this |\n| \"Get efficient kernels\" | torch.compile is catching up |\n| \"Control tiling/fusion\" | Most people don't need this |\n| \"Tensor core access\" | cuBLAS/cuDNN already hit them |\n\n**Maybe still useful for:**\n\n1. **Truly novel ops** — before someone figures out the PyTorch trick\n2. **Research DSL** — cleaner notation than einsum chains, documents intent\n3. **Non-PyTorch targets** — if you wanted to target something else (JAX, custom hardware)\n4. **Learning exercise** — understanding how contractions map to GPU execution\n\nBut as a \"make my model faster\" tool? Probably not, unless you're doing something really unusual and perf-critical.\n\n---\n\nSeems like the honest conclusion is: tensor logic is nice notation, cuTile is cool tech, but PyTorch + torch.compile has eaten most of the lunch here. The neuro-symbolic angle doesn't pan out, and the performance angle is marginal.\n\nWas this mostly exploratory for you, or were you hoping to build something production-useful?"
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"content": "cutile is not gonna work w/ non nvidia, the thing would control tilling / fusion but idk hwo you would expose that to a user in a sane way"
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"thinking": "The user is making more good points:\n\n1. cuTile is NVIDIA-only, so it's not a general solution\n2. Even if you can control tiling/fusion, how do you expose that to users in a sensible way? Like, what knobs would you give them? Tile sizes? Fusion decisions? That's pretty low-level and requires understanding GPU architecture.\n\nThis is getting at a real problem with \"compiler\" approaches - the abstraction has to be at the right level. Too high and you lose control, too low and it's just CUDA with extra steps.\n\nLet me engage with this honestly. Maybe the project doesn't have strong legs.",
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"text": "Right, two good points:\n\n**1. NVIDIA lock-in**\n\ncuTile only runs on Blackwell+ GPUs. So you're building for a narrow target. Not \"tensor logic for AI\" — more like \"tensor logic for people with RTX 5090s.\"\n\n**2. The UX problem**\n\nSay you expose tiling/fusion controls. What does that even look like?\n\n```python\n# Option A: Magic numbers nobody understands\ncompile_rule(\"C[i,k] <- A[i,j], B[j,k]\", \n tile_i=64, tile_j=32, tile_k=128,\n fuse_with=[\"softmax\"])\n\n# Option B: Just trust the compiler\ncompile_rule(\"C[i,k] <- A[i,j], B[j,k]\") # picks defaults\n\n# Option C: Hint-based?\ncompile_rule(\"C[i,k] <- A[i,j], B[j,k]\",\n hints=[\"memory_bound\", \"small_k\"])\n```\n\nOption A is expert-only. Option B is what torch.compile already does. Option C is vague and hard to act on.\n\n**The hard question:** Who is the user?\n\n| User | What they want | Does this help? |\n|------|----------------|-----------------|\n| ML researcher | \"Just make it fast\" | No — use torch.compile |\n| Kernel engineer | Full control over GPU | No — they'd write CUDA/Triton directly |\n| Somewhere in between? | ??? | Maybe, but that's a tiny niche |\n\nThe problem is that tiling/fusion decisions require understanding:\n- Memory hierarchy (L1, L2, HBM bandwidth)\n- Tensor core tile shapes (16×16×16 etc)\n- Occupancy tradeoffs\n- Your specific tensor sizes\n\nThat's not something you expose through a DSL. It's autotuning territory (like Triton does) or expert hand-tuning.\n\n---\n\nI think the project was a reasonable idea to explore, but the conclusion might be: \"this space is already served by existing tools, and the remaining gaps are either tiny or require expertise that a DSL can't abstract away.\"\n\nDoes that match your read?"
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"content": "cutile is kind of nvidia's answer to triton?"
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"content": "Web search results for query: \"NVIDIA cuTile vs Triton comparison tile-based GPU programming\"\n\nLinks: [{\"title\":\"CUDA Tile | NVIDIA Developer\",\"url\":\"https://developer.nvidia.com/cuda/tile\"},{\"title\":\"Focus on Your Algorithm—NVIDIA CUDA Tile Handles the Hardware | NVIDIA Technical Blog\",\"url\":\"https://developer.nvidia.com/blog/focus-on-your-algorithm-nvidia-cuda-tile-handles-the-hardware\"},{\"title\":\"Block/Tile-Based GPU Programming (not Scratch) - GPU - Julia Programming Language\",\"url\":\"https://discourse.julialang.org/t/block-tile-based-gpu-programming-not-scratch/127731\"},{\"title\":\"NVIDIA CUDA 13.1 Powers Next-Gen GPU Programming with NVIDIA CUDA Tile and Performance Gains | NVIDIA Technical Blog\",\"url\":\"https://developer.nvidia.com/blog/nvidia-cuda-13-1-powers-next-gen-gpu-programming-with-nvidia-cuda-tile-and-performance-gains\"},{\"title\":\"Nvidia Tilus: A Tile-Level GPU Kernel Programming Language | Hacker News\",\"url\":\"https://news.ycombinator.com/item?id=44906985\"},{\"title\":\"cuTile Python: Tile-Based GPU Programming in CUDA 13.1 | by CodeBun | Coding Nexus | Dec, 2025 | Medium\",\"url\":\"https://medium.com/coding-nexus/cutile-python-tile-based-gpu-programming-in-cuda-13-1-41c28a4d3b8f\"},{\"title\":\"Marco Mascorro @ NeurIPS 2025 on X: \\\"Nvidia launched an OpenAI’s Triton competitor for GPU kernels: cuTile, a tile programming model for CUDA:\\\" / X\",\"url\":\"https://x.com/Mascobot/status/1902543270519173478\"},{\"title\":\"cuTile, the New/Old Kid on the Block: Python Programming Models for GPUs :: SciPy 2025 :: pretalx\",\"url\":\"https://cfp.scipy.org/scipy2025/talk/PBLESZ/\"},{\"title\":\"Nvidia\",\"url\":\"https://docs.nvidia.com/cuda/tile-ir/\"},{\"title\":\"CUDA Toolkit - Free Tools and Training | NVIDIA Developer\",\"url\":\"https://developer.nvidia.com/cuda/toolkit\"}]\n\nBased on the search results, here's a comparison between NVIDIA cuTile and Triton for tile-based GPU programming:\n\n## Overview\n\ncuTile is \"an array-based paradigm where the compiler automates mem movement, pipelining & tensor core utilization, making GPU programming easier & more portable.\"(no content)It was developed by NVIDIA's team and is positioned as a competitor to OpenAI's Triton.\n\nCUDA 13.1 launched CUDA Tile as \"its largest advancement since the NVIDIA CUDA platform was invented in 2006.\" It introduces a virtual instruction set for tile-based parallel programming, focusing on the ability to write algorithms at a higher level and abstract away the details of specialized hardware, such as tensor cores.\n\n## Both Are Part of a Broader Trend\n\nThere has been an \"explosion of interest in block-based Python programming models targeting GPUs, driven by the machine learning community.\" Multiple frameworks have been developed, including Triton, JAX/Pallas, and Warp.(no content)In March 2025, NVIDIA announced cuTile and Tile IR as their entry into this space.\n\n## Key Features of cuTile\n\n- CUDA Tile is \"a tile-based GPU programming model that targets portability for NVIDIA Tensor Cores\" and \"unlocks peak GPU performance with a programming model that simplifies the creation of optimized, tile-based kernels across NVIDIA platforms.\"\n\n- CUDA 13.1 introduces CUDA Tile, providing \"a tile-based programming model and Virtual ISA (CUDA Tile IR), along with cuTile Python DSL, abstracting tensor core usage and enabling forward compatibility for NVIDIA Blackwell (compute capability 10.x and 12.x) GPUs.\"\n\n- cuTile is \"a new block-based programming model for NVIDIA's CUDA platform\" and is \"implemented with Numba and a novel compiler stack and intermediate representation called Tile IR.\"\n\n## Why This Programming Model Matters\n\nThe trend towards block-based models is driven by several factors: \"More and more scientists are programming GPUs, including those who are not experts in concurrency and hardware performance.\" Block-based code is \"simpler to design, write, and debug for data-parallel GPU applications,\" compilers can \"reason about block-based programs without more complex and brittle analysis,\" and these frameworks \"offer better portability even as GPU architectures change.\"(no content)Tile-based programming represents \"one of the most significant changes to GPU programming since CUDA was first released\" and \"allows developers to write GPU programs at a much higher level, without worrying about low-level thread management.\"\n\n## Coexistence with Traditional CUDA\n\nTile programming on GPUs coexists with traditional SIMT programming—\"you don't have to choose between SIMT and tile programming.\" When you need SIMT, you write kernels as before; when you want to operate using tensor cores, you write tile kernels.\n\n## Hardware Support\n\nCUDA Tile is \"supported on NVIDIA Blackwell (compute capability 10.x and 12.x) products only\" currently, with future versions of CUDA planned to add support for more architectures.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"thinking": "So cuTile is NVIDIA's answer to Triton. The key differences seem to be:- cuTile is NVIDIA-only, Blackwell+ only (for now)- Both are tile-based Python DSLs\n- Triton is open source, works on NVIDIA and AMD (with effort)\n- cuTile promises tighter hardware integration (tensor memory accelerator)\n\nThe user's point stands - if you're targeting cuTile, you're locked to recent NVIDIA hardware. And the space is getting crowded: Triton, cuTile, JAX/Pallas, Warp...\n\nLet me summarize this for the user.",
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"text": "Yeah, exactly. cuTile is NVIDIA's answer to Triton.\n\n## The Landscape\n\n| Framework | Who | Hardware | Maturity |\n|-----------|-----|----------|----------|\n| **Triton** | OpenAI | NVIDIA (+ AMD with effort) | Battle-tested, used in torch.compile |\n| **cuTile** | NVIDIA | Blackwell+ only (for now) | Brand new, CUDA 13.1 |\n| **JAX/Pallas** | Google | TPU, GPU | Mature for TPU |\n| **Warp** | NVIDIA | NVIDIA GPUs | Simulation-focused |\n\n## Why NVIDIA Made Their Own\n\nFrom the [NVIDIA blog](https://developer.nvidia.com/blog/focus-on-your-algorithm-nvidia-cuda-tile-handles-the-hardware):\n\n- Tighter integration with their hardware (tensor memory accelerator, etc.)\n- \"Forward compatibility\" — write once, run on future NVIDIA architectures\n- Control the stack end-to-end\n- First-class tensor core support without workarounds\n\n## The Cynical Read\n\nTriton got popular → NVIDIA wants to own this layer → cuTile\n\nThey're pitching it as \"largest advancement since CUDA\" but it's really playing catch-up to what Triton already does, with the advantage of deeper hardware integration (since they *are* the hardware).\n\n## For Your Project\n\nThis makes tensor logic → cuTile even more niche:\n- cuTile is already competing with Triton\n- Triton has mindshare and torch.compile integration\n- Adding another layer on top (tensor logic) competes with both\n\nThe stack would be:\n```\nTensor Logic DSL\n ↓\n cuTile\n ↓\n Tile IR\n ↓\n Blackwell GPU\n```\n\nvs what most people do:\n```\nPyTorch + torch.compile\n ↓\n Triton\n ↓\n Any NVIDIA GPU\n```\n\nHard to see tensor logic winning users away from that."
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"thinking": "The user seems to be questioning or wanting more detail on what \"deeper hardware integration\" actually means for cuTile vs Triton. Let me explain what NVIDIA can do with cuTile that Triton can't easily do because NVIDIA controls the hardware.",
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"text": "Fair to be skeptical of that claim. Let me unpack what it actually means.\n\n## What NVIDIA Has That Triton Doesn't\n\n**1. Tensor Memory Accelerator (TMA)**\n\nBlackwell has dedicated hardware for async bulk memory copies. cuTile can use it directly:\n\n```python\n# cuTile: TMA handles this automatically\ntile = ct.load(A, (i, j), shape=(64, 64)) # Hardware-accelerated bulk load\n```\n\nTriton has to orchestrate this with software — it works, but more overhead.\n\n**2. Undocumented hardware details**\n\nNVIDIA knows the *actual* tensor core tile shapes, cache behavior, warp scheduler quirks. Triton reverse-engineers this from PTX docs and experimentation.\n\n**3. Tile IR as a stable target**\n\ncuTile compiles to Tile IR, which NVIDIA promises will be forward-compatible. They control both sides — if Blackwell++ changes something, they update the Tile IR backend transparently.\n\nTriton compiles to PTX, which is documented but not designed for tile-level abstraction.\n\n**4. Guaranteed tensor core mapping**\n\ncuTile's `ct.mma()` is *guaranteed* to hit tensor cores if shapes match. Triton usually does, but it's heuristic-based — sometimes you get regular CUDA cores by accident.\n\n## How Much Does This Matter?\n\nHonestly? **Maybe 10-20% perf difference** in the best case.\n\nTriton already gets 90%+ of cuBLAS on matmuls. cuTile claims the same. The gap is in:\n\n- Edge cases (weird shapes, non-standard layouts)\n- Memory-bound kernels where TMA helps\n- Future hardware where Triton needs to catch up\n\n## The Real \"Integration\"\n\nThe deeper integration is **business**, not just technical:\n\n- NVIDIA can deprecate PTX features that only Triton relies on\n- NVIDIA can optimize their driver for Tile IR paths\n- NVIDIA can make cuTile the blessed way to access new hardware features\n\nIt's platform control, not just perf."
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"text": "Ha, good catch. I did say that.\n\nSo the logic would be:\n\n1. CTM is memory-bound (rolling histories, sync matrices)\n2. cuTile has TMA (hardware-accelerated memory ops)\n3. Therefore cuTile might actually help CTM?\n\n## Let Me Think About This More Carefully\n\n**CTM's memory patterns:**\n- Rolling history buffer: `A[b, d, t] → A[b, d, t+1]` (FIFO shift + append)\n- Sync update: `S = r * S + z_i × z_j` (incremental, not full recompute)\n- Attention: standard cross-attention (FlashAttention exists)\n\n**Where TMA helps:**\n- Bulk async loads while compute happens\n- Prefetching next tiles during current computation\n- Avoiding explicit synchronization barriers\n\n**But:**\n- CTM already does incremental updates (not bulk loads)\n- The histories are small (M ≈ 50 steps, not thousands)\n- The sync is pairwise samples, not full D×D\n\n## Revised Take\n\nTMA helps when you're streaming large tiles through the GPU. CTM's pattern is more like:\n\n```\nsmall update → small compute → small update → small compute → ...\n```\n\nNot:\n\n```\nload giant tile → massive compute → store giant tile\n```\n\nSo the memory-bound bottleneck might not be the kind TMA solves. It's more \"lots of small accesses\" than \"need to move huge blocks efficiently.\"\n\n---\n\nI think I talked myself back to \"marginal gains.\" But you're right to push on it — I was being sloppy connecting memory-bound → TMA automatically."
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"text": "Yeah, basically:\n\n## Where Massive Tile Movement Happens\n\n**1. LLM Training**\n\n```\nHidden states: [batch, seq, 4096+]\nWeight matrices: [4096, 16384] (MLP)\nAttention: [batch, heads, seq, seq]\n```\n\nMatmuls moving gigabytes per layer. This is where memory bandwidth is life or death.\n\n**2. Large Batch Inference**\n\nThroughput-optimized serving — batch 256+ sequences, same story.\n\n**3. Long Context Attention**\n\nSequence length 32K+ → attention matrix is huge. FlashAttention exists precisely because naive attention was memory-bound.\n\n**4. Large Convolutions**\n\nImage models, video models — big spatial dimensions × many channels.\n\n## But Here's the Thing\n\nThese workloads already use:\n\n| Workload | What People Use |\n|----------|-----------------|\n| Matmul | cuBLAS (98% peak) |\n| Attention | FlashAttention (fused, memory-efficient) |\n| Conv | cuDNN |\n| Custom fusion | Triton / torch.compile |\n\ncuTile's pitch is basically: \"What if you could write **custom** kernels that are almost as good as these libraries, without being a CUDA wizard?\"\n\n## The Gap cuTile Targets\n\n```\n │ Standard ops (matmul, attention)\n │ → cuBLAS, FlashAttention (already optimal)\n │\n │ Custom ops that look like standard ops\n │ → torch.compile/Triton (pretty good)\n │\n │ Weird custom ops (CTM, novel architectures) ← cuTile's niche\n │ → Currently: slow PyTorch fallbacks or expert CUDA\n │\n ▼\n```\n\nSo it's for the third category — novel research architectures where no optimized kernel exists yet. But that's a small (though influential) user base.\n\n---\n\nThe tensor logic angle would be: make it even easier to express those weird ops, compile to cuTile, get decent perf. But as we've discussed, the market for that is narrow, and torch.compile keeps getting better at handling weird stuff automatically."
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"content": "Web search results for query: \"DeepSeek paper 2026 offloading RAM engram memory VRAM\"\n\nLinks: [{\"title\":\"Deepseek research touts memory breakthrough, decoupling compute power and RAM pools to bypass GPU & HBM constraints — Engram conditional memory module commits static knowledge to system RAM | Tom's Hardware\",\"url\":\"https://www.tomshardware.com/tech-industry/artificial-intelligence/deepseek-touts-memory-breakthrough-engram\"},{\"title\":\"DeepSeek’s conditional memory fixes silent LLM waste: GPU cycles lost to static lookups | VentureBeat\",\"url\":\"https://venturebeat.com/data/deepseeks-conditional-memory-fixes-silent-llm-waste-gpu-cycles-lost-to\"},{\"title\":\"DeepSeek Thinks VRAM Isn’t the Real LLM Bottleneck Anymore | Hardware Corner\",\"url\":\"https://www.hardware-corner.net/deepseek-ngram-system-ram-vs-vram-20260142153/\"},{\"title\":\"DeepSeek-V4 Leaked: The 1-Trillion Parameter “Engram” Monster That Changes Everything | by Dinmay kumar Brahma | Jan, 2026 | Medium\",\"url\":\"https://dinmaybrahma.medium.com/deepseek-v4-leaked-the-1-trillion-parameter-engram-monster-that-changes-everything-2495061d82a2\"},{\"title\":\"GitHub - deepseek-ai/Engram: Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models\",\"url\":\"https://github.com/deepseek-ai/Engram\"},{\"title\":\"DeepSeek Engram Explained: How Conditional Memory and O(1) Lookups Fix LLM Amnesia | by Pan Xinghan | Jan, 2026 | Medium\",\"url\":\"https://medium.com/@sampan090611/deepseek-engram-explained-how-conditional-memory-and-o-1-lookups-fix-llm-amnesia-50fce3937bb8\"},{\"title\":\"DeepSeek V4: New Engram AI Beats GPT-4o in Coding | Mid-Feb 2026 Release\",\"url\":\"https://vertu.com/lifestyle/deepseek-v4-guide-engram-architecture-release-date-and-coding-benchmarks/\"},{\"title\":\"DeepSeek Once Again Revolutionizes AI Compute\",\"url\":\"https://enertuition.substack.com/p/deepseek-once-again-revolutionizes\"},{\"title\":\"Deepseek research touts memory breakthrough, decoupling compute power and RAM pools to bypass GPU & HBM constraints — Engram conditional memory module commits static knowledge to system RAM\",\"url\":\"https://startupnews.fyi/2026/01/14/deepseek-research-touts-memory-breakthrough-decoupling-compute-power-and-ram-pools-to-bypass-gpu-hbm-constraints-engram-conditional-memory-module-commits-static-knowledge-to-system-ram/\"},{\"title\":\"DeepSeek's Engram: A Memory Breakthrough That Could Redefine AI Efficiency and Ease GPU Bottlenecks - BigGo News\",\"url\":\"https://biggo.com/news/202601132021_DeepSeek-Engram-AI-Memory-Breakthrough\"}]\n\nHere's what I found about the DeepSeek paper on Engram and memory offloading:\n\n## DeepSeek's Engram Paper (January 2026)\n\nDeepSeek-AI released a research paper titled \"Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models\" that proposes a significant architectural innovation for AI models.\n\n### Key Concept\n\nThe proposed solution is a module called Engram, which fundamentally changes the architecture by separating the \"reasoning\" capability from the \"memory\" capability. The core reasoning engine remains a standard neural network that requires high-speed VRAM, while static knowledge (facts, dates, and common phrases) is moved into a massive, separate N-gram lookup table.\n\n### How It Works\n\nEngram acts as a massive, read-only external memory that sits between Transformer layers. Instead of predicting the next token solely via neural weights, the model hashes the input text (N-grams) and performs an O(1) lookup in a massive table stored in System RAM (DRAM), not VRAM.\n\n### RAM Offloading Results\n\nThe researchers demonstrated this with a 100B-parameter embedding table entirely offloaded to host DRAM, achieving throughput penalties below 3%. This decoupling of storage from compute addresses a critical enterprise constraint as GPU high-bandwidth memory remains expensive and scarce.(no content)We could see a future where we prioritize NVMe speeds and RAM capacity over raw VRAM quantity for specific model types. If the Engram table can be 95 percent offloaded to RAM or even fast NVMe storage without crippling the token generation speed, the barrier to entry for running \"SOTA-class\" models drops significantly.\n\n### Optimal Memory/Compute Balance\n\nThrough systematic experiments DeepSeek found the optimal balance between computation and memory with 75% of sparse model capacity allocated to dynamic reasoning and 25% to static lookups.\n\n### Performance Improvements\n\nIn knowledge-intensive tasks, the Engram model outperformed its MoE counterpart by 3.4 to 4 percentage points. It also excelled in reasoning tasks, showing a 3.7 to 5-point uplift. The most dramatic improvement was in long-context processing—on the \"Needle in a Haystack\" (NIAH) benchmark, Engram achieved 97% accuracy compared to the MoE model's 84.2%.\n\n### Technical Innovations\n\nThis is enabled through tokenizer compression, which compresses equivalent tokens (such as the same word with different forms of capitalization) as the same, canonical concept. This allowed Deepseek to reduce the vocabulary size for the conditional memory module by 23%.(no content)They employ a methodology named Hashing, which allows the model to apply a number to a series of words. Engram adds to this with Multi-Head Hashing, where you can put several hashes onto multiple numbers to avoid erroneously adding the wrong context.\n\n### Future Implications\n\nDeepSeek has a history of implementing their research into production models quickly. Speculation is already mounting that this architecture could be the foundation for their next major release, potentially DeepSeek V4.\n\nThe paper and code are available on GitHub at deepseek-ai/Engram.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "# Engram Technical Architecture Summary\n\nBased on the GitHub repository content, here's what's documented about Engram's technical approach:\n\n## Core Concept\n\nEngram modernizes \"classic $N$-gram embeddings for $\\mathcal{O}(1)$ lookup,\" providing conditional memory as a sparsity mechanism complementary to Mixture-of-Experts (MoE) in transformer models.\n\n## Integration Approach\n\nThe module \"augments the backbone by retrieving static $N$-gram memory and fusing it with dynamic hidden states.\" This allows transformers to incorporate knowledge lookups alongside their standard computation.\n\n## Key Technical Features\n\n**Deterministic Addressing**: The system uses \"deterministic addressing, enabling the offloading of massive embedding tables to host memory with minimal inference overhead.\" This design choice allows separation of memory storage from GPU computation.\n\n**System Design**: The architecture demonstrates how to balance \"neural computation (MoE) and static memory (Engram),\" with findings showing a U-shaped scaling law that indicates optimal capacity allocation between these two approaches.\n\n## Mechanistic Insight\n\nAnalysis suggests Engram \"relieves early layers from static pattern reconstruction, potentially preserving effective depth for complex reasoning.\" This indicates the module handles routine pattern matching, freeing transformer depth for complex tasks.\n\n## Limitations of Available Information\n\nThe repository provides architectural diagrams and performance results but lacks detailed specifications on the exact hashing mechanism, memory access patterns, or complete implementation of the lookup algorithm—the provided demo code \"mocks standard components\" rather than implementing full details."
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"text": "This is clever. Different approach to memory than what we were discussing.\n\n## Engram: The Core Idea\n\nSeparate **static knowledge** from **dynamic reasoning**:\n\n```\nTraditional LLM:\n Everything in VRAM → expensive, limited\n\nEngram:\n Static knowledge (facts, n-grams) → System RAM (cheap, huge)\n Dynamic reasoning (attention, MLP) → VRAM (fast, expensive)\n```\n\n## How It Works\n\n```\nInput: \"The capital of France is ___\"\n\n1. Hash the n-gram context → O(1) lookup in RAM\n2. Retrieve static embedding from giant table (100B+ params in DRAM)\n3. Fuse with transformer hidden states\n4. Continue normal forward pass\n```\n\nThe n-gram table is **read-only** and **deterministically addressed** — no attention, no learned routing, just hash and fetch.\n\n## Why This Matters\n\n| Resource | Engram's Use |\n|----------|-------------|\n| VRAM (expensive) | Only reasoning layers |\n| System RAM (cheap) | Static knowledge table |\n| Throughput penalty | <3% for 100B table offload |\n\nThey found optimal split is **75% compute / 25% memory lookup**.\n\n## Results\n\n- Knowledge tasks: +3.4-4% over MoE\n- Reasoning tasks: +3.7-5% over MoE \n- Needle-in-haystack: 97% vs 84% (huge win for long context)\n\n## Connection to Our Discussion\n\nThis is a different kind of memory optimization than TMA/cuTile:\n\n| Approach | What It Optimizes |\n|----------|------------------|\n| cuTile/TMA | GPU tile movement (VRAM ↔ compute) |\n| Engram | Offload static data to system RAM entirely |\n\nEngram says: \"Stop putting everything in VRAM. Some stuff doesn't need GPU bandwidth — just hash and fetch from DDR.\"\n\n---\n\nInteresting that they're attacking the memory problem from the **architecture** side rather than the **kernel** side. Instead of making VRAM access faster, just use less VRAM.\n\n**Sources:**\n- [DeepSeek Engram GitHub](https://github.com/deepseek-ai/Engram)\n- [Tom's Hardware coverage](https://www.tomshardware.com/tech-industry/artificial-intelligence/deepseek-touts-memory-breakthrough-engram)\n- [VentureBeat analysis](https://venturebeat.com/data/deepseeks-conditional-memory-fixes-silent-llm-waste-gpu-cycles-lost-to)"
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"content": "this what happens when we limit their hardware and they're still cracked. \n\nbut you're saying for regualr llm inference we store n-gram data in vram?"
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"text": "No — regular LLMs don't have n-gram tables at all. That's the insight.\n\n## Traditional LLMs\n\nEverything is **implicit** in the weights:\n\n```\nInput: \"The capital of France is\"\n ↓\n [Attention + MLP layers] ← All in VRAM\n ↓\n Probability distribution over vocab\n ↓\n Output: \"Paris\"\n```\n\nThe model \"knows\" Paris because the weights encode that pattern — learned during training, stored implicitly across billions of parameters.\n\n**No lookup table.** The network re-derives \"capital of France → Paris\" through matrix multiplies every single time.\n\n## DeepSeek's Observation\n\nThat's wasteful. Some knowledge is:\n- Static (facts don't change)\n- Frequent (common phrases, patterns)\n- Doesn't need \"reasoning\" to recall\n\nWhy burn GPU cycles recomputing \"The capital of France is Paris\" when you could just... look it up?\n\n## Engram's Fix\n\n```\nInput: \"The capital of France is\"\n ↓\n Hash(\"capital of France\") → O(1) lookup in RAM table\n ↓\n Retrieved: embedding for \"Paris\"-type completions\n ↓\n Fuse with transformer hidden states\n ↓\n Output: \"Paris\"\n```\n\nThe n-gram table is **new** — they're adding explicit memory that traditional LLMs don't have. And since it's just a hash table (no compute), it can live in cheap system RAM.\n\n## The Punchline\n\nTraditional LLMs use VRAM for everything because everything is neural computation.\n\nEngram says: \"Some of that 'computation' is really just memorized lookup. Make it actual lookup, move it to RAM, free up VRAM for real reasoning.\"\n\n---\n\nIt's kind of obvious in retrospect — why would you use matrix multiplies to recall \"Paris is in France\" when a hash table works?"
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"content": "sure but how does it work in practice? is there some sort of ngram probability that triggers actually doing real inference? when you \"fuse with hidden states\" that just means what? update the context w/ the word from the ngrams?"
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Conditional Memory via Scalable Lookup… | by Mehul Gupta | Data Science in Your Pocket | Jan, 2026 | Medium\",\"url\":\"https://medium.com/data-science-in-your-pocket/what-is-deepseek-engram-d9b48f4abe37\"}]\n\n# DeepSeek Engram: Fusion Mechanism & Gating Hidden States\n\nBased on the search results, here's how DeepSeek's Engram fusion mechanism and context-aware gating works:\n\n## Overview\n\nThe Engram module augments the backbone by retrieving static N-gram memory and fusing it with dynamic hidden states.(no content)Engram is a conditional memory module aiming to enhance the Transformer backbone network by structurally separating the storage of static patterns from dynamic computation.\n\n## The Context-Aware Gating Mechanism\n\nEngram employs a context-aware gating mechanism inspired by Attention. Specifically, they utilize the current hidden state h_t—which has aggregated global context via preceding attention layers—as a dynamic Query, while the retrieved memory e_t serves as the source for both Key and Value projections.(no content)The retrieved memory is projected to key/value representations and then modulated by a scalar gate α_t computed from the current hidden state and the projected memory key (RMSNorm-based scaled dot product passed through σ).\n\n### How the Gate Works\n\nThe gated output is defined as ṽ_t = α_t · v_t. This design enforces semantic alignment: if the retrieved memory e_t contradicts the current context h_t, the gate α_t tends toward zero, effectively suppressing the memory.(no content)The current hidden state of the transformer acts as a query that decides whether the retrieved memory should be trusted. If the memory aligns with the current context, the gate opens and the memory contributes strongly. If it conflicts, the gate suppresses the memory almost completely.\n\n### Why Gating is Critical\n\nThis gating is a central modeling safeguard because it provides a mechanism to suppress irrelevant or collided lookups when local N-gram patterns do not align with broader context.(no content)Without gating, you'd either always use the memory (bad for novel contexts) or never use it (defeating the purpose). The gating mechanism lets the model learn when N-gram patterns are trustworthy.\n\n## Post-Gating Processing\n\nA lightweight depthwise causal convolution (with SiLU and a residual formulation) is applied to refine the fused signal, with convolution weights initialized to 0 to preserve identity at initialization and reduce early optimization instability.(no content)Using a kernel size w (set to 4), dilation δ (set to the max N-gram order) and SiLU activation, the final output Y is computed through this causal convolution.\n\n## Integration with Backbone\n\nEngram does not replace attention or feed-forward networks. It augments them. The memory output is lightly processed using a small causal convolution and then added back into the hidden state via a residual connection. After this, the transformer proceeds normally with attention and MoE.\n\n## Optimal Allocation\n\nDeepSeek's key finding: the optimal split is 75-80% for computation and 20-25% for memory.(no content)Engram-27B uses 2+55 experts (top-6) vs MoE-27B's 2+72 experts. By replacing 17 experts with Engram memory, they got better performance with fewer computational experts.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"text": "Okay so here's how the fusion actually works:\n\n## The Mechanism\n\n```\nFor each token position t:\n\n1. Hash recent context → retrieve embedding e_t from RAM table\n\n2. Project to key/value:\n k_t = W_k · e_t\n v_t = W_v · e_t\n\n3. Compute gate (does memory match context?):\n α_t = σ(h_t · k_t / √d) # scalar 0-1\n \n4. Gated output:\n ṽ_t = α_t · v_t # if mismatch, α→0, memory ignored\n\n5. Light convolution + residual:\n output = h_t + Conv(ṽ_t) # add back to hidden state\n```\n\nSo it's **not** \"either lookup or inference.\" It's:\n- Always do the lookup\n- Always do inference \n- Gate decides how much the lookup contributes\n\n---\n\nBut you're right to push on the VRAM question. Let me address that directly:\n\n## Where's the VRAM Savings?\n\nThe n-gram table in RAM doesn't directly save VRAM. What saves VRAM is:\n\n**Fewer MoE experts needed.**\n\n| Model | Experts | Why |\n|-------|---------|-----|\n| MoE-27B | 2 + 72 experts | All knowledge in neural weights |\n| Engram-27B | 2 + 55 experts | Static patterns offloaded to RAM table |\n\nThey replaced **17 experts** with the Engram lookup. Fewer experts = fewer parameters in VRAM.\n\nThe n-gram table is huge (100B+ params) but lives in **system RAM**, which is:\n- 10x cheaper per GB than HBM\n- Available in much larger quantities (256GB+ easy)\n\n## So It's Really\n\n```\nTraditional: 72 experts in VRAM (expensive)\nEngram: 55 experts in VRAM + giant hash table in RAM (cheaper overall)\n```\n\nThe \"memory breakthrough\" isn't free — you need lots of system RAM. But RAM is cheap, HBM is scarce.\n\n**Sources:**\n- [Engram GitHub](https://github.com/deepseek-ai/Engram)\n- [rewire.it technical breakdown](https://rewire.it/blog/engram-how-deepseek-added-second-brain-to-llm/)"
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"thinking": "The user is pushing back on my explanation, and they're right to be skeptical. Let me think about this more carefully.\n\nMy claim was: fewer MoE experts = less VRAM. But that's not really \"running the same model with less VRAM\" — that's \"using a different, smaller model.\"\n\nWhat the user is getting at: if you have a fixed N billion parameter model, Engram doesn't magically let you fit it in less VRAM. The weights still need to live somewhere.\n\nThe actual claim DeepSeek is making seems to be:\n1. For equivalent PERFORMANCE, you need fewer neural parameters\n2. The n-gram table compensates for the reduced neural capacity\n3. So you can hit the same benchmark scores with less VRAM (because fewer experts)\n\nBut that's different from \"run GPT-4 with less VRAM.\" It's \"build a different architecture that matches GPT-4 performance with less VRAM.\"\n\nThe user is right — this is really about compute/parameter efficiency, not about magically compressing an existing model.\n\nLet me clarify this.",
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"text": "You're right, I was being sloppy.\n\n## What Engram Actually Does\n\nIt's **not**: \"Take existing 70B model, run it with less VRAM\"\n\nIt **is**: \"Build a different architecture that hits same performance with fewer neural parameters\"\n\n```\nTraditional 70B model:\n 70B params in VRAM → X performance\n\nEngram equivalent:\n 50B params in VRAM + 100B n-gram table in RAM → X performance (or better)\n```\n\nYou're not compressing anything. You're trading:\n- Neural parameters (expensive, in VRAM)\n- For lookup table (cheap, in RAM)\n\n## You're Right About Compute\n\nThe real win might be compute, not memory:\n\n| Operation | Cost |\n|-----------|------|\n| MoE expert forward pass | Matrix multiply, expensive |\n| N-gram hash lookup | O(1), basically free |\n\nIf 20% of your \"knowledge\" can be served by hash lookup instead of expert computation, you save 20% FLOPs on those patterns.\n\n## The VRAM Claim Is Indirect\n\nDeepSeek's pitch: \"You can build a model that matches [bigger model] performance without needing [bigger model] VRAM.\"\n\nBut that's architectural efficiency, not compression. You'd have to train a new model with Engram from scratch — you can't bolt it onto an existing LLM and use less VRAM.\n\n---\n\nSo yeah — \"memory breakthrough\" is marketing. The real story is: \"we found a way to offload memorization to a hash table, which lets us allocate more of our VRAM budget to reasoning.\"\n\nDoes that framing make more sense?"
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"content": "hash table memorization seems easier to expand / train continously? what exactly is it [what are the keys and values]? a hash map of embeddings to something?"
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"thinking": "The user is asking about the specifics of the hash table in Engram:\n1. What are the keys?\n2. What are the values?\n3. Is it easier to expand/train continuously?\n\nLet me think about what I know:\n- Keys are n-grams (sequences of tokens) hashed\n- Values are embeddings (vectors)\n\nSo it's like:\n- Key: hash(\"capital of France\") → some integer\n- Value: embedding vector at that index\n\nFor continuous learning, you could:\n- Add new n-gram entries without retraining the whole model\n- Update specific embeddings for new facts\n- No catastrophic forgetting because it's just a lookup table\n\nThis is actually a good point - it's more modular than having everything baked into weights.\n\nLet me search for more details on the actual structure.",
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"content": "# N-gram Hash Table Analysis\n\nI cannot extract the requested details about the n-gram hash table from the provided PDF content. The document appears to be a research paper titled \"Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models,\" but the PDF metadata and structure you've provided consists primarily of technical PDF formatting information rather than readable text content.\n\nThe content includes:\n- PDF object references and encoding declarations\n- Font definitions and page structure metadata\n- Link annotations and cross-references\n- Compressed content streams\n\nTo properly extract information about:\n- N-gram hash table keys and values\n- Training methodology\n- Multi-head hashing mechanisms\n- Continuous learning approaches\n\nI would need the actual decoded text content of the paper sections discussing these technical components. The compressed streams in the PDF would need to be decompressed and decoded to access the human-readable text.\n\n**Recommendation**: Consider converting this PDF to plain text or providing the specific sections of the paper that discuss the n-gram hash table implementation details."
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"content": "Web search results for query: \"DeepSeek Engram n-gram hash table keys values embeddings multi-head hashing training\"\n\nLinks: [{\"title\":\"GitHub - deepseek-ai/Engram: Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models\",\"url\":\"https://github.com/deepseek-ai/Engram\"},{\"title\":\"N-gram Embeddings and Scalable Lookup | deepseek-ai/Engram | DeepWiki\",\"url\":\"https://deepwiki.com/deepseek-ai/Engram/2.2-n-gram-embeddings-and-scalable-lookup\"},{\"title\":\"Deepseek research touts memory breakthrough, decoupling compute power and RAM pools to bypass GPU & HBM constraints — Engram conditional memory module commits static knowledge to system RAM | Tom's Hardware\",\"url\":\"https://www.tomshardware.com/tech-industry/artificial-intelligence/deepseek-touts-memory-breakthrough-engram\"},{\"title\":\"DeepSeek Engram Explained: How Conditional Memory and O(1) Lookups Fix LLM Amnesia | by Pan Xinghan | Jan, 2026 | Medium\",\"url\":\"https://medium.com/@sampan090611/deepseek-engram-explained-how-conditional-memory-and-o-1-lookups-fix-llm-amnesia-50fce3937bb8\"},{\"title\":\"DeepSeek Engram: The Future of Memory-Augmented Language Models\",\"url\":\"https://www.analyticsvidhya.com/blog/2026/01/deepseek-engram/\"},{\"title\":\"Conditional Memory via Scalable Lookup:\",\"url\":\"https://www.arxiv.org/pdf/2601.07372\"},{\"title\":\"Engram: How DeepSeek Added a Second Brain to Their LLM | rewire.it | rewire.it Blog\",\"url\":\"https://rewire.it/blog/engram-how-deepseek-added-second-brain-to-llm/\"},{\"title\":\"DeepSeek AI Researchers Introduce Engram: A Conditional Memory Axis For Sparse LLMs - MarkTechPost\",\"url\":\"https://www.marktechpost.com/2026/01/14/deepseek-ai-researchers-introduce-engram-a-conditional-memory-axis-for-sparse-llms/\"},{\"title\":\"What is DeepSeek Engram ?. Conditional Memory via Scalable Lookup… | by Mehul Gupta | Data Science in Your Pocket | Jan, 2026 | Medium\",\"url\":\"https://medium.com/data-science-in-your-pocket/what-is-deepseek-engram-d9b48f4abe37\"},{\"title\":\"DeepSeek-V4 Leaked: The 1-Trillion Parameter “Engram” Monster That Changes Everything | by Dinmay kumar Brahma | Jan, 2026 | Medium\",\"url\":\"https://dinmaybrahma.medium.com/deepseek-v4-leaked-the-1-trillion-parameter-engram-monster-that-changes-everything-2495061d82a2\"}]\n\nBased on the search results, here's what I found about DeepSeek's **Engram** architecture and its use of N-gram hash tables, embeddings, and multi-head hashing:\n\n## Overview\n\nEngram introduces a complementary axis of sparsity called Conditional Memory. It acts as a module that modernizes classic N-gram embeddings to perform constant-time O(1) lookups alongside the neural backbone.\n\n## Core Architecture\n\n### N-gram Embeddings and Hash Tables\nThe module augments the backbone by retrieving static N-gram memory and fusing it with dynamic hidden states via context-aware gating.(no content)At each token position, Engram first looks at the local suffix context, usually 2-grams and 3-grams. These short windows are ideal for capturing entities, phrases, and syntax patterns.(no content)The normalized N-grams are then passed through multiple hash functions. Each hash deterministically maps the N-gram to a location in a large embedding table. Multiple hash heads are used to reduce collision noise. The retrieved embeddings are concatenated to form a memory vector.\n\n### Multi-Head Hashing\nEngram uses K independent hash functions to map each N-gram to K different embedding vectors, then concatenates them. This reduces the impact of hash collisions since it's unlikely that two N-grams will collide across all K hash functions simultaneously.(no content)To prevent collisions (i.e., different patterns mapping to the same location), Engram employs multiple hash functions. For example, think of it as having several different phone books – if one gives you the wrong number, the others will have your back.\n\n### Unified Embedding Table with Offset-Based Indexing\nAll N-gram types and heads share one nn.Embedding layer with total_vocab = Σ(all prime moduli). Offset-based partitioning means each head's indices are offset by the sum of all preceding vocabulary sizes.\n\n### Keys and Values\nSpecifically, a parameter-sharing strategy is implemented: a single sparse embedding table and a Value projection matrix W_V are shared across all M branches, whereas M distinct Key projection matrices are used.\n\n## Training Details\n\nDuring the Training Phase, the massive embedding tables are sharded across available GPUs. An All-to-All communication primitive is employed to retrieve active embedding rows across devices.(no content)The approach uses the DeepSeek V3 tokenizer with a 128k vocabulary and pre-trains on 262B tokens. The backbone is a 30-block Transformer with hidden size 2560. Each block uses Multi-head Latent Attention with 32 heads.(no content)The Engram module uses N equal to {2,3}, 8 Engram heads, dimension 1280 and is inserted at layers 2 and 15.\n\n### Tokenizer Compression\nText normalization converts tokens to canonical form (NFKC → NFD → strip accents → lowercase → whitespace collapse). The DeepSeek-V3 vocabulary (129,280 tokens) compresses to approximately 30-40% of the original size, significantly reducing the N-gram hash space.\n\n## Context-Aware Gating\nThis is the \"conditional\" in Conditional Memory. Retrieved embeddings are not blindly added to the residual stream. They are gated by the current hidden state (αt). If the retrieved memory conflicts with the global context, the gate suppresses the noise.\n\n## System Efficiency (Inference)\nDeepSeek demonstrated offloading a 100B-parameter embedding table entirely to host memory (CPU DRAM). Because indices are known ahead of time, the system asynchronously prefetches embeddings via PCIe. Result: The throughput penalty is negligible (< 3%).\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Based on the search results, here's what I found about DeepSeek's **Engram** architecture and its use of N-gram hash tables, embeddings, and multi-head hashing:\n\n## Overview\n\nEngram introduces a complementary axis of sparsity called Conditional Memory. It acts as a module that modernizes classic N-gram embeddings to perform constant-time O(1) lookups alongside the neural backbone.\n\n## Core Architecture\n\n### N-gram Embeddings and Hash Tables\nThe module augments the backbone by retrieving static N-gram memory and fusing it with dynamic hidden states via context-aware gating.(no content)At each token position, Engram first looks at the local suffix context, usually 2-grams and 3-grams. These short windows are ideal for capturing entities, phrases, and syntax patterns.(no content)The normalized N-grams are then passed through multiple hash functions. Each hash deterministically maps the N-gram to a location in a large embedding table. Multiple hash heads are used to reduce collision noise. The retrieved embeddings are concatenated to form a memory vector.\n\n### Multi-Head Hashing\nEngram uses K independent hash functions to map each N-gram to K different embedding vectors, then concatenates them. This reduces the impact of hash collisions since it's unlikely that two N-grams will collide across all K hash functions simultaneously.(no content)To prevent collisions (i.e., different patterns mapping to the same location), Engram employs multiple hash functions. For example, think of it as having several different phone books – if one gives you the wrong number, the others will have your back.\n\n### Unified Embedding Table with Offset-Based Indexing\nAll N-gram types and heads share one nn.Embedding layer with total_vocab = Σ(all prime moduli). Offset-based partitioning means each head's indices are offset by the sum of all preceding vocabulary sizes.\n\n### Keys and Values\nSpecifically, a parameter-sharing strategy is implemented: a single sparse embedding table and a Value projection matrix W_V are shared across all M branches, whereas M distinct Key projection matrices are used.\n\n## Training Details\n\nDuring the Training Phase, the massive embedding tables are sharded across available GPUs. An All-to-All communication primitive is employed to retrieve active embedding rows across devices.(no content)The approach uses the DeepSeek V3 tokenizer with a 128k vocabulary and pre-trains on 262B tokens. The backbone is a 30-block Transformer with hidden size 2560. Each block uses Multi-head Latent Attention with 32 heads.(no content)The Engram module uses N equal to {2,3}, 8 Engram heads, dimension 1280 and is inserted at layers 2 and 15.\n\n### Tokenizer Compression\nText normalization converts tokens to canonical form (NFKC → NFD → strip accents → lowercase → whitespace collapse). The DeepSeek-V3 vocabulary (129,280 tokens) compresses to approximately 30-40% of the original size, significantly reducing the N-gram hash space.\n\n## Context-Aware Gating\nThis is the \"conditional\" in Conditional Memory. Retrieved embeddings are not blindly added to the residual stream. They are gated by the current hidden state (αt). If the retrieved memory conflicts with the global context, the gate suppresses the noise.\n\n## System Efficiency (Inference)\nDeepSeek demonstrated offloading a 100B-parameter embedding table entirely to host memory (CPU DRAM). Because indices are known ahead of time, the system asynchronously prefetches embeddings via PCIe. Result: The throughput penalty is negligible (< 3%)."
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"thinking": "Now I have good details. Let me summarize the hash table structure and address the continuous learning question.",
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"text": "Okay, now I have the details:\n\n## The Hash Table Structure\n\n**Keys:** N-grams (2-grams and 3-grams) of tokens, hashed\n\n```\nInput: \"capital of France\"\n ↓\nNormalized: lowercase, strip accents, etc.\n ↓\nExtract n-grams: [\"capital of\", \"of france\", \"capital of france\"]\n ↓\nHash each n-gram → integer index into embedding table\n```\n\n**Values:** Learned embedding vectors\n\n```\nhash(\"capital of\") → index 847291 → embedding[847291] ∈ ℝ^1280\n```\n\n**Multi-head hashing:** K different hash functions per n-gram\n\n```\nn-gram: \"capital of\"\n hash_1(\"capital of\") → embedding_1\n hash_2(\"capital of\") → embedding_2\n ...\n hash_K(\"capital of\") → embedding_K\n \n concat([embedding_1, ..., embedding_K]) → final memory vector\n```\n\nThis reduces collision damage — unlikely two different n-grams collide on ALL K hashes.\n\n## You're Right About Continuous Learning\n\nThis is way more modular than neural weights:\n\n| Operation | Neural Weights | Engram Table |\n|-----------|---------------|--------------|\n| Add new fact | Retrain or fine-tune (catastrophic forgetting risk) | Insert new row |\n| Update fact | Same problem | Update specific embedding |\n| Expand knowledge | Grow model size | Grow table (cheap) |\n| Localize knowledge | Impossible (distributed) | Hash tells you exactly where |\n\nYou could literally:\n```python\n# \"Paris is no longer capital of France\" (hypothetically)\nidx = hash(\"capital of france\")\ntable[idx] = new_embedding # Done. No retraining.\n```\n\n## The Training\n\nDuring training, embeddings are learned end-to-end via backprop. The hash indices are fixed (deterministic), but the embedding vectors at those indices are learned.\n\nAt inference, table is frozen and can live in RAM.\n\n---\n\nThis is actually a return to **explicit memory** that neural nets threw away. Old-school AI had knowledge bases you could edit. Neural nets made everything implicit and uneditable. Engram brings back editability for the static knowledge portion.\n\n**Sources:**\n- [DeepWiki: N-gram Embeddings and Scalable Lookup](https://deepwiki.com/deepseek-ai/Engram/2.2-n-gram-embeddings-and-scalable-lookup)\n- [Engram GitHub](https://github.com/deepseek-ai/Engram)\n- [MarkTechPost analysis](https://www.marktechpost.com/2026/01/14/deepseek-ai-researchers-introduce-engram-a-conditional-memory-axis-for-sparse-llms/)"
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"content": "did they publish the hashtable?"
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"content": "Web search results for query: \"DeepSeek Engram weights download pretrained embedding table release huggingface\"\n\nLinks: [{\"title\":\"deepseek-ai/DeepSeek-V3.2 · Hugging Face\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3.2\"},{\"title\":\"deepseek-ai/DeepSeek-V3 · Hugging Face\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3\"},{\"title\":\"DeepSeek-V3\",\"url\":\"https://huggingface.co/docs/transformers/model_doc/deepseek_v3\"},{\"title\":\"GitHub - deepseek-ai/Engram: Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models\",\"url\":\"https://github.com/deepseek-ai/Engram\"},{\"title\":\"deepseek-ai/DeepSeek-V3.2-Exp · Hugging Face\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3.2-Exp\"},{\"title\":\"deepseek-ai/DeepSeek-V3.2-Speciale · Hugging Face\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3.2-Speciale\"},{\"title\":\"How to download deep-seek weights for v3? - Beginners - Hugging Face Forums\",\"url\":\"https://discuss.huggingface.co/t/how-to-download-deep-seek-weights-for-v3/161861\"},{\"title\":\"README_WEIGHTS.md · deepseek-ai/DeepSeek-V3 at main\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3/blob/main/README_WEIGHTS.md\"},{\"title\":\"DeepSeek-V2\",\"url\":\"https://huggingface.co/docs/transformers/main/model_doc/deepseek_v2\"},{\"title\":\"deepseek-ai/DeepSeek-V3.1 · Hugging Face\",\"url\":\"https://huggingface.co/deepseek-ai/DeepSeek-V3.1\"}]\n\nBased on my search, here's what I found regarding DeepSeek Engram:\n\n## DeepSeek Engram\n\nDeepSeek Engram is described as \"Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models.\"\n\n### Key Features\n- The module employs deterministic addressing, enabling the offloading of massive embedding tables to host memory with minimal inference overhead.\n- The Engram module augments the backbone by retrieving static N-gram memory and fusing it with dynamic hidden states.\n\n### Code Availability\nThe GitHub repository is available at: https://github.com/deepseek-ai/Engram\n\nThe provided code is a demonstration version intended to illustrate the data flow. It mocks standard components (like Attention/MoE/mHC) to focus on the Engram module.(no content)The use of Engram models is subject to the Model License.\n\n### Pretrained Weights Status\n**I did not find any search results indicating that DeepSeek has released pretrained Engram embedding table weights on Hugging Face.** The search results show various DeepSeek-V3 model weights are available on Hugging Face, but no specific Engram embedding tables appear to be publicly released at this time.\n\nIf you're looking for the pretrained embedding tables specifically, you may want to:\n1. Check the GitHub repository's issues or discussions\n2. Contact DeepSeek directly at service@deepseek.com if you have questions\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Based on my search, here's what I found regarding DeepSeek Engram:\n\n## DeepSeek Engram\n\nDeepSeek Engram is described as \"Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models.\"\n\n### Key Features\n- The module employs deterministic addressing, enabling the offloading of massive embedding tables to host memory with minimal inference overhead.\n- The Engram module augments the backbone by retrieving static N-gram memory and fusing it with dynamic hidden states.\n\n### Code Availability\nThe GitHub repository is available at: https://github.com/deepseek-ai/Engram\n\nThe provided code is a demonstration version intended to illustrate the data flow. It mocks standard components (like Attention/MoE/mHC) to focus on the Engram module.(no content)The use of Engram models is subject to the Model License.\n\n### Pretrained Weights Status\n**I did not find any search results indicating that DeepSeek has released pretrained Engram embedding table weights on Hugging Face.** The search results show various DeepSeek-V3 model weights are available on Hugging Face, but no specific Engram embedding tables appear to be publicly released at this time.\n\nIf you're looking for the pretrained embedding tables specifically, you may want to:\n1. Check the GitHub repository's issues or discussions\n2. Contact DeepSeek directly at service@deepseek.com if you have questions"
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"text": "Doesn't look like it. From the [GitHub repo](https://github.com/deepseek-ai/Engram):\n\n> \"The provided code is a demonstration version intended to illustrate the data flow. It **mocks standard components** (like Attention/MoE/mHC) to focus on the Engram module.\"\n\nSo they released:\n- Paper\n- Architecture code (demo only)\n- The idea\n\nThey did **not** release:\n- Trained embedding table\n- Full model weights with Engram integrated\n- Production-ready implementation\n\n## Classic DeepSeek Move\n\n| What They Share | What They Keep |\n|-----------------|----------------|\n| Architecture papers | Trained weights |\n| Demo code | Production code |\n| Training recipes | Actual trained models (sometimes) |\n\nThe hash table is the valuable part — it's the \"knowledge\" learned from their training data. Releasing it would be giving away the moat.\n\n## What This Means\n\nIf you wanted to use Engram, you'd have to:\n1. Train your own embedding table from scratch\n2. On your own data\n3. With significant compute\n\nThe architecture is open, the knowledge isn't.\n\n---\n\nSpeculation: if DeepSeek-V4 uses Engram (as rumored), they might release weights then. But a 100B+ parameter embedding table is a big download regardless — that's ~200GB+ at fp16."
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"content": "I don't really get how this shit works tbh, \"the bank is ____\" depends on the previous context [$ bank vs sand bank etc.] how do you drop that down to a hashtable of ngrams"
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"thinking": "This is a really good question. The user is pointing out that n-grams are local context, but disambiguation often requires long-range context. \"The bank is\" could be:\n- \"The bank is lending money\" (financial)\n- \"The bank is eroding\" (river)\n\nA 2-gram or 3-gram hash table can't see the earlier context that would disambiguate. So how does Engram handle this?\n\nThe answer is in the gating mechanism - the n-gram lookup happens, but the GATE (which uses the full hidden state h_t from attention) decides whether to trust it. The hidden state has seen the full context via attention layers.\n\nSo:\n1. N-gram lookup retrieves an embedding for \"bank is\" - might be ambiguous\n2. The hidden state h_t (which has attended to earlier tokens mentioning \"river\" or \"money\") acts as query\n3. Gate α = σ(h_t · k_t) - if the retrieved memory doesn't match the global context, gate → 0\n\nBut the user's point still stands - if \"bank\" is ambiguous, what embedding does the hash table even store? Is it some average of all meanings? That seems useless.\n\nLet me think about this more carefully...\n\nActually, the key insight might be:\n1. The n-gram table doesn't store disambiguated meanings\n2. It stores something like \"common continuations given this local pattern\"\n3. The gate decides whether that's relevant given global context\n4. If ambiguous, gate suppresses, and neural network handles it\n\nSo for \"the bank is\", the table might store a blend of financial/river associations. If context clearly indicates one meaning (via h_t), the gate either accepts or rejects the memory contribution.\n\nBut this seems... weak? If the memory is ambiguous, what's the point of retrieving it?\n\nMaybe the value is in UNAMBIGUOUS cases:\n- \"capital of France is\" → clearly \"Paris\" territory\n- \"2 + 2 =\" → clearly \"4\" territory\n\n\n- \"the bank is\" → ambiguous, so the gate suppresses the memory and the neural network takes over\n\nThis would explain the 75% compute / 25% memory split - Engram accelerates straightforward cases where local patterns are reliable, but the neural component still carries the load for genuinely ambiguous or context-dependent predictions.",
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"text": "You're hitting on the real limitation.\n\n## The Problem You're Identifying\n\n```\nContext A: \"I deposited $500. The bank is ___\" → \"closed\" / \"lending\"\nContext B: \"The river flooded. The bank is ___\" → \"eroding\" / \"muddy\"\n\nN-gram \"the bank is\" → ???\n```\n\nA hash table keyed on \"the bank is\" can't distinguish these. The n-gram is identical, but the answer depends on tokens from way earlier.\n\n## How Engram \"Handles\" This\n\nIt doesn't, really. The gating mechanism is the escape hatch:\n\n```\n1. Hash(\"the bank is\") → retrieve embedding e_t\n (some ambiguous blend of financial/river meanings)\n\n2. Hidden state h_t has already attended to \"deposited $500\" or \"river flooded\"\n (h_t knows which meaning is active)\n\n3. Gate: α_t = σ(h_t · k_t)\n - If e_t aligns with h_t → gate opens, memory contributes\n - If e_t is ambiguous/wrong → gate closes, memory suppressed\n\n4. When gate closes → neural network handles it the normal way\n```\n\n## So What's the Point?\n\nEngram helps with **unambiguous** cases:\n\n| N-gram | Ambiguous? | Engram helps? |\n|--------|------------|---------------|\n| \"capital of France is\" | No | Yes — just lookup \"Paris\" |\n| \"2 + 2 =\" | No | Yes — just lookup \"4\" |\n| \"the bank is\" | Yes | No — gate suppresses, neural net does it |\n| \"once upon a\" | No | Yes — \"time\" is predictable |\n\nThe claim is: a lot of language is predictable/formulaic. Engram handles that, freeing the neural network for genuinely ambiguous cases.\n\n## The Honest Answer\n\nYou're right to be skeptical. The n-gram table is fundamentally limited:\n\n- Can't handle long-range dependencies\n- Can't disambiguate\n- Can only help when local context is sufficient\n\nIt's not magic. It's just \"cache common patterns, fall back to neural network for hard stuff.\"\n\nThe 75/25 compute/memory split suggests: **75% of predictions still need real neural inference.** Engram just skims off the easy 25%.\n\n---\n\nDoes that framing make more sense? It's less \"revolutionary new architecture\" and more \"clever caching layer.\""
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"content": "the thing that gets hashed is different from the hidden state how exactly? in your example it's abunch of tokens? si it really 1 token IRL and hidden state has the rest?"
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"text": "Let me be precise:\n\n## What Gets Hashed\n\n**Raw token IDs** from a small window, not embeddings:\n\n```\nSequence: [\"The\", \"river\", \"flooded\", \".\", \"The\", \"bank\", \"is\"]\n ↑─────────↑\n 2-gram: (bank, is)\n 3-gram: (The, bank, is)\n\nhash((token_id[bank], token_id[is])) → index into table\n```\n\nIt's literally just the last 2-3 token IDs concatenated and hashed. No semantics, no context — just \"what tokens appeared recently.\"\n\n## What the Hidden State Has\n\nThe hidden state `h_t` at position t has been through attention layers, so it's seen **everything before it**:\n\n```\nh_t = Attention([\"The\", \"river\", \"flooded\", \".\", \"The\", \"bank\", \"is\"])\n ↑\n Knows about \"river\", \"flooded\" — has global context\n```\n\n## The Split\n\n| Component | Input | Sees |\n|-----------|-------|------|\n| Hash lookup | Last 2-3 token IDs | Local only |\n| Hidden state | Full sequence through attention | Global context |\n| Gate | Both | Decides if local lookup matches global context |\n\n## So Your Intuition Is Right\n\nThe hash is dumb. It's just:\n```python\ndef get_ngram_embedding(tokens, n=3):\n recent = tokens[-n:] # Last 3 tokens\n key = hash(tuple(recent)) # Just their IDs\n return embedding_table[key]\n```\n\nAll the \"intelligence\" about whether to use that embedding comes from the hidden state + gate.\n\n## Why This Works At All\n\nBecause a lot of language is locally predictable:\n\n```\n\"New York ___\" → \"City\" / \"Times\" / etc. (don't need global context)\n\"kind of ___\" → common continuations (formulaic phrase)\n\"= 4\" after any math → obvious (local pattern sufficient)\n```\n\nFor these, the hash works. For \"the bank is\" type ambiguity, the gate learns to suppress and let the neural network handle it.\n\n---\n\nIt's surprisingly dumb at the lookup level. The sophistication is in knowing when to trust the dumb lookup."
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"content": "and how exactly does MOE come into play"
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"thinking": "The user is asking how MoE (Mixture of Experts) fits into the Engram architecture. From what I read earlier:1. Engram is complementary to MoE - they're both sparsity mechanisms\n2. Engram replaces some MoE experts with the hash table3. Engram-27B uses 57 experts vs MoE-27B's 74 experts\n\nLet me explain how they work together.",
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"text": "Good question. They're both **sparsity** mechanisms that work together:\n\n## MoE Recap\n\nInstead of one big FFN, you have N experts (small FFNs), and a router picks top-k per token:\n\n```\nToken → Router → picks 2 of 64 experts → only those 2 compute\n ↓\n \"This token is about math\" → Expert 12, Expert 47\n \"This token is about code\" → Expert 3, Expert 31\n```\n\nSparsity = you have 64 experts but only run 2. Capacity without full compute cost.\n\n## How Engram Fits\n\nEngram is **another axis of sparsity** alongside MoE:\n\n```\nStandard MoE block:\n Token → Attention → Router → Top-k Experts → Output\n\nEngram + MoE block:\n Token → Attention → [Engram lookup + Gate] → Router → Top-k Experts → Output\n ↑\n If pattern is in table and gate opens,\n some \"work\" is already done before experts\n```\n\n## The Trade-off They Found\n\nYou can substitute some expert capacity for Engram capacity:\n\n```\nMoE-27B: 2 + 72 experts (top-6) + no Engram\nEngram-27B: 2 + 55 experts (top-6) + Engram table\n\nSame parameter budget, different allocation.\n```\n\n17 fewer experts, but the hash table compensates.\n\n## Why This Works\n\n| What Handles It | Example |\n|-----------------|---------|\n| Engram (static lookup) | \"New York ___\" → common completions |\n| MoE experts (dynamic compute) | Reasoning, novel combinations |\n\nEngram handles the memorization, experts handle the thinking. Before Engram, experts had to do both.\n\n## The Architecture\n\n```\n┌─────────────────────────────────────────────┐\n│ Transformer Block │\n├─────────────────────────────────────────────┤\n│ Input │\n│ ↓ │\n│ Attention │\n│ ↓ │\n│ ┌─────────┐ │\n│ │ Engram │ ← Hash lookup from RAM │\n│ │ + Gate │ ← Fuse with hidden state │\n│ └────┬────┘ │\n│ ↓ │\n│ ┌─────────┐ │\n│ │ Router │ → Pick top-k experts │\n│ └────┬────┘ │\n│ ↓ │\n│ ┌─────────┐ │\n│ │ Experts │ ← Only k of N run │\n│ └────┬────┘ │\n│ ↓ │\n│ Output │\n└─────────────────────────────────────────────┘\n```\n\nEngram runs **before** the MoE router, so by the time experts see the hidden state, static patterns have already been mixed in (if the gate allowed it).\n\n## The U-Shaped Scaling Law\n\nDeepSeek found there's an optimal balance:\n\n```\nToo much Engram: Can't reason, just regurgitates\nToo much MoE: Wastes compute on memorization\nSweet spot: ~25% Engram, ~75% MoE\n```\n\n---\n\nSo MoE and Engram are complementary sparsity mechanisms:\n- MoE = sparse **compute** (only some experts run)\n- Engram = sparse **memory** (only some lookups hit)\n\nBoth save resources by not doing everything for every token."
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"content": "kindof cool because it maybe helps get at just reasoning / non memorizing LLMs, just drop the hash table on an engram model [karpathy wanted this]\n\nback to the neurosymbolic thing, could you maybe do a categorical deep learning thing that could do math as some of the experts 2402.15332? is this something you could describe in tensor logic?"
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"thinking": "The user is asking about:\n1. Engram potentially enabling \"pure reasoning\" LLMs by separating memorization (interesting point about Karpathy)\n2. A paper on categorical deep learning (2402.15332) - they want to know if you could have some experts do symbolic math\n3. Whether this could be described in tensor logicLet me search for that paper (arXiv 2402.15332) on categorical deep learning.",
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"content": "Web search results for query: \"arxiv 2402.15332 categorical deep learning\"\n\nLinks: [{\"title\":\"[2402.15332] Position: Categorical Deep Learning is an Algebraic Theory of All Architectures\",\"url\":\"https://arxiv.org/abs/2402.15332\"},{\"title\":\"Categorical Deep Learning is an Algebraic Theory of All ...\",\"url\":\"https://arxiv.org/pdf/2402.15332\"},{\"title\":\"Categorical Deep Learning: An Algebraic Theory\",\"url\":\"https://www.emergentmind.com/papers/2402.15332\"},{\"title\":\"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures | Cool Papers - Immersive Paper Discovery\",\"url\":\"https://papers.cool/arxiv/2402.15332\"},{\"title\":\"Categorical Deep Learning: An Algebraic Theory of Architectures – ScienceOpen\",\"url\":\"https://www.scienceopen.com/document?vid=cc03bece-b782-4c14-8f66-da291ae27d56\"},{\"title\":\"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures | Bytez\",\"url\":\"https://bytez.com/docs/arxiv/2402.15332/paper\"},{\"title\":\"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures | alphaXiv\",\"url\":\"https://www.alphaxiv.org/abs/2402.15332\"},{\"title\":\"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures | AI Research Paper Details\",\"url\":\"https://www.aimodels.fyi/papers/arxiv/position-categorical-deep-learning-is-algebraic-theory\"},{\"title\":\"Categorical Deep Learning: An Algebraic Theory of Architectures\",\"url\":\"https://paperreading.club/page?id=210940\"},{\"title\":\"Statistics Feb 2024\",\"url\":\"https://arxiv.org/list/stat/2024-02?skip=1090&show=50\"}]\n\nHere are the search results for arxiv paper 2402.15332 on Categorical Deep Learning:\n\n## Paper Overview\n\nThe paper \"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures\" presents a position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures.\n\n### Authors\nThe paper is authored by Bruno Gavranović, Paul Lessard, Andrew Dudzik, Tamara von Glehn, João G.M. Araújo, and Petar Veličković.\n\n### Key Ideas\n\nThe authors' opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations.(no content)To address this, they propose to apply category theory — precisely, the universal algebra of monads valued in a 2-category of parametric maps — as a single theory elegantly subsuming both of these flavours of neural network design.(no content)To defend their position, they show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. They also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.\n\n### Publication Details\n\nThe paper appears in ICML 2024.(no content)The paper is categorized under cs.LG, cs.AI, math.CT, math.RA, and stat.ML, covering disciplines including Machine learning, General mathematics, Artificial intelligence, and Algebra.\n\n### Links\n- arXiv page: https://arxiv.org/abs/2402.15332\n- PDF: https://arxiv.org/pdf/2402.15332\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "# Key Technical Ideas from the Paper\n\nBased on the arXiv abstract provided, I can outline the main concepts, though the full technical details aren't available in this page content:\n\n## Core Framework\n\nThe paper proposes applying **category theory—specifically universal algebra of monads valued in 2-categories of parametric maps** as a unified theoretical foundation. The authors argue this bridges the gap between specifying architectural constraints and their implementations.\n\n## Parametric Maps & Monads\n\nWhile the abstract doesn't detail parametric maps explicitly, it indicates they form the foundational structure. The use of \"monads valued in a 2-category of parametric maps\" suggests this framework maps neural operations as mathematical objects that can be composed and studied algebraically.\n\n## Connections to Existing Approaches\n\nThe theory reportedly \"recovers constraints induced by geometric deep learning\" and implements \"many architectures drawn from the diverse landscape of neural networks, such as RNNs.\"\n\n## Symbolic Integration\n\nNotably, the abstract states the theory \"naturally encodes many standard constructs in computer science and automata theory,\" suggesting an inherent bridge between neural and symbolic computation through categorical structures.\n\n## Status\n\nThe position paper was \"to appear in ICML 2024\" with additional information at categoricaldeeplearning.com. For technical specifics on parametric maps, monad-architecture relationships, and concrete unification examples, the full PDF would be necessary."
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"content": "Web search results for query: \"categorical deep learning symbolic computation automata expert modules Bruno Gavranović\"\n\nLinks: [{\"title\":\"[2402.15332] Position: Categorical Deep Learning is an Algebraic Theory of All Architectures\",\"url\":\"https://arxiv.org/abs/2402.15332\"},{\"title\":\"Bruno Gavranović\",\"url\":\"https://www.brunogavranovic.com/\"},{\"title\":\"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures\",\"url\":\"https://proceedings.mlr.press/v235/gavranovic24a.html\"},{\"title\":\"Position: categorical deep learning is an algebraic theory ...\",\"url\":\"https://dl.acm.org/doi/10.5555/3692070.3692679\"},{\"title\":\"Bruno Gavranović\",\"url\":\"https://scholar.google.com/citations?user=ofP7CgYAAAAJ&hl=en\"},{\"title\":\"Categorical Deep Learning: An Algebraic Theory of ...\",\"url\":\"https://www.cl.cam.ac.uk/events/syco/12/static/slides/gavranovic.pdf\"},{\"title\":\"bgavran (Bruno Gavranović) · GitHub\",\"url\":\"https://github.com/bgavran\"},{\"title\":\"Categorical Deep Learning: An Algebraic Theory of Architectures – ScienceOpen\",\"url\":\"https://www.scienceopen.com/document?vid=cc03bece-b782-4c14-8f66-da291ae27d56\"},{\"title\":\"dblp: Bruno Gavranovic\",\"url\":\"https://dblp.org/pid/245/2917.html\"},{\"title\":\"BrunoGavranović JANUARY, 2025 London, United Kingdom Ó info@brunogavranovic.com\",\"url\":\"https://www.brunogavranovic.com/assets/Bruno_Gavranovic_CV.pdf\"}]\n\nBased on the search results, here's what I found about Bruno Gavranović's work on categorical deep learning:\n\n## Bruno Gavranović and Categorical Deep Learning\n\nBruno Gavranović is building neural networks that process structured data based on the Categorical Deep Learning programme. He defended his PhD thesis \"Fundamental Components of Deep Learning: A category-theoretic approach\" in the Mathematically Structured Programming Group at the University of Strathclyde in Glasgow.\n\n### Key Paper: \"Categorical Deep Learning is an Algebraic Theory of All Architectures\"\n\nThis paper was authored by Bruno Gavranović, Paul Lessard, Andrew Joseph Dudzik, Tamara Von Glehn, João Guilherme Madeira Araújo, and Petar Veličković, and was published in the Proceedings of the 41st International Conference on Machine Learning (ICML), 2024.\n\nThe paper addresses \"the elusive quest for a general-purpose framework for specifying and studying deep learning architectures.\" The authors argue that key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations.(no content)To build this bridge, they propose applying category theory—precisely, the universal algebra of monads valued in a 2-category of parametric maps—as a single theory elegantly subsuming both flavours of neural network design.(no content)The theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. The theory also naturally encodes many standard constructs in computer science and automata theory.\n\n### Related Work\n\nBruno Gavranović has published extensively in this area, including:\n\n- \"Categorical Foundations of Gradient-Based Learning\" (ESOP 2022)\n- \"Graph Convolutional Neural Networks as Parametric CoKleisli morphisms\"\n- \"Towards Foundations of Categorical Cybernetics\" (ACT 2021)\n- \"Deep Learning with Parametric Lenses\"(no content)He is affiliated with Symbolica, and describes his work as \"applying Category theory, the science of structure, to the field of deep learning, which aims to learn structure from data.\"\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "Web search results for query: \"Symbolica AI neurosymbolic categorical deep learning math reasoning\"\n\nLinks: [{\"title\":\"AI Reasoning in Deep Learning Era: From Symbolic AI to Neural–Symbolic AI\",\"url\":\"https://www.mdpi.com/2227-7390/13/11/1707\"},{\"title\":\"Neuro-symbolic AI - Wikipedia\",\"url\":\"https://en.wikipedia.org/wiki/Neuro-symbolic_AI\"},{\"title\":\"A review of neuro-symbolic AI integrating reasoning and learning for advanced cognitive systems - ScienceDirect\",\"url\":\"https://www.sciencedirect.com/science/article/pii/S2667305325000675\"},{\"title\":\"A Beginner's Guide to Symbolic Reasoning (Symbolic AI) & Deep Learning - Deeplearning4j: Open-source, Distributed Deep Learning for the JVM\",\"url\":\"https://mgubaidullin.github.io/deeplearning4j-docs/symbolicreasoning.html\"},{\"title\":\"Neuro-symbolic AI - IBM Research\",\"url\":\"https://research.ibm.com/topics/neuro-symbolic-ai\"},{\"title\":\"Neuro-Symbolic AI: Integrating Symbolic Reasoning with Deep Learning | IEEE Conference Publication | IEEE Xplore\",\"url\":\"https://ieeexplore.ieee.org/document/10434380/\"},{\"title\":\"Symbolic artificial intelligence - Wikipedia\",\"url\":\"https://en.wikipedia.org/wiki/Symbolic_artificial_intelligence\"},{\"title\":\"Neuro-Symbolic AI: Combining Neural Networks And Symbolic AI\",\"url\":\"https://www.crossml.com/neuro-symbolic-ai-combining-neural-networks/\"},{\"title\":\"[1912.01412] Deep Learning for Symbolic Mathematics\",\"url\":\"https://arxiv.org/abs/1912.01412\"},{\"title\":\"Neuro-symbolic data generation for math reasoning | Proceedings of the 38th International Conference on Neural Information Processing Systems\",\"url\":\"https://dl.acm.org/doi/10.5555/3737916.3738656\"}]\n\nThe search results don't specifically mention \"Symbolica AI\" as a company. Let me search more specifically for the company:\n\nLinks: [{\"title\":\"Symbolica AI\",\"url\":\"https://www.symbolica.ai/\"},{\"title\":\"Symbolica AI | LinkedIn\",\"url\":\"https://www.linkedin.com/company/symbolica-ai\"},{\"title\":\"Move over, deep learning: Symbolica's structured approach could transform AI | VentureBeat\",\"url\":\"https://venturebeat.com/ai/move-over-deep-learning-symbolicas-structured-approach-could-transform-ai\"},{\"title\":\"Symbolica\",\"url\":\"https://www.symbolica.ai/research\"},{\"title\":\"Symbolica Raises $31M to Redesign AI with Structured Reasoning\",\"url\":\"https://www.maginative.com/article/symbolica-raises-31m-to-redesign-ai-with-structured-reasoning/\"},{\"title\":\"Machine Learning Jobs for Category Theorists | The n-Category Café\",\"url\":\"https://golem.ph.utexas.edu/category/2024/04/machine_learning_jobs_for_cate.html\"},{\"title\":\"Symbolica - 2025 Company Profile, Team, Funding & Competitors - Tracxn\",\"url\":\"https://tracxn.com/d/companies/symbolica/__cBPKhbzD9YyVw_MrYWVY_vdSLMS9AAuWPXQYQBvw5vs\"},{\"title\":\"Symbolica AI - Crunchbase Company Profile & Funding\",\"url\":\"https://www.crunchbase.com/organization/symbolica-ai\"},{\"title\":\"Contact Paul Lessard, Email: ****@symbolica.ai & Phone Number | Categorical Deep Learning Principal Scientist at Symbolica - ZoomInfo\",\"url\":\"https://www.zoominfo.com/p/Paul-Lessard/3593164117\"},{\"title\":\"Symbolica AI 2026 Company Profile: Valuation, Funding & Investors | PitchBook\",\"url\":\"https://pitchbook.com/profiles/company/528117-40\"}]\n\nHere's what I found about Symbolica AI and their neurosymbolic categorical deep learning approach:\n\n## Symbolica AI Overview\n\nSymbolica is a pioneering AI research lab founded in 2022. They are a team of software engineers and mathematicians leveraging a form of abstract mathematics, known as category theory, to deliver exceptional breakthroughs in modern artificial intelligence research.(no content)Symbolica is an artificial intelligence research laboratory pioneering the application of category theory and type theory to design architectures that are capable of structured reasoning.\n\n## Funding and Background\n\nFounded by former Tesla senior autopilot engineer George Morgan, the startup has raised $31 million in a Series A funding round led by Khosla Ventures, with participation from Day One Ventures, General Catalyst, Abstract Ventures, and Buckley Ventures.(no content)Early OpenAI investor Vinod Khosla said Symbolica offers a \"very innovative approach\" to one of the biggest challenges facing AI: How to develop smaller, more efficient models that can reason more like humans do.\n\n## Technical Approach: Categorical Deep Learning\n\nAt the heart of Symbolica's approach is \"category theory,\" a branch of mathematics that formalizes mathematical structures and their relationships.(no content)Symbolica recently co-authored a paper with Google DeepMind on \"categorical deep learning,\" which mathematically demonstrated how its approach could supersede previous work on geometric deep learning to build structurally-aware models.(no content)They can now go beyond the constraints induced by Geometric Deep Learning (GDL), by looking at arbitrary endofunctors—giving them 'equivariance constraints' for neural networks processing data living on lists, trees, and many kinds of automata. They can also generalize these concepts to 2-categories, allowing them to describe the weight tying across several layers of many standard models, such as RNNs.\n\n## Research Focus: Math Reasoning & Program Synthesis\n\nAt Symbolica, they believe that the narrow domain most likely to yield generalized reasoning capability is program synthesis and theorem proving. A general purpose program synthesis engine or, alternatively, a sufficiently advanced theorem prover, can be configured to perform many different kinds of combinatorial optimization problems.(no content)Their research program is focused on designing a fully general-purpose program synthesis engine on their proprietary non-LLM categorical architecture. They believe that future progress in artificial intelligence depends on the unification of the reliability of symbolic program execution with the adaptability of neural optimization. The way they build this bridge is by leveraging the language of category theory.\n\n## Key Differentiators\n\nThe lack of rigor in current AI, Morgan argued, means current AI models are essentially black boxes. Symbolica's approach aims to open up that black box, enabling interpretability.(no content)Their approach is based on categorical deep learning, which categorizes and encodes the underlying structure of data, allowing for less computational power and data usage compared to large unstructured models.(no content)Symbolica's pivot to categorical deep learning started about five months before their public launch. Bruno Gavranović had just finished his Ph.D. thesis laying the foundations for the topic and they began refining a roadmap to develop and apply CDL, and writing a position paper in collaboration with researchers at Google DeepMind.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Here's what I found about Symbolica AI and their neurosymbolic categorical deep learning approach:\n\n## Symbolica AI Overview\n\nSymbolica is a pioneering AI research lab founded in 2022. They are a team of software engineers and mathematicians leveraging a form of abstract mathematics, known as category theory, to deliver exceptional breakthroughs in modern artificial intelligence research.(no content)Symbolica is an artificial intelligence research laboratory pioneering the application of category theory and type theory to design architectures that are capable of structured reasoning.\n\n## Funding and Background\n\nFounded by former Tesla senior autopilot engineer George Morgan, the startup has raised $31 million in a Series A funding round led by Khosla Ventures, with participation from Day One Ventures, General Catalyst, Abstract Ventures, and Buckley Ventures.(no content)Early OpenAI investor Vinod Khosla said Symbolica offers a \"very innovative approach\" to one of the biggest challenges facing AI: How to develop smaller, more efficient models that can reason more like humans do.\n\n## Technical Approach: Categorical Deep Learning\n\nAt the heart of Symbolica's approach is \"category theory,\" a branch of mathematics that formalizes mathematical structures and their relationships.(no content)Symbolica recently co-authored a paper with Google DeepMind on \"categorical deep learning,\" which mathematically demonstrated how its approach could supersede previous work on geometric deep learning to build structurally-aware models.(no content)They can now go beyond the constraints induced by Geometric Deep Learning (GDL), by looking at arbitrary endofunctors—giving them 'equivariance constraints' for neural networks processing data living on lists, trees, and many kinds of automata. They can also generalize these concepts to 2-categories, allowing them to describe the weight tying across several layers of many standard models, such as RNNs.\n\n## Research Focus: Math Reasoning & Program Synthesis\n\nAt Symbolica, they believe that the narrow domain most likely to yield generalized reasoning capability is program synthesis and theorem proving. A general purpose program synthesis engine or, alternatively, a sufficiently advanced theorem prover, can be configured to perform many different kinds of combinatorial optimization problems.(no content)Their research program is focused on designing a fully general-purpose program synthesis engine on their proprietary non-LLM categorical architecture. They believe that future progress in artificial intelligence depends on the unification of the reliability of symbolic program execution with the adaptability of neural optimization. The way they build this bridge is by leveraging the language of category theory.\n\n## Key Differentiators\n\nThe lack of rigor in current AI, Morgan argued, means current AI models are essentially black boxes. Symbolica's approach aims to open up that black box, enabling interpretability.(no content)Their approach is based on categorical deep learning, which categorizes and encodes the underlying structure of data, allowing for less computational power and data usage compared to large unstructured models.(no content)Symbolica's pivot to categorical deep learning started about five months before their public launch. Bruno Gavranović had just finished his Ph.D. thesis laying the foundations for the topic and they began refining a roadmap to develop and apply CDL, and writing a position paper in collaboration with researchers at Google DeepMind."
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"text": "Okay, now this is getting interesting. Let me connect the threads:\n\n## The Vision You're Pointing At\n\n```\nEngram: Handles memorization (hash table)\n ↓\n Frees up experts for reasoning\n ↓\nCategorical DL: Some experts could be SYMBOLIC (not neural)\n ↓\n Math expert = actual algebra, not learned approximation\n```\n\n## What Symbolica Is Doing\n\nFrom [VentureBeat](https://venturebeat.com/ai/move-over-deep-learning-symbolicas-structured-approach-could-transform-ai/):\n\n> \"Future progress in AI depends on the unification of the reliability of symbolic program execution with the adaptability of neural optimization.\"\n\nThey're building **non-LLM architectures** where:\n- Structure is encoded via category theory (types, composition rules)\n- Some operations are **provably correct** (not learned)\n- Neural parts handle fuzzy perception, symbolic parts handle exact reasoning\n\n## Could You Have a \"Math Expert\" That Actually Does Math?\n\nIn principle, yes. The categorical framework lets you describe:\n\n```\nNeural expert: f: Tensor → Tensor (learned weights)\nSymbolic expert: g: Expression → Expression (algebraic rewrite rules)\n```\n\nBoth are just **morphisms** in the right category. You could route:\n- \"What's 2+2?\" → Symbolic expert (exact)\n- \"What's the vibe of this sentence?\" → Neural expert (learned)\n\n## Tensor Logic Connection?\n\nHere's where it gets speculative:\n\nTensor logic says: predicates are tensors, inference is contraction.\n\nCategory theory says: operations are morphisms, composition is functorial.\n\nThey're both trying to find **algebraic structure** in computation. The question is whether tensor logic's \"everything is einsum\" is rich enough to capture categorical deep learning's \"everything is a morphism in a 2-category.\"\n\nMy guess: **tensor logic is a special case**. It handles the linear algebra part well, but categorical DL is more general — it can encode trees, automata, type systems, things that aren't naturally tensors.\n\n## The Hybrid You're Imagining\n\n```\n┌─────────────────────────────────────────────────────────┐\n│ Hybrid Architecture │\n├─────────────────────────────────────────────────────────┤\n│ Engram (hash table) → Static knowledge (RAM) │\n│ ↓ │\n│ Router → Pick expert type │\n│ ↓ │\n│ ┌─────────────┬────────────────┬──────────────────┐ │\n│ │ Neural MoE │ Symbolic Math │ Program Synthesis│ │\n│ │ (learned) │ (algebraic) │ (categorical) │ │\n│ └─────────────┴────────────────┴──────────────────┘ │\n│ ↓ │\n│ Output │\n└─────────────────────────────────────────────────────────┘\n```\n\nThe Karpathy point: if you drop the Engram table, you get a \"pure reasoning\" model that can't memorize, only compute.\n\n---\n\nThis is the real neurosymbolic dream — not \"neural nets doing logic badly\" but \"neural and symbolic as peers, routed appropriately.\" Whether tensor logic is the right language for it... I'm skeptical. Category theory seems more expressive for the symbolic parts.\n\n**Sources:**\n- [Categorical Deep Learning paper](https://arxiv.org/abs/2402.15332)\n- [Symbolica research](https://www.symbolica.ai/research)\n- [VentureBeat on Symbolica](https://venturebeat.com/ai/move-over-deep-learning-symbolicas-structured-approach-could-transform-ai/)"
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"content": "Web search results for query: \"categorical deep learning implementation parametric lens optics code Gavranović\"\n\nLinks: [{\"title\":\"Bruno Gavranović\",\"url\":\"https://www.brunogavranovic.com/\"},{\"title\":\"DEEP LEARNING WITH PARAMETRIC LENSES\",\"url\":\"https://arxiv.org/pdf/2404.00408\"},{\"title\":\"Position: categorical deep learning is an algebraic theory ...\",\"url\":\"https://dl.acm.org/doi/10.5555/3692070.3692679\"},{\"title\":\"Papers\",\"url\":\"https://www.brunogavranovic.com/papers.html\"},{\"title\":\"BrunoGavranović JANUARY, 2025 London, United Kingdom Ó info@brunogavranovic.com\",\"url\":\"https://www.brunogavranovic.com/assets/Bruno_Gavranovic_CV.pdf\"},{\"title\":\"Fundamental Components of Deep Learning A category-theoretic approach\",\"url\":\"https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf\"},{\"title\":\"1 Categorical Foundations of Gradient-Based Learning\",\"url\":\"https://www.cl.cam.ac.uk/events/act2021/papers/ACT_2021_paper_11.pdf\"},{\"title\":\"[2404.00408] Deep Learning with Parametric Lenses\",\"url\":\"https://arxiv.org/abs/2404.00408\"},{\"title\":\"bgavran (Bruno Gavranović) · GitHub\",\"url\":\"https://github.com/bgavran\"},{\"title\":\"[2403.13001] Fundamental Components of Deep Learning: A category-theoretic approach\",\"url\":\"https://arxiv.org/abs/2403.13001\"}]\n\nHere are the search results for categorical deep learning implementation with parametric lenses and optics by Bruno Gavranović:\n\n## Bruno Gavranović's Work on Categorical Deep Learning\n\nBruno Gavranović defended his PhD thesis \"Fundamental Components of Deep Learning: A category-theoretic approach\" in the Mathematically Structured Programming Group at the University of Strathclyde in Glasgow.(no content)He is building neural networks that process structured data based on the Categorical Deep Learning programme.\n\n### Key Papers\n\n**Deep Learning with Parametric Lenses**\nThis paper proposes a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, and different architectures.(no content)They demonstrate the practical significance of their framework with an implementation in Python.\n\n**PhD Thesis: Fundamental Components of Deep Learning**\nIn Part I, two main properties of deep learning systems are identified and modeled—parametricity and bidirectionality—by expanding on actegories and Para to study the former, and defining weighted optics to study the latter. Combining them yields parametric weighted optics, a categorical model of artificial neural networks.(no content)The work provides a lens-theoretic axiomatisation of differentiation, covering not just smooth spaces, but discrete settings of boolean circuits as well. It surveys existing and develops new categorical models of neural network architectures, formalises the notion of optimisers, and combines all concepts into a uniform and compositional framework for supervised learning.\n\n### Code and Resources\n\nGavranović is applying category theory to deep learning. His most recent project is TensorType: a framework for type-safe, pure functional and non-cubical tensor processing, written in Idris 2.(no content)He maintains a curated list of papers on category theory and machine learning at: https://github.com/bgavran/Category_Theory_Machine_Learning\n\n### Related Work\n\nHe has also defined a bicategory of Graph Convolutional Neural Networks (GCNN), factoring it using categorical constructions for deep learning: Para and Lens, on the CoKleisli category of the product comonad. This gives a high-level categorical account of the inductive bias of GCNNs.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Here are the search results for categorical deep learning implementation with parametric lenses and optics by Bruno Gavranović:\n\n## Bruno Gavranović's Work on Categorical Deep Learning\n\nBruno Gavranović defended his PhD thesis \"Fundamental Components of Deep Learning: A category-theoretic approach\" in the Mathematically Structured Programming Group at the University of Strathclyde in Glasgow.(no content)He is building neural networks that process structured data based on the Categorical Deep Learning programme.\n\n### Key Papers\n\n**Deep Learning with Parametric Lenses**\nThis paper proposes a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, and different architectures.(no content)They demonstrate the practical significance of their framework with an implementation in Python.\n\n**PhD Thesis: Fundamental Components of Deep Learning**\nIn Part I, two main properties of deep learning systems are identified and modeled—parametricity and bidirectionality—by expanding on actegories and Para to study the former, and defining weighted optics to study the latter. Combining them yields parametric weighted optics, a categorical model of artificial neural networks.(no content)The work provides a lens-theoretic axiomatisation of differentiation, covering not just smooth spaces, but discrete settings of boolean circuits as well. It surveys existing and develops new categorical models of neural network architectures, formalises the notion of optimisers, and combines all concepts into a uniform and compositional framework for supervised learning.\n\n### Code and Resources\n\nGavranović is applying category theory to deep learning. His most recent project is TensorType: a framework for type-safe, pure functional and non-cubical tensor processing, written in Idris 2.(no content)He maintains a curated list of papers on category theory and machine learning at: https://github.com/bgavran/Category_Theory_Machine_Learning\n\n### Related Work\n\nHe has also defined a bicategory of Graph Convolutional Neural Networks (GCNN), factoring it using categorical constructions for deep learning: Para and Lens, on the CoKleisli category of the product comonad. This gives a high-level categorical account of the inductive bias of GCNNs."
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"content": "Web search results for query: \"Symbolica AI architecture implementation category theory neural network\"\n\nLinks: [{\"title\":\"Symbolica AI\",\"url\":\"https://www.symbolica.ai/\"},{\"title\":\"Unifying Neural Network Design with Category Theory: A Comprehensive Framework for Deep Learning Architecture - MarkTechPost\",\"url\":\"https://www.marktechpost.com/2024/04/06/unifying-neural-network-design-with-category-theory-a-comprehensive-framework-for-deep-learning-architecture/\"},{\"title\":\"Symbolica\",\"url\":\"https://www.symbolica.ai/research\"},{\"title\":\"Neuro-symbolic AI - Wikipedia\",\"url\":\"https://en.wikipedia.org/wiki/Neuro-symbolic_AI\"},{\"title\":\"Categorical Deep Learning is an Algebraic Theory of All ...\",\"url\":\"https://arxiv.org/pdf/2402.15332\"},{\"title\":\"Early OpenAI investor bets on alternative Sam Altman’s approach to AI\",\"url\":\"https://www.semafor.com/article/04/12/2024/early-openai-investor-now-bets-on-sam-altmans-firm-being-wrong\"},{\"title\":\"Symbolic artificial intelligence - Wikipedia\",\"url\":\"https://en.wikipedia.org/wiki/Symbolic_artificial_intelligence\"},{\"title\":\"Symbolica AI | LinkedIn\",\"url\":\"https://www.linkedin.com/company/symbolica-ai\"},{\"title\":\"Unifying Neural Network Design with Category Theory: A Comprehensive Framework for Deep Learning Architecture – Every Intel\",\"url\":\"https://everyintel.ai/unifying-neural-network-design-with-category-theory-a-comprehensive-framework-for-deep-learning-architecture/\"},{\"title\":\"Move over, deep learning: Symbolica's structured approach could transform AI | VentureBeat\",\"url\":\"https://venturebeat.com/ai/move-over-deep-learning-symbolicas-structured-approach-could-transform-ai\"}]\n\nBased on my search, here's what I found about Symbolica AI's architecture implementation using category theory for neural networks:\n\n## Symbolica AI Overview\n\nSymbolica is an artificial intelligence research lab that is pioneering the application of category theory and type theory to build a symbolic reasoning engine.(no content)Founded in 2022, they are a team of software engineers and mathematicians leveraging a form of abstract mathematics, known as category theory, to deliver exceptional breakthroughs in modern artificial intelligence research.\n\n## Core Architecture Approach\n\nProgram execution, search, and synthesis all occur in the same architecture, fundamentally blurring the line between neural, symbolic, and reasoning. By leveraging category theory to inform the construction of their architectures, they are able to seamlessly unify discrete concepts like types and programs with fuzzy concepts like vector spaces and optimization with a degree of sophistication inaccessible to other architectures.(no content)Symbolica's research program is focused on designing a fully general-purpose program synthesis engine, from the ground up, on their proprietary non-LLM categorical architecture.\n\n## Key Technical Details\n\nThey integrate the ability to run arbitrary, recursive computation directly into the model architecture. The ability to execute programs is a fundamental building block of symbolic systems. By leveraging categorical semantics, they ensure that learned representations are always informed by symbolic execution.\n\n## Research Paper: \"Categorical Deep Learning\"\n\nResearchers from Symbolica AI, University of Edinburgh, Google DeepMind, and University of Cambridge collaborated on a significant paper. They introduce a theoretical framework that unites the specification of constraints with their implementations through monads valued in a 2-category of parametric maps, aiming to create a more integrated and coherent methodology for neural network design.(no content)This approach can go beyond the constraints induced by Geometric Deep Learning (GDL), by looking at arbitrary endofunctors—giving 'equivariance constraints' for neural networks processing data living on lists, trees, and many kinds of automata. The concepts can also be generalized to 2-categories, allowing description of weight tying across several layers of many standard models, such as RNNs.\n\n## Benefits and Goals\n\nUsing category theory, Symbolica hopes its method will lead to AI with guardrails and rules baked in from the beginning. In contrast, foundation models based on transformer architecture require those factors to be added on later.(no content)Symbolica's approach also promises AI systems that can perform complex reasoning tasks with far less training data and computing power than today's data-hungry models. Their CEO stated: \"If we build an architecture that's natively capable of reasoning, it takes much less data to get that model to perform as well as completely unstructured models.\"\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Based on my search, here's what I found about Symbolica AI's architecture implementation using category theory for neural networks:\n\n## Symbolica AI Overview\n\nSymbolica is an artificial intelligence research lab that is pioneering the application of category theory and type theory to build a symbolic reasoning engine.(no content)Founded in 2022, they are a team of software engineers and mathematicians leveraging a form of abstract mathematics, known as category theory, to deliver exceptional breakthroughs in modern artificial intelligence research.\n\n## Core Architecture Approach\n\nProgram execution, search, and synthesis all occur in the same architecture, fundamentally blurring the line between neural, symbolic, and reasoning. By leveraging category theory to inform the construction of their architectures, they are able to seamlessly unify discrete concepts like types and programs with fuzzy concepts like vector spaces and optimization with a degree of sophistication inaccessible to other architectures.(no content)Symbolica's research program is focused on designing a fully general-purpose program synthesis engine, from the ground up, on their proprietary non-LLM categorical architecture.\n\n## Key Technical Details\n\nThey integrate the ability to run arbitrary, recursive computation directly into the model architecture. The ability to execute programs is a fundamental building block of symbolic systems. By leveraging categorical semantics, they ensure that learned representations are always informed by symbolic execution.\n\n## Research Paper: \"Categorical Deep Learning\"\n\nResearchers from Symbolica AI, University of Edinburgh, Google DeepMind, and University of Cambridge collaborated on a significant paper. They introduce a theoretical framework that unites the specification of constraints with their implementations through monads valued in a 2-category of parametric maps, aiming to create a more integrated and coherent methodology for neural network design.(no content)This approach can go beyond the constraints induced by Geometric Deep Learning (GDL), by looking at arbitrary endofunctors—giving 'equivariance constraints' for neural networks processing data living on lists, trees, and many kinds of automata. The concepts can also be generalized to 2-categories, allowing description of weight tying across several layers of many standard models, such as RNNs.\n\n## Benefits and Goals\n\nUsing category theory, Symbolica hopes its method will lead to AI with guardrails and rules baked in from the beginning. In contrast, foundation models based on transformer architecture require those factors to be added on later.(no content)Symbolica's approach also promises AI systems that can perform complex reasoning tasks with far less training data and computing power than today's data-hungry models. Their CEO stated: \"If we build an architecture that's natively capable of reasoning, it takes much less data to get that model to perform as well as completely unstructured models.\""
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"content": "Web search results for query: \"\"parametric lens\" deep learning Python implementation code github\"\n\nLinks: [{\"title\":\"DEEP LEARNING WITH PARAMETRIC LENSES\",\"url\":\"https://arxiv.org/pdf/2404.00408\"},{\"title\":\"[2404.00408] Deep Learning with Parametric Lenses\",\"url\":\"https://arxiv.org/abs/2404.00408\"}]\n\nLet me search more specifically for a GitHub repository:\n\nLinks: [{\"title\":\"DEEP LEARNING WITH PARAMETRIC LENSES\",\"url\":\"https://arxiv.org/pdf/2404.00408\"},{\"title\":\"GitHub - HarrisonKramer/LensAI: Integrated Machine and Deep Learning for Optical Design\",\"url\":\"https://github.com/HarrisonKramer/LensAI\"},{\"title\":\"GitHub - darwaishx/Deep-Learning-With-Deep-Lens\",\"url\":\"https://github.com/darwaishx/Deep-Learning-With-Deep-Lens\"},{\"title\":\"[2404.00408] Deep Learning with Parametric Lenses\",\"url\":\"https://arxiv.org/abs/2404.00408\"},{\"title\":\"GitHub - singer-yang/awesome-deep-optics: A paper list for deep optics design.\",\"url\":\"https://github.com/singer-yang/awesome-deep-optics\"},{\"title\":\"GitHub - twoolf/DeepLens: using Deep Lens for project on individuals self-monitoring of a skin condition\",\"url\":\"https://github.com/twoolf/DeepLens\"},{\"title\":\"GitHub - vccimaging/DiffOptics: This is the open source repository for our IEEE Transactions on Computational Imaging 2022 paper \\\"dO: A differentiable engine for Deep Lens design of computational imaging systems\\\".\",\"url\":\"https://github.com/vccimaging/DiffOptics\"},{\"title\":\"GitHub - McWilliamsCenter/CMUDeepLens: Deep learning strong lens finder\",\"url\":\"https://github.com/McWilliamsCenter/CMUDeepLens\"},{\"title\":\"GitHub - dvlab-research/Parametric-Contrastive-Learning: Parametric Contrastive Learning (ICCV2021) & GPaCo (TPAMI 2023)\",\"url\":\"https://github.com/dvlab-research/Parametric-Contrastive-Learning\"},{\"title\":\"lens · GitHub Topics\",\"url\":\"https://github.com/topics/lens\"}]\n\nLinks: [{\"title\":\"[2404.00408] Deep Learning with Parametric Lenses\",\"url\":\"https://arxiv.org/abs/2404.00408\"},{\"title\":\"Torch Lens Maker\",\"url\":\"https://victorpoughon.github.io/torchlensmaker/\"},{\"title\":\"GitHub - quartiq/rayopt: Python optics and lens design, raytracing\",\"url\":\"https://github.com/quartiq/rayopt\"},{\"title\":\"8. Computer practical. — LightPipes for Python 2.1.5 documentation\",\"url\":\"https://opticspy.github.io/lightpipes/computerprac.html\"},{\"title\":\"Introduction to pyOpTools - pyOpTools 0.3.8.dev4+gf14428e1b documentation\",\"url\":\"https://pyoptools.readthedocs.io/en/latest/notebooks/basic/00-Intro.html\"},{\"title\":\"GitHub - ingolemo/python-lenses: A python lens library for manipulating deeply nested immutable structures\",\"url\":\"https://github.com/ingolemo/python-lenses\"},{\"title\":\"pyMOE: Mask design and modeling for micro optical elements and flat optics - ScienceDirect\",\"url\":\"https://www.sciencedirect.com/science/article/pii/S0010465524002546\"},{\"title\":\"GitHub - HarrisonKramer/optiland: Comprehensive optical design, optimization, and analysis in Python, including GPU-accelerated and differentiable ray tracing via PyTorch.\",\"url\":\"https://github.com/HarrisonKramer/optiland\"},{\"title\":\"Optics — Lenses 1.2.0 documentation\",\"url\":\"https://python-lenses.readthedocs.io/en/latest/tutorial/optics.html\"},{\"title\":\"Python: module optics.lens\",\"url\":\"https://www2.ph.ed.ac.uk/~wjh/teaching/Python/optics.lens.html\"}]\n\nBased on my search results, here's what I found regarding \"parametric lens\" deep learning implementations:\n\n## Main Research Paper\n\nThe paper \"Deep Learning with Parametric Lenses\" by Geoffrey S. H. Cruttwell and 4 other authors proposes a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy.(no content)The authors demonstrate the practical significance of their framework with an implementation in Python.\n\nThe paper is available at: https://arxiv.org/abs/2404.00408 (published March 30, 2024)\n\n## Related GitHub Repositories Found\n\nWhile I couldn't find the specific GitHub repository for the parametric lenses paper, I found several related projects:\n\n1. **python-lenses** (https://github.com/ingolemo/python-lenses) - A Python lens library for manipulating deeply nested immutable structures. Comprehensions only work with lists, dictionaries, and sets, whereas the lenses library can work with arbitrary python objects.\n\n2. **DiffOptics** (https://github.com/vccimaging/DiffOptics) - Implemented in PyTorch, a memory- and computation-efficient differentiable ray tracing system for optical designs, for design applications in freeform, Deep Lens, metrology, and more.\n\n3. **Optiland** (https://github.com/HarrisonKramer/optiland) - Comprehensive optical design, optimization, and analysis in Python, including GPU-accelerated and differentiable ray tracing via PyTorch. Compatible with PyTorch pipelines for deep learning, differentiable modeling, and end-to-end training.\n\n4. **Torch Lens Maker** (https://victorpoughon.github.io/torchlensmaker/) - An open-source Python library for differentiable geometric optics based on PyTorch. Currently a very experimental project, the goal is to be able to design complex real-world optical systems (lenses, mirrors, etc.)\n\nIf you're looking specifically for the categorical/mathematical \"parametric lens\" implementation from the Cruttwell paper, the code may be included as supplementary material in the arXiv submission, or you may need to contact the authors directly.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"title": "GitHub - singer-yang/awesome-deep-optics: A paper list for deep optics design.",
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"content": "# Bruno Gavranović's Relevant Repositories\n\nBased on the GitHub profile content, here are the repositories related to category theory, deep learning, lenses, optics, or neural networks:\n\n1. **TensorType** - \"Framework for type-safe pure functional and non-cubical tensor processing, written in Idris 2\"\n\n2. **Category_Theory_Machine_Learning** - \"List of papers studying machine learning through the lens of category theory\"\n\n3. **Category_Theory_Resources** - \"List of resources for learning Category Theory\"\n\n4. **autodiff** - \"Rudimentary automatic differentiation framework\"\n\n5. **DNC** - \"Implementation of the Differentiable Neural Computer in Tensorflow\"\n\n6. **Compositional_Deep_Learning** - \"Deep learning via category theory and functional programming\"\n\nThese repositories reflect Gavranović's stated focus on building neural networks that generate provably correct code using category theory as a foundational mathematical framework."
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"content": "# Compositional Deep Learning: Repository Analysis\n\n## Project Overview\n\nThis Haskell-based research project explores neural networks through **category theory and functional programming**. The creator notes: \"understanding and reimplementing neural networks from first principles using the language of category theory.\"\n\n## Core Concepts Implemented\n\nThe project structures neural network abstractions hierarchically:\n\n1. **Automatic Differentiation (AD)**\n - Forward and reverse mode differentiation\n - Based on the SimpleAD paper framework\n - \"Generalized, mode-independent automatic differentiation with rudimentary functions\"\n\n2. **Para Category**\n - Parametrized function composition\n - Derived from the Backprop as Functor paper\n - Enables composable parametrized transformations\n\n3. **Learners**\n - Represent trainable components\n - Combine cost functions with update rules\n - Support meta-learning architectures\n\n## Code Structure\n\nThe repository contains:\n- **src/** - Core implementations of categorical abstractions\n- **app/** - Application-level code and examples\n- **test/** - Testing suite\n- **Examples.hs** - Linear regression demonstration\n\n## Key Architectural Insight\n\nThe project addresses a fundamental problem: \"As more and more components stop being fixed throughout training, there is increasingly larger need for more precise formal specification.\" Category theory provides the mathematical framework for describing what *remains invariant* across learning systems.\n\n## Current Status\n\nAs of May 2020, development shifted toward theoretical work. The creator states this is \"research level stuff and at the moment not really usable for anything other than playing around,\" with plans to return to practical implementation.\n\n## Related Theoretical Foundations\n\n- The Simple Essence of Automatic Differentiation (SimpleAD)\n- Backprop as Functor\n- Lenses and Learners framework"
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"content": "Web search results for query: \"\"backprop as functor\" lens category theory neural network code example\"\n\nLinks: [{\"title\":\"arXiv:1711.10455v3 [math.CT] 1 May 2019 1 Backprop as Functor:\",\"url\":\"https://arxiv.org/pdf/1711.10455\"},{\"title\":\"[1711.10455] Backprop as Functor: A compositional perspective on supervised learning\",\"url\":\"https://arxiv.org/abs/1711.10455\"},{\"title\":\"Backprop as Functor: A compositional perspective on supervised learning | Request PDF\",\"url\":\"https://www.researchgate.net/publication/338738998_Backprop_as_Functor_A_compositional_perspective_on_supervised_learning\"},{\"title\":\"GitHub - bgavran/Category_Theory_Machine_Learning: List of papers studying machine learning through the lens of category theory\",\"url\":\"https://github.com/bgavran/Category_Theory_Machine_Learning\"},{\"title\":\"GitHub - mandubian/neurocat: From neural networks to the Category of composable supervised learning algorithms in Scala with compile-time matrix checking based on singleton-types\",\"url\":\"https://github.com/mandubian/neurocat\"},{\"title\":\"Backprop as functor | Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science\",\"url\":\"https://dl.acm.org/doi/10.5555/3470152.3470163\"},{\"title\":\"Fundamental Components of Deep Learning A category-theoretic approach\",\"url\":\"https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf\"},{\"title\":\"Backprop as Functor: A compositional perspective on supervised learning - ADS\",\"url\":\"https://ui.adsabs.harvard.edu/abs/2017arXiv171110455F/abstract\"},{\"title\":\"Deep Learning for the Working Category Theorist Jade Master\",\"url\":\"https://math.ucr.edu/home/baez/mathematical/ACTUCR/Master_Deep_Learning_for_the_Working_Category_Theorist.pdf\"},{\"title\":\"Backprop As Functor | PDF | Artificial Neural Network | Function (Mathematics)\",\"url\":\"https://www.scribd.com/document/487849574/Backprop-as-Functor\"}]\n\nHere are the results for your search on \"backprop as functor\" and the lens/category theory connection to neural networks:\n\n## Key Paper: \"Backprop as Functor\"\n\nA supervised learning algorithm searches over a set of functions A → B parametrised by a space P to find the best approximation to some ideal function f: A → B. It does this by taking examples (a,f(a)) ∈ A×B, and updating the parameter according to some rule. The authors define a category where these update rules may be composed, and show that gradient descent—with respect to a fixed step size and an error function satisfying a certain property—defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.\n\nThe paper is available on arXiv: [arXiv:1711.10455](https://arxiv.org/abs/1711.10455), authored by B. Fong, D. I. Spivak, and R. Tuyéras.\n\n## Key Categorical Structures\n\nThe category **NNet** of neural networks has natural numbers as objects and neural networks of type (m, n) as morphisms. Composition is given by concatenation of neural networks.(no content)The paper defines functors Iσ: NNet → Para and Lε,e: Para → Learn. In particular, Lε,e implements the usual backpropagation algorithm with quadratic error and step size ε on a neural network with activation function σ.\n\n## Connection to Lenses\n\nAsymmetric lenses are simply learners with trivial state spaces, and learners themselves are open games obeying a certain singleton best response condition. This gives a chain of functors: **Lens ↪ Learn ↪ Game**.\n\n## Code Examples\n\nThe **neurocat** GitHub project implements the concepts from the \"Backprop as Functor\" paper in Scala, including compile-time matrix dimension checking using singleton-types.(no content)By simple functor composition, you have a Functor from neural networks to supervised learning algorithms: **NNet → Learn : (ParaFn → Learn) ∘ (NNet → ParaFn)**.\n\n## Additional Resources\n\nThe GitHub repository **Category_Theory_Machine_Learning** maintains a list of related papers including: Backprop as Functor, Lenses and Learners, Reverse Derivative Ascent, Dioptics, and Learning Functors using Gradient Descent.(no content)Bruno Gavranović's PhD thesis \"Fundamental Components of Deep Learning: A category-theoretic approach\" develops a novel mathematical foundation for deep learning based on category theory, creating a framework that is end-to-end, uniform, and amenable to direct implementation in programming languages.\n\n### Useful Links:\n- **Paper**: https://arxiv.org/abs/1711.10455\n- **Scala Implementation (neurocat)**: https://github.com/mandubian/neurocat\n- **Paper Collection**: https://github.com/bgavran/Category_Theory_Machine_Learning\n- **Bruno Gavranović's Thesis**: https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "# Neurocat: Category Theory Meets Neural Networks\n\n## Core Categorical Concepts\n\nThe library explores the mathematical structure unifying neural networks and supervised learning through category theory. Key concepts include:\n\n**The Learn Category**: \"The set of supervised learning algorithms equipped with 3 functions (implement, update-params, request-input) forms a symmetric monoidal category `Learn`\" enabling algorithm composition.\n\n**Functor Composition Chain**: The architecture establishes multiple functors:\n- `ParaFn -> Learn`: Maps parametrized functions to learning algorithms\n- `NNet -> ParaFn`: Transforms neural networks into parametrized functions \n- `NNet -> Learn`: Composition of the above two functors\n\n## Key Abstractions\n\n**Parametrized Functions**: Represented as `P -> A -> B`, where `P` contains trainable parameters, `A` is input, and `B` is output. These form the bridge between abstract mathematics and concrete neural network operations.\n\n**Supervised Learning**: Characterized as \"a structure able to approximate a function `A -> B` relying on parameters `P` which are updated through an optimization/training process.\"\n\n**Neural Networks as Functors**: A trained neural network approximating `InputLayer -> OutputLayer` using weights `W` becomes a functor mapping to parametrized function space.\n\n## Technical Implementation\n\nThe library uses **singleton-types** (Scala's SIP-23) for compile-time matrix dimension checking, enabling type-safe matrix operations without runtime overhead. The implementation leverages the `singleton-ops` library and ND4J for actual numerical computation.\n\nThis approach enables composing neural network layers as mathematical objects while maintaining dimensional correctness at compile time."
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"result": "# Neurocat: Category Theory Meets Neural Networks\n\n## Core Categorical Concepts\n\nThe library explores the mathematical structure unifying neural networks and supervised learning through category theory. Key concepts include:\n\n**The Learn Category**: \"The set of supervised learning algorithms equipped with 3 functions (implement, update-params, request-input) forms a symmetric monoidal category `Learn`\" enabling algorithm composition.\n\n**Functor Composition Chain**: The architecture establishes multiple functors:\n- `ParaFn -> Learn`: Maps parametrized functions to learning algorithms\n- `NNet -> ParaFn`: Transforms neural networks into parametrized functions \n- `NNet -> Learn`: Composition of the above two functors\n\n## Key Abstractions\n\n**Parametrized Functions**: Represented as `P -> A -> B`, where `P` contains trainable parameters, `A` is input, and `B` is output. These form the bridge between abstract mathematics and concrete neural network operations.\n\n**Supervised Learning**: Characterized as \"a structure able to approximate a function `A -> B` relying on parameters `P` which are updated through an optimization/training process.\"\n\n**Neural Networks as Functors**: A trained neural network approximating `InputLayer -> OutputLayer` using weights `W` becomes a functor mapping to parametrized function space.\n\n## Technical Implementation\n\nThe library uses **singleton-types** (Scala's SIP-23) for compile-time matrix dimension checking, enabling type-safe matrix operations without runtime overhead. The implementation leverages the `singleton-ops` library and ND4J for actual numerical computation.\n\nThis approach enables composing neural network layers as mathematical objects while maintaining dimensional correctness at compile time.",
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"text": "Okay, let me break down how this actually works:\n\n## The Core Insight: Lenses Model Forward + Backward\n\nA **lens** is a bidirectional thing — it has a \"get\" (forward) and a \"put\" (backward):\n\n```\nTraditional lens (functional programming):\n get: S → A # Extract a part\n put: S × A → S # Update with new value\n\nNeural network lens:\n forward: Input → Output\n backward: (Input, Gradient) → (Updated Input, Param Gradient)\n```\n\nBackprop is literally a lens! You go forward, then backward with the same structure.\n\n## Para: Adding Parameters\n\n**Para** (parametric) wraps functions to include learnable weights:\n\n```\nRegular function: A → B\nParametric function: P × A → B (P = parameters)\n\nIn code:\n f(x) # regular\n f(θ, x) # parametric, θ are weights\n```\n\nA neural network layer is a parametric function:\n```\nlinear_layer: (Weights × Bias) × Input → Output\n```\n\n## The Functor Chain\n\n```\nNNet → Para → Learn\n\n1. NNet: Neural network architecture (layers, connections)\n ↓ functor\n2. Para: Parametrized functions (weights + computation)\n ↓ functor \n3. Learn: Full learning algorithm (forward, backward, update)\n```\n\nEach arrow is a **functor** — structure-preserving map. Composing layers? Just compose functors.\n\n## Concrete Code (from neurocat in Scala)\n\n```scala\n// A parametric function: P => A => B\ntrait ParaFn[P, A, B] {\n def apply(p: P)(a: A): B\n}\n\n// A learner adds update rule\ntrait Learner[P, A, B] {\n def implement(p: P)(a: A): B // forward\n def updateParams(p: P, a: A, b: B): P // gradient step\n def requestInput(p: P, a: A, b: B): A // backprop to input\n}\n\n// Composition is just functor composition\ndef compose[P, Q, A, B, C](\n f: Learner[P, A, B],\n g: Learner[Q, B, C]\n): Learner[(P, Q), A, C]\n```\n\n## Why This Matters\n\n**Compositionality**: Build complex networks by composing simple pieces — the math guarantees it works.\n\n```\nlayer1: Learn[W1, Image, Features]\nlayer2: Learn[W2, Features, Logits]\nnetwork = layer1 >>> layer2 // Composition in Learn category\n```\n\n**Correctness**: Types enforce shapes. Can't compose incompatible layers — compiler catches it.\n\n**Generalization**: Same framework handles:\n- Standard backprop\n- RNNs (weight tying = specific functor)\n- GNNs (equivariance = specific functor)\n- Potentially symbolic modules (different category)\n\n## The Symbolic Connection\n\nHere's where it gets interesting for your question. A **symbolic math module** would be:\n\n```\nNeural expert: Para[Weights, Input, Output] (learned)\nSymbolic expert: Para[Unit, Expression, Expression] (no weights, just rules)\n```\n\nBoth fit in the same categorical framework! The symbolic one has no learnable parameters — it's a pure function. But it composes the same way.\n\n```scala\n// Neural: learned transformation\nval neural: Learner[Weights, Vector, Vector] = ...\n\n// Symbolic: algebraic simplification (no weights to learn)\nval symbolic: Learner[Unit, Expr, Expr] = Learner(\n implement = (_, expr) => simplify(expr),\n updateParams = (_, _, _) => (), // nothing to update\n requestInput = (_, expr, _) => derivative(expr) // chain rule symbolically\n)\n\n// They compose!\nval hybrid = neural >>> symbolic\n```\n\n---\n\n**Sources:**\n- [Backprop as Functor](https://arxiv.org/abs/1711.10455)\n- [neurocat implementation](https://github.com/mandubian/neurocat)\n- [Gavranović's thesis](https://www.brunogavranovic.com/assets/FundamentalComponentsOfDeepLearning.pdf)"
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"content": "idk seems like a nothing burger. so you can think of algebraic functions and learned functions as functions. what am I missing"
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"thinking": "The user is pushing back and asking what's actually novel or useful here beyond \"functions are functions.\" They're right to be skeptical - I need to explain what the categorical framework actually buys you that you don't get from just \"everything is a function.\"\n\nLet me think about what the actual value proposition is:\n\n1. **Compositionality with correctness guarantees** - but you get this from types in any typed language\n\n2. **Automatic differentiation structure** - but PyTorch/JAX already do this\n\n3. **Weight tying / equivariance constraints** - this is actually interesting, the category theory tells you what weight sharing is valid\n\n4. **Unified framework for different architectures** - but in practice, people just implement them in PyTorch\n\nThe honest answer might be that it's mostly:\n- Theoretically elegant\n- Good for understanding/proving things\n- Not obviously better for building things\n\nUnless Symbolica has something concrete that shows practical benefits...\n\nLet me be honest with the user that they might be right - this could be more theoretical elegance than practical advantage.",
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"text": "You're not missing much. Let me be honest.\n\n## What Category Theory Actually Buys You\n\n| Claimed Benefit | Reality |\n|-----------------|---------|\n| \"Compositionality\" | You already have this — `nn.Sequential` composes layers |\n| \"Type safety\" | Useful, but PyTorch shape errors are fine in practice |\n| \"Unified framework\" | Academically nice, doesn't make models better |\n| \"Proves correctness\" | Of what? That backprop works? We knew that |\n\n## The Actual Value (Maybe)\n\n**1. Equivariance constraints become automatic**\n\nCategory theory can tell you: \"If your data has X symmetry, your network must have Y structure.\"\n\n```\nData lives on a graph → Use GNN (equivariant to permutation)\nData lives on a grid → Use CNN (equivariant to translation)\nData is a sequence → Use RNN (equivariant to shift)\n```\n\nBut... people figured this out without category theory. It's a nice formalization, not a discovery.\n\n**2. Weight tying falls out naturally**\n\nRNNs share weights across time. Category theory says: \"That's a specific functor.\" \n\nBut again — we already knew how to build RNNs.\n\n**3. Maybe: principled hybrid architectures?**\n\nThe hope is: if you have neural + symbolic components, category theory tells you how they compose correctly.\n\nBut Symbolica hasn't shown this working at scale. It's promissory.\n\n## The Cynical Read\n\nCategory theory for deep learning is:\n- Great for papers\n- Great for understanding existing architectures post-hoc\n- Not shown to produce better models\n- Not shown to enable things that weren't already possible\n\nIt's like saying \"physics is just differential geometry.\" True, elegant, not helpful for building a bridge.\n\n## What Would Change My Mind\n\nIf Symbolica showed:\n- A model that beats transformers on reasoning\n- Built using categorical principles\n- That couldn't have been built without them\n\nThey haven't. They've raised $31M on the promise, but the results aren't public.\n\n---\n\nSo yeah — you might be right. It could be a theoretically beautiful nothing burger. The test is whether Symbolica ships something that works better, not whether the math is pretty."
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"content": " Data lives on a graph → Use GNN (equivariant to permutation)\n Data lives on a grid → Use CNN (equivariant to translation)\n Data is a sequence → Use RNN (equivariant to shift)\n\n\nbut we're using transformers for all of these at this point"
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"thinking": "The user makes an excellent point. The categorical deep learning folks talk about how different data structures require different architectures with different equivariance properties, but in practice:- Vision Transformers (ViT) replaced CNNs for images\n- Transformers replaced RNNs for sequences- Graph Transformers are competitive with GNNs\n\nAnd transformers are basically \"attention over everything\" - they don't bake in the structural assumptions. They just learn whatever structure is useful from data.\n\nThis is actually a deep point against the categorical approach. The whole pitch is \"match your architecture to your data's structure.\" But transformers say \"who cares, we'll learn the structure.\"\n\nAnd transformers won.",
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"text": "Lmao yeah, that's the thing isn't it.\n\n## The Categorical Pitch\n\n> \"Match your architecture to your data's structure for proper inductive bias\"\n\n## What Actually Happened\n\n```\nImages: CNN → ViT (transformer)\nSequences: RNN → Transformer\nGraphs: GNN → Graph Transformer\nAudio: CNN/RNN → Transformer\nVideo: 3D CNN → Transformer\nProtein: Custom → AlphaFold (transformer + structure)\n```\n\nTransformers just... won. Everywhere. Despite having \"wrong\" inductive bias for most of these domains.\n\n## Why This Is Awkward for Categorical DL\n\nThe whole framework says: \"Your architecture should respect the symmetries of your data.\"\n\nTransformers say: \"Everything is a set of tokens. Attention learns whatever structure matters.\"\n\nAnd attention **learns** equivariance-ish behavior when it's useful, without being constrained to it.\n\n```\nCategorical approach: Bake in the right structure → efficient, principled\nTransformer approach: Learn structure from data → works better in practice\n```\n\n## The Counter-Argument (Steel-Man)\n\nMaybe transformers only work because:\n- Massive scale (trillion tokens, billions of params)\n- Huge compute (thousands of GPUs)\n\nIf you had the \"right\" architecture, maybe you'd need less data and compute?\n\nThis is Symbolica's bet:\n> \"If we build an architecture that's natively capable of reasoning, it takes much less data\"\n\nBut... no evidence yet. Meanwhile transformers keep scaling.\n\n## The Uncomfortable Truth\n\nTransformers are basically: \"Throw out inductive bias, let attention figure it out, scale up.\"\n\nAnd it keeps working. The categorical framework is elegant, but empirically, brute force attention + scale beats principled architecture design.\n\n---\n\nMaybe the categorical stuff matters for small-data regimes or formal verification? But for \"make AI that works\" — transformers won by ignoring the theory."
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"text": "A transformer applied to graph data. Instead of \"sequence of tokens,\" you have \"nodes in a graph.\"\n\n## Standard Transformer\n\n```\nInput: [token_1, token_2, ..., token_n] (sequence)\n\nAttention: every token attends to every other token\n O(n²) attention matrix\n```\n\n## Graph Transformer\n\n```\nInput: nodes with edges between them\n\nOption 1 - Full attention (ignore edges):\n Every node attends to every node\n Edges just add features / bias to attention scores\n\nOption 2 - Sparse attention (respect edges):\n Nodes only attend to neighbors (or k-hop neighbors)\n Attention pattern = graph structure\n```\n\n## The Basic Idea\n\n```\n# Regular transformer attention\nQ, K, V = project(tokens)\nattn = softmax(Q @ K.T / sqrt(d))\nout = attn @ V\n\n# Graph transformer attention \nQ, K, V = project(node_features)\nattn = softmax(Q @ K.T / sqrt(d) + edge_bias) # edges modify attention\n# ↑ learned or structural bias\nout = attn @ V\n```\n\nSome variants:\n- Add edge features to the attention computation\n- Use adjacency matrix as attention mask\n- Encode graph structure via positional encodings (eigenvectors of Laplacian, random walk, etc.)\n\n## Why It's Funny\n\nGNNs were designed with careful message-passing to respect graph structure.\n\nGraph Transformers said: \"What if we just... do full attention and let it figure out the graph?\"\n\nAnd it often works better. Same story as ViT killing CNNs.\n\n## Examples\n\n- **Graphormer** (Microsoft) — won molecular prediction benchmarks\n- **GPS** (General Powerful Scalable) — hybrid GNN + Transformer\n- **TokenGT** — treats edges as tokens too\n\n---\n\nIt's the \"transformers win again\" story. GNNs had principled message passing. Graph Transformers said \"attention go brr\" and beat them on benchmarks."
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"content": "seems like the cool part of Engram is not so much the vram or compute savings as much as the ability to offload memorization completely from the inference weights into a standard sharable, expandable thing. That seems more bitter lesson pilled than some of this other stuff. \n\nBtw wasn't the point of the MLP in the transformer architecture to \"memorize\" stuff? "
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"text": "Yeah, you're onto something.\n\n## Engram's Real Value\n\nNot this:\n> \"Save 20% VRAM\"\n\nBut this:\n> \"Memorization becomes a **data structure** instead of **implicit in weights**\"\n\nThat means:\n- **Shareable**: Same hash table works with different models\n- **Expandable**: Add new facts without retraining\n- **Editable**: Fix a wrong fact by updating one row\n- **Inspectable**: You can look at what's stored\n\nThat's a different kind of win than efficiency. It's architectural clarity.\n\n## You're Right About MLPs\n\nThe interpretability view (from Anthropic, etc.):\n\n```\nTransformer block:\n Attention: Routes information (\"what's relevant to what\")\n MLP: Stores facts (\"retrieves knowledge\")\n```\n\nThere's research showing MLPs act like key-value stores:\n- Input activations = query\n- MLP weights = stored key-value pairs\n- Output = retrieved knowledge\n\nFamous example: \"The Eiffel Tower is in [Paris]\" — specific MLP neurons fire for this fact.\n\n## Engram Makes This Explicit\n\n```\nTraditional Transformer:\n Attention: routing\n MLP: implicit key-value memory (learned, frozen, opaque)\n\nEngram Transformer:\n Attention: routing\n MLP: reduced (just reasoning?)\n Engram: explicit key-value memory (hash table, editable, inspectable)\n```\n\nYou're literally factoring out what the MLP was doing implicitly and making it a proper data structure.\n\n## The Bitter Lesson Connection\n\nSutton's bitter lesson: simple methods + scale beat clever methods.\n\nEngram is bitter lesson pilled because:\n- Hash tables are dumb and old\n- But they scale trivially (just add RAM)\n- And they're simple (O(1) lookup)\n- And they separate concerns cleanly\n\nvs. clever approaches like:\n- \"Learn a neural memory module\"\n- \"Use structured knowledge graphs with neural retrieval\"\n- \"Category-theoretic knowledge representation\"\n\nDeepSeek just said: \"What if we used a hash table like it's 1995, but really big?\"\n\n---\n\nThe pattern: **make implicit structure explicit, use boring scalable primitives**. That's more bitter lesson than \"design elegant architectures with the right inductive bias.\""
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"content": "so you think you still need MLP for reasoning?"
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"text": "Honestly, not sure. Let me think through this.\n\n## What Do MLPs Actually Do?\n\nTwo theories:\n\n**Theory 1: MLPs = Memory**\n- Store key-value pairs of facts\n- Input pattern matches → output associated knowledge\n- Basically a soft lookup table\n\n**Theory 2: MLPs = Computation**\n- Learn nonlinear transformations\n- Combine features in complex ways\n- \"Think\" vs \"recall\"\n\n## If MLPs Are Just Memory...\n\nThen Engram could replace them entirely:\n\n```\nCurrent: Attention → MLP (memory + compute?) → Attention → MLP ...\nEngram: Attention → Hash lookup → Attention → Hash lookup ...\nExtreme: Attention only + giant hash table?\n```\n\n## But What Is \"Reasoning\"?\n\nHere's the hard question. When you do multi-step reasoning:\n\n```\n\"If all mammals are warm-blooded, and dogs are mammals, then dogs are ___\"\n```\n\nIs that:\n1. **Lookup**: Retrieve \"dogs are warm-blooded\" from memory\n2. **Computation**: Actually apply the logical rule at inference time\n\nIf (1), MLPs are just memory and Engram could replace them.\n\nIf (2), you need something that *computes*, not just retrieves.\n\n## The Attention-Only Hypothesis\n\nSome researchers have asked: what if transformers without MLPs?\n\n```\nAttention-only transformer:\n- Just attention layers\n- No MLPs at all\n- All \"computation\" is routing/copying\n```\n\nResult: Works surprisingly well for some tasks, but degrades on others.\n\nThe tasks that break tend to require **composing** information in ways that aren't just retrieval.\n\n## My Guess\n\n```\nMemorization: MLP ≈ hash table (Engram can replace)\nReasoning: ??? \n\nMaybe:\n- Simple reasoning = just retrieval chains (hash table works)\n- Complex reasoning = needs actual computation (MLPs? symbolic? something else?)\n```\n\n## The Radical Position\n\nWhat if reasoning is *also* just lookup, but over a space of reasoning patterns?\n\n```\n\"If X then Y\" + \"X is true\" → look up \"modus ponens pattern\" → output \"Y\"\n```\n\nThen everything is memory, and you just need a big enough hash table.\n\nThis would be very bitter lesson: reasoning isn't special, it's just retrieval from a sufficiently large table of patterns.\n\n---\n\nDeepSeek's 75/25 split (compute/memory) suggests they think you still need *some* MLPs. But maybe that ratio shifts as tables get bigger?\n\nWhat's your intuition?"
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"content": "Web search results for query: \"transformer MLP memory vs computation mechanistic interpretability circuits Anthropic\"\n\nLinks: [{\"title\":\"Transformer-circuits\",\"url\":\"https://transformer-circuits.pub/\"},{\"title\":\"A Mathematical Framework for Transformer Circuits\",\"url\":\"https://transformer-circuits.pub/2021/framework/index.html\"},{\"title\":\"Anthropic drops an amazing report on LLM interpretability | by Lee Fischman | Medium\",\"url\":\"https://medium.com/@lee.fischman/anthropic-drops-an-amazing-report-on-llm-interpretability-d3fbcd5ba762\"},{\"title\":\"Mechanistic Interpretability On the Internals of LLMs\",\"url\":\"https://krmopuri.github.io/xml/static_files/presentations/MI-Pranav.pdf\"},{\"title\":\"A Practical Review of Mechanistic Interpretability for Transformer-Based Language Models\",\"url\":\"https://arxiv.org/html/2407.02646v2\"},{\"title\":\"Circuit Tracing: Revealing Computational Graphs in Language Models\",\"url\":\"https://transformer-circuits.pub/2025/attribution-graphs/methods.html\"},{\"title\":\"Chapter 1 | AI Safety — ARENA\",\"url\":\"https://www.arena.education/chapter1\"},{\"title\":\"Circuit Tracing: A Step Closer to Understanding Large Language Models | Towards Data Science\",\"url\":\"https://towardsdatascience.com/circuit-tracing-a-step-closer-to-understanding-large-language-models/\"},{\"title\":\"Anthropic's Circuits Updates: All Releases (Updated May 2024) - Begins w/ AI\",\"url\":\"https://beginswithai.com/anthropic-circuits-updates/\"},{\"title\":\"Understanding LLMs: Insights from Mechanistic ...\",\"url\":\"https://www.lesswrong.com/posts/XGHf7EY3CK4KorBpw/understanding-llms-insights-from-mechanistic\"}]\n\nHere are the search results on transformer MLP memory vs computation in mechanistic interpretability circuits from Anthropic:\n\n## Key Findings\n\n### Anthropic's Transformer Circuits Research\n\nThe Transformer Circuits website hosts research from the Anthropic Interpretability Team, whose present research aims to create a foundation for mechanistic interpretability research.(no content)One avenue for addressing issues with unexpected model behaviors is mechanistic interpretability, \"attempting to reverse engineer the detailed computations performed by transformers, similar to how a programmer might try to reverse engineer complicated binaries into human-readable source code.\"\n\n### MLP Layers and Residual Stream\n\nAll components of a transformer (the token embedding, attention heads, MLP layers, and unembedding) communicate with each other by reading and writing to different subspaces of the residual stream. Rather than analyze the residual stream vectors, it can be helpful to decompose the residual stream into all these different communication channels, corresponding to paths through the model.(no content)There are generally far more \"computational dimensions\" (such as neurons and attention head result dimensions) than the residual stream has dimensions to move information. Just a single MLP layer typically has four times more neurons than the residual stream has dimensions. For example, at layer 25 of a 50 layer transformer, the residual stream has 100 times more neurons trying to communicate through superposition.\n\n### Circuits and Features\n\nWhile the study of features helps understand what information is encoded in a model's activations, it does not inform us of how these features are extracted from the input, their interactions, or how they are used by the model. MI bridges this gap by investigating circuits – \"meaningful computational pathways that connect features and facilitate specific LM behaviors.\"\n\n### Recent Work: Circuit Tracing with Transcoders\n\nA transcoder is a neural network designed to replace the MLP block in a transformer model with a more interpretable, functionally equivalent component (feature).(no content)A transformer block typically consists of two main components: an attention block and an MLP block (feedforward network). To build a replacement model, the MLP block in the original transformer model is replaced with a transcoder.(no content)In Anthropic's approach, they first identify features (interpretable building blocks that the model uses in its computations), then describe the processes, or circuits, by which these features interact to produce model outputs.\n\n### Attention vs MLP Focus\n\nIn the original Mathematical Framework paper, Anthropic made a significant simplification by focusing on \"attention-only\" transformers without MLP layers. This was motivated by the fact that circuits with attention heads present new challenges not faced by earlier circuits work.(no content)Many Circuits Updates dive deep into the behavior of individual attention heads and the computational \"circuits\" they form with other heads, MLPs, and skip connections.\n\nFor the most authoritative sources, you can explore:\n- **transformer-circuits.pub** - Anthropic's dedicated research website\n- **\"A Mathematical Framework for Transformer Circuits\"** (2021) - foundational paper\n- **\"Circuit Tracing\"** (2025) - recent work on attribution graphs\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "Web search results for query: \"\"MLP layers\" transformers \"key-value memory\" facts knowledge storage neurons\"\n\nLinks: [{\"title\":\"Transformer Feed-Forward Layers Are Key-Value Memories | Request PDF\",\"url\":\"https://www.researchgate.net/publication/357126029_Transformer_Feed-Forward_Layers_Are_Key-Value_Memories\"},{\"title\":\"How LLMs Store and Use Knowledge? This AI Paper Introduces Knowledge Circuits: A Framework for Understanding and Improving Knowledge Storage in Transformer-Based LLMs - MarkTechPost\",\"url\":\"https://www.marktechpost.com/2024/12/14/how-llms-store-and-use-knowledge-this-ai-paper-introduces-knowledge-circuits-a-framework-for-understanding-and-improving-knowledge-storage-in-transformer-based-llms/\"}]\n\nBased on the search results, here's what I found about MLP layers in transformers functioning as key-value memory for knowledge storage:\n\n## Key Findings\n\nResearch by Geva et al. (2021) and Meng et al. (2022a) reveals that Transformer FFNs (Feed-Forward Networks) function as key-value memories and are effective targets for knowledge editing.(no content)Studies have hypothesized that LLMs store parametric information within the Feed Forward Network (FFN) layers, which function as a key-value memory.\n\n### Structure and Function\n\nFor each MLP neuron i, the value vector (rows of the weight matrix) controls the scaling of the value vector through an activation score.(no content)These neurons are presumed to store factual information, acting as key-value memory. Knowledge editing techniques have been developed to refine this stored data, addressing inaccuracies and updating biases.\n\n### Knowledge Editing Applications\n\nROME (Meng et al., 2022) utilizes causal tracing techniques to localize factual knowledge within MLP layers and performs parameter editing by solving a constrained least-squares problem, under the assumption that MLP layers function as key-value memory modules.(no content)Prior work has shown that not all neurons contribute equally to the computation of specific knowledge during inference.\n\n### Scale of FFN Parameters\n\nAmong transformer components, FFNs dominate the parameter budgets as they can account for as much as 67% of the total parameters in decoder-only models. FFN width is typically set by rules of thumb rather than design principles, e.g., 4× expansion in GPT-2 and 2.67× in LLaMA.\n\n### Challenges\n\nTechniques like ROME and fine-tuning layer methods successfully added new knowledge but often disrupted unrelated areas. For example, when modifying a model to associate \"Intel\" with specific hardware, unrelated queries about \"Windows servers\" erroneously reflected the edited knowledge, highlighting the risk of overfitting.(no content)Research emphasizes the importance of considering the broader context of knowledge circuits rather than focusing solely on individual layers or neurons, offering a comprehensive framework for analyzing and improving transformer-based models.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"content": "Web search results for query: \"transformer MLP computation reasoning \"grokking\" algorithm learning mechanistic interpretability\"\n\nLinks: [{\"title\":\"Published as a conference paper at ICLR 2023 PROGRESS MEASURES FOR GROKKING VIA\",\"url\":\"https://arxiv.org/pdf/2301.05217\"},{\"title\":\"Understanding Mechanistic Interpretability in AI Models | IntuitionLabs\",\"url\":\"https://intuitionlabs.ai/articles/mechanistic-interpretability-ai-llms\"},{\"title\":\"Progress measures for grokking via mechanistic interpretability | alphaXiv\",\"url\":\"https://www.alphaxiv.org/overview/2301.05217\"},{\"title\":\"Grokked Transformers are Implicit Reasoners:\",\"url\":\"https://arxiv.org/pdf/2405.15071\"},{\"title\":\"A Mechanistic Interpretability Analysis of Grokking\",\"url\":\"https://www.alignmentforum.org/posts/N6WM6hs7RQMKDhYjB/a-mechanistic-interpretability-analysis-of-grokking\"},{\"title\":\"Grokking in Neural Networks & Mechanistic Interpretability | by Ashkan Golgoon | Medium\",\"url\":\"https://medium.com/@ashkangolgoon/grokking-in-neural-networks-mechanistic-interpretability-6d830367ad1d\"},{\"title\":\"Progress measures for grokking via mechanistic interpretability\",\"url\":\"https://arxiv.org/html/2301.05217\"},{\"title\":\"Do Machine Learning Models Memorize or Generalize?\",\"url\":\"https://pair.withgoogle.com/explorables/grokking/\"},{\"title\":\"Aclanthology\",\"url\":\"https://aclanthology.org/2025.acl-long.235.pdf\"},{\"title\":\"Grokking\",\"url\":\"https://sidn.baulab.info/grokking/\"}]\n\nHere are the search results on transformer MLP computation, reasoning, grokking, algorithm learning, and mechanistic interpretability:\n\n## Key Research Findings\n\n### Foundational Work on Grokking\n\"Grokking\" was discovered by Power et al. (perhaps initially accidentally) as an unusual behavior while training a two-layer transformer on a small algorithmically generated dataset well past the point of overfitting. The phenomenon refers to the model making the switch from memorization of training set to perfect generalization to unseen data.\n\n### Mechanistic Interpretability Analysis\n\nProgress measures can be found via mechanistic interpretability: reverse-engineering learned behaviors into their individual components. As a case study, researchers investigated the \"grokking\" phenomenon exhibited by small transformers trained on modular addition tasks. They fully reverse-engineered the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle.\n\n### MLP and Attention Head Computations\n\nThe MLP and attention heads can be approximated well using 2nd degree polynomials of trigonometric functions of a single frequency. Attention heads and most neurons are well approximated by 2nd degree polynomials of sines and cosines of a single frequency. The corresponding direction in WL also contains the same frequency. Hence model computations are localized across all the frequencies.\n\n### Three Training Phases\n\nUsing their mechanistic understanding, the authors identify three distinct phases in the learning process: Phase 1 (Memorization, 0-1.4k epochs): The network quickly memorizes the training data. Phase 2 (Circuit Formation, 1.4k-9.4k epochs): This is the crucial hidden progress phase where the network gradually builds the generalizing Fourier circuit. Phase 3 (Cleanup, 9.4k-14k epochs): The sudden grokking transition occurs as weight decay aggressively prunes the less efficient memorization components.\n\n### Role of Weight Decay\n\nGrokking is a contingent phenomenon — it goes away if model size, weight decay, data size and other hyperparameters aren't just right. With too little weight decay, the model can't escape overfitting the training data. Adding more weight decay pushes the model to generalize after memorizing. Increasing weight decay even more causes test and train loss to fall together; the model goes straight to generalizing. And with too much weight decay the model will fail to learn anything.\n\n### Implications for Reasoning\n\nMechanistic analysis into the model's internals reveals the configuration and gradual formation of the generalizing circuit, and also explains the different levels of systematicity the model acquires across tasks. These findings guide data and training setup to better induce implicit reasoning, and suggest potential improvements to the transformer architecture to further unlock its generalization.\n\n### Broader Significance\n\nThe main impact is proving that mechanistic interpretability is actually possible—that we can take a trained neural network and reverse-engineer non-trivial and unexpected algorithms from it. By focusing on grokking, researchers selected a problem where mechanistic interpretability was unusually tractable (small models, clean algorithmic tasks), and found real insights about grokking as a direct result.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"text": "Good evidence exists for both views:\n\n## Evidence: MLPs as Memory\n\nFrom [Geva et al.'s paper](https://www.researchgate.net/publication/357126029_Transformer_Feed-Forward_Layers_Are_Key-Value_Memories) \"Transformer Feed-Forward Layers Are Key-Value Memories\":\n\n```\nMLP structure:\n Input → W₁ (keys) → activation → W₂ (values) → Output\n\nEach row of W₁ = a \"key\" pattern\nEach column of W₂ = a \"value\" to output when key matches\n```\n\n**Supporting evidence:**\n- [ROME](https://www.marktechpost.com/2024/12/14/how-llms-store-and-use-knowledge-this-ai-paper-introduces-knowledge-circuits-a-framework-for-understanding-and-improving-knowledge-storage-in-transformer-based-llms/) (knowledge editing) can surgically edit facts by modifying specific MLP weights\n- \"The Eiffel Tower is in ___\" → specific neurons fire, can be traced\n- FFNs are 67% of parameters — mostly \"storage\"\n\n## Evidence: MLPs as Computation\n\nFrom [grokking research](https://arxiv.org/abs/2301.05217) (Neel Nanda et al.):\n\n```\nTask: Learn modular addition (a + b mod 113)\n\nWhat the MLP learned:\n - Discrete Fourier transforms\n - Trigonometric identities\n - Converts addition to rotation on a circle\n```\n\nThis is actual **algorithm**, not lookup. The MLP learned to **compute** modular arithmetic using Fourier basis.\n\n**Key insight from grokking:**\n```\nPhase 1: Memorize (store input-output pairs)\nPhase 2: Form circuit (learn the algorithm)\nPhase 3: Cleanup (weight decay kills memorization, keeps algorithm)\n```\n\nThe model transitions from memory → computation with enough regularization.\n\n## The Synthesis\n\nMaybe MLPs do **both**, depending on the task:\n\n| Task Type | What MLP Does |\n|-----------|---------------|\n| Facts (\"Paris is capital of...\") | Memory (key-value lookup) |\n| Algorithms (\"17 + 38 = ...\") | Computation (learned circuit) |\n| Fuzzy reasoning | Probably mix of both? |\n\n## What This Means for Engram\n\nEngram can replace the **memory** part:\n- Factual recall → hash table\n- Pattern completion → hash table\n\nBut the **computation** part might still need MLPs:\n- Algorithmic reasoning → learned circuits\n- Novel combinations → can't just look up\n\n## The Open Question\n\nWhen humans \"reason,\" are we:\n1. Retrieving cached reasoning patterns (memory)\n2. Actually computing something new (computation)\n3. Both?\n\nIf mostly (1), Engram + small MLP works.\nIf (2) matters, you need real compute capacity.\n\n---\n\n**Sources:**\n- [Transformer Feed-Forward Layers Are Key-Value Memories](https://www.researchgate.net/publication/357126029_Transformer_Feed-Forward_Layers_Are_Key-Value_Memories)\n- [Progress Measures for Grokking](https://arxiv.org/abs/2301.05217)\n- [Transformer Circuits](https://transformer-circuits.pub/)"
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"content": " Key insight from grokking:\n Phase 1: Memorize (store input-output pairs)\n Phase 2: Form circuit (learn the algorithm)\n Phase 3: Cleanup (weight decay kills memorization, keeps algorithm)\n\n The model transitions from memory → computation with enough regularization.\n\n\nO sick so this is double decent shit?"
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"thinking": "The user is connecting grokking to double descent! That's a great observation. Double descent is the phenomenon where:1. As model complexity increases, test error first decreases (underfitting → good fit)\n2. Then increases (overfitting)\n3. Then decreases AGAIN past the interpolation threshold (double descent)Grokking is related but different:- Grokking is about training TIME (not model size)\n- Model memorizes first, then generalizes with more training\n- Weight decay is crucial to force the transitionBut they're both about the relationship between memorization and generalization, and how sometimes you have to go THROUGH overfitting to get to generalization.\n\nLet me explain the connection and distinction.",
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"text": "Yeah, related phenomena! Both are about the weird relationship between memorization and generalization.\n\n## Double Descent (model complexity)\n\n```\nTest error vs model size:\n\nError\n │\n │ ╲ ╱╲\n │ ╲ ╱ ╲\n │ ╲ ╱ ╲___________\n │ ╲╱\n └──────────────────────────→ Model size\n ↑ ↑\n Classic Interpolation threshold\n U-curve (memorizes training set perfectly)\n Then gets BETTER again\n```\n\nPast the point where you can memorize everything, adding more capacity helps generalization. Counterintuitive.\n\n## Grokking (training time)\n\n```\nAccuracy vs training steps:\n\nAcc\n │ train ━━━━━━━━━━━━━━━━━━━━\n │ ╱\n │ ╱\n │ ╱ test ─────────\n │ ╱ ╱\n │ ╱ ╱\n │ ╱ ╱\n └────────────────────────────────────→ Steps\n ↑ ↑\n Memorize Generalize\n (overfit) (grok!)\n │ │\n └──────────────┘\n HUGE gap (sometimes 10x more training)\n```\n\nModel memorizes perfectly, then **way later** suddenly generalizes.\n\n## The Connection\n\nBoth suggest:\n\n> **Memorization might be a stepping stone to generalization, not an alternative**\n\nThe model needs to \"see\" the patterns via memorization first, then compresses them into an algorithm.\n\n## Weight Decay is the Key for Grokking\n\n```\nNo weight decay: Stays in memorization forever\nSome weight decay: Memorizes → eventually groks\nMore weight decay: Goes straight to generalizing\nToo much: Can't learn at all\n```\n\nWeight decay penalizes large weights → forces compression → memorization is \"expensive\" → algorithm is \"cheap\" → model finds the algorithm.\n\n## The Bitter Lesson Angle\n\nThis suggests:\n- Memorization and generalization aren't opposites\n- You might need capacity for BOTH\n- Regularization pressure forces the transition\n\nFor Engram:\n```\nWhat if: Engram handles early-phase \"memorization\"\n MLPs focus on \"algorithm formation\"\n \nSeparate the storage from the compute explicitly?\n```\n\n## The Wild Speculation\n\nMaybe large LLMs are **partially grokked**:\n- Some circuits are algorithms (generalize)\n- Some circuits are memorization (don't)\n- Scaling + training time shifts the ratio\n\n---\n\nThe grokking folks at [PAIR](https://pair.withgoogle.com/explorables/grokking/) have a nice interactive explorer if you want to play with it."
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"content": "so does the memorization have to happen in situ in the mlp for grokking to work? Or does plugging in a huge hash table let you skip right to generalizing"
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"thinking": "This is a really sharp question. The user is asking: if grokking requires going through memorization FIRST to then learn the algorithm, what happens if you externalize the memorization to a hash table?\n\nTwo possibilities:\n\n1. Memorization is a necessary stepping stone - the model needs to \"see\" the patterns encoded in its own weights before it can compress them into an algorithm. If you externalize to hash table, maybe you lose the gradient signal that drives algorithm formation?\n\n2. Memorization is just providing the training signal - the model needs to be able to produce correct outputs (via lookup) while it slowly forms the algorithm. In that case, an external hash table might work, and you'd grok faster because you don't waste capacity on memorization.\n\nThis is actually an open empirical question. I don't think anyone has tested this exactly. But it's a great research question.\n\nLet me think about the mechanics:\n\nIn grokking:\n- Phase 1: Model memorizes (MLP stores input-output pairs)\n- Phase 2: Algorithm forms in parallel, but memorization dominates\n- Phase 3: Weight decay kills memorization, algorithm remains\n\nThe key insight is that during Phase 2, the algorithm is forming WHILE memorization is happening. The gradients are still flowing because the model is getting things right (via memorization), but weight decay is pushing toward simpler solutions.\n\nIf you externalize to hash table:\n- Hash table handles correctness on training data\n- MLP can focus on finding the algorithm\n- But... does the MLP get the right gradient signal?\n\nThe problem might be: if the hash table is doing all the work, what gradient does the MLP get? If the hash table always gives the right answer, the MLP might not learn anything.\n\nUnless... you use something like the gating mechanism in Engram. The gate learns when to trust the hash table vs compute. During training, maybe:\n- Easy cases: hash table handles it, MLP gets no gradient\n- Hard cases: MLP has to compute, gets gradient\n\n\n- The algorithm would emerge from those harder cases where the model actually needs to figure things out.\n\nThis is speculative but touches on a genuinely open research question.",
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"text": "That's a genuinely sharp question that I don't think has been tested.\n\n## The Tension\n\n**Grokking story:**\n```\nMemorize in weights → gradients keep flowing → algorithm forms slowly → weight decay kills memorization → algorithm remains\n```\n\n**The key part:** During memorization, the model is still getting gradients (because it's correct on training data). The algorithm forms *in parallel* with memorization, then takes over.\n\n## If You Externalize Memorization...\n\n```\nHash table gives correct answer → MLP gets no gradient??? → nothing to grok?\n```\n\nThe problem: if Engram's hash table is handling all the training examples correctly, what signal does the MLP get to form an algorithm?\n\n## Two Possibilities\n\n**Pessimistic:**\nMemorization has to happen *in situ* because the gradients through the memorized weights are what slowly sculpt the algorithm. Externalize the memory → lose the gradient signal → no grokking.\n\n**Optimistic:**\nMemorization is just providing \"correct outputs while you figure it out.\" An external hash table could do the same job. The MLP still gets gradients because:\n- Hash table might not cover all cases\n- Gate learns when to trust hash vs compute\n- Novel combinations require MLP → gradients flow there\n\n## The Engram Gating Angle\n\nRemember the gate:\n```\nα = σ(h_t · k_t) # trust memory or not?\n```\n\nDuring training, maybe:\n```\nSeen pattern: α → 1, use hash table, MLP gradient ≈ 0\nNovel pattern: α → 0, MLP computes, MLP gets gradient\n```\n\nThe MLP *only* gets trained on cases the hash table can't handle. Those are exactly the cases requiring generalization.\n\n**This might actually accelerate grokking:**\n- Don't waste MLP capacity on memorizable stuff\n- Force MLP to focus on algorithmic patterns\n- Grok faster?\n\n## The Experiment No One Has Run (I Think)\n\n```\nSetup:\n - Modular addition task (standard grokking setup)\n - Engram-style architecture: hash table + small MLP + gate\n \nQuestions:\n - Does it grok faster?\n - Does it grok at all?\n - What forms in the MLP?\n```\n\n## My Guess\n\nIt might work *better* because you're forcing separation of concerns:\n- Hash table: \"I've seen this exact input before\"\n- MLP: \"I need to actually compute this\"\n\nThe MLP can't be lazy and memorize — it only sees hard cases. Might grok faster with less capacity.\n\n---\n\nBut this is speculation. Would be a cool paper: \"Does Externalized Memorization Accelerate Grokking?\""
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"thinking": "The user is making a connection between Engram's gating mechanism and the concept of \"surprise\" or \"free energy\" in energy-based models / predictive coding / active inference. This is actually a deep connection.In those frameworks:\n- Low surprise = prediction matches observation\n- High surprise = mismatch, need to update\n\nIn Engram:- Gate high (α → 1) = memory matches context, use it\n- Gate low (α → 0) = memory doesn't match, compute instead\n\nThe parallel is:- \"Surprise\" = mismatch between expectation and reality\n- Engram gate = mismatch between retrieved memory and current contextThis connects to Karl Friston's free energy principle, predictive coding, and the idea that the brain minimizes surprise.Let me explore this connection.",
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"text": "Yeah, there's a real connection there.\n\n## The Parallel\n\n**Energy-based models / Free Energy Principle:**\n```\nSurprise = -log P(observation | model)\n\nLow surprise: Prediction matches reality → chill\nHigh surprise: Mismatch → update beliefs / do more processing\n```\n\n**Engram gate:**\n```\nα = σ(h_t · k_t)\n\nHigh α: Memory aligns with context → trust it\nLow α: Memory conflicts with context → suppress, compute instead\n```\n\nThe gate is basically measuring \"does this retrieved memory surprise me given what I know?\"\n\n## Predictive Coding Connection\n\nIn predictive coding (Friston et al.):\n```\nBrain hierarchy:\n Higher layers: Send predictions down\n Lower layers: Send prediction errors up\n \nOnly errors propagate — if prediction matches, no signal\n```\n\nEngram is similar:\n```\nHash table: \"I predict the answer is X\"\nContext: \"Does X make sense here?\"\n\nMatch: Use X, minimal compute\nMismatch: Prediction error → route to MLP for real processing\n```\n\n## The \"Attend Only to Surprising Things\" Idea\n\nThis is why attention might work so well — it's routing compute to surprising/relevant tokens:\n\n```\nStandard attention: Attend to everything (expensive)\nSparse attention: Attend to surprising/relevant things (efficient)\nEngram gate: Only invoke memory when it's not surprising\n```\n\n## Energy-Based Framing of Engram\n\nYou could frame Engram as:\n```\nE(context, memory) = -h_t · k_t # energy function\n\nLow energy: Memory fits context → use it\nHigh energy: Memory conflicts → suppress, find better explanation\n```\n\nThe sigmoid gate is basically a Boltzmann factor:\n```\nα = σ(h · k / T) ≈ exp(-E/T) / Z\n```\n\nTemperature controls how \"strict\" the match needs to be — exactly like Domingos' tensor logic temperature for fuzzy vs crisp reasoning.\n\n## The Bigger Picture\n\nMaybe the right framing is:\n```\nLLM = energy-based model where:\n - Engram: Proposes low-energy (familiar) completions\n - Attention: Routes based on surprisal\n - MLP: Computes when no low-energy solution found\n - Output: Minimize free energy (maximize probability)\n```\n\nTransformers might already be doing approximate free energy minimization — Engram just makes part of it explicit.\n\n---\n\nHave you looked at the active inference / free energy takes on transformers? There's some work trying to connect them."
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"content": "Web search results for query: \"transformers active inference free energy principle Friston LLM predictive coding\"\n\nLinks: [{\"title\":\"Free energy principle - Wikipedia\",\"url\":\"https://en.wikipedia.org/wiki/Free_energy_principle\"},{\"title\":\"Predictive coding under the free-energy principle - PMC\",\"url\":\"https://pmc.ncbi.nlm.nih.gov/articles/PMC2666703/\"},{\"title\":\"GitHub - BerenMillidge/FEP_Active_Inference_Papers: A repository for major/influential FEP and active inference papers.\",\"url\":\"https://github.com/BerenMillidge/FEP_Active_Inference_Papers\"},{\"title\":\"What does the free energy principle tell us about the brain?\",\"url\":\"https://gershmanlab.com/pubs/free_energy.pdf\"},{\"title\":\"Active Inference and the Free Energy Principle - Tasshin\",\"url\":\"https://tasshin.com/blog/active-inference-and-the-free-energy-principle/\"},{\"title\":\"THE FREE ENERGY PRINCIPLE FOR PERCEPTION AND ACTION:\",\"url\":\"https://arxiv.org/pdf/2207.06415\"},{\"title\":\"The Free Energy Principle for Perception and Action: A Deep Learning Perspective - PMC\",\"url\":\"https://pmc.ncbi.nlm.nih.gov/articles/PMC8871280/\"},{\"title\":\"The Missing Reward: Active Inference in the Era of Experience\",\"url\":\"https://arxiv.org/html/2508.05619v1\"},{\"title\":\"Active Inference: A Process Theory - Free Energy Principle\",\"url\":\"https://activeinference.github.io/papers/process_theory.pdf\"},{\"title\":\"Incorporating (variational) free energy models into mechanisms: the case of predictive processing under the free energy principle | Synthese\",\"url\":\"https://link.springer.com/article/10.1007/s11229-023-04292-2\"}]\n\nHere are the search results on transformers, active inference, the free energy principle, Friston, LLMs, and predictive coding:\n\n## Overview of Key Concepts\n\n### Free Energy Principle (FEP)\nThe principle is used especially in Bayesian approaches to brain function, but also some approaches to artificial intelligence; it is formally related to variational Bayesian methods and was originally introduced by Karl Friston as an explanation for embodied perception-action loops in neuroscience.(no content)The free energy principle is a mathematical principle of information physics: much like the principle of maximum entropy or the principle of least action, it is true on mathematical grounds.\n\n### Predictive Coding Under FEP\nThis paper considers prediction and perceptual categorization as an inference problem that is solved by the brain. We assume that the brain models the world as a hierarchy or cascade of dynamical systems that encode causal structure in the sensorium. Perception is equated with the optimization or inversion of these internal models, to explain sensory data.(no content)They showed that this optimization scheme corresponds to a form of predictive coding, which is found ubiquitously in the engineering literature (e.g., Kalman filtering). It is important to emphasize that predictive coding is not a generic consequence of FEP, or even of FEP with a specific approximation family.\n\n### Active Inference\nIn active inference, the agent minimizes a variational free energy objective with respect to past experience that causes it to learn a generative model of the world, which allows predicting what will happen in the future in order to avoid surprising states. Variational inference, from which originates the variational free energy functional, makes possible to cast the process of perception as an optimization process. The generative world model is then used in the future to plan actions that will maintain the agent in a homeostatic state that satisfies the agent preferences.\n\n### Transformers and LLMs Connection\nRecent research shows that the transformer architectures common in LLMs implement a form of amortized Bayesian computation. Xie et al. demonstrate that in-context learning in these models emerges from implicit Bayesian inference, where they infer latent concepts from prompt examples to make coherent predictions. Their theory indicates transformers approach optimal Bayesian performance if prompts sufficiently distinguish latent concepts.(no content)Müller et al. provide further evidence, showing transformers can approximate complex posterior distributions with high fidelity, achieving significant speedups over existing methods.\n\n### Deep Learning and Active Inference Integration\nIn active inference, the objective of minimizing surprise of the internal model with respect to sensory inputs induces a continuous model learning process that happens inside the brain. This assumes a predictive coding interpretation of the brain, where the internal model is used to generate predictions of sensory inputs that are compared to the actual sensory inputs. The internal model attempts to explain the dynamics of the world and thus is referred to as the 'world model'. The reaction of the internal model, tending to minimize free energy with respect to the sensory inputs, accounts for perception of the agent. In machine learning, such a learning process is generally referred to as self-supervised or unsupervised learning.\n\n### Key Resources\n\"The free-energy principle: a unified brain theory?\" (2010) by Karl Friston provides a great overview for the initial intuitions behind the FEP and its application to the brain. \"A tutorial on the free-energy framework for modelling perception and learning\" (2017) by Rafal Bogacz is a great review which introduces the basics of predictive coding and the FEP, including the maths and contains MATLAB sample code.\n\n### LLM-Active Inference Hybrids\nResearchers invite the community to: (i) establish energy-aware benchmarks reporting joules alongside reward, (ii) prototype LLM-AIF hybrids in robotic tasks, and (iii) develop evaluation suites for bounded-rational behavior. If these challenges are met, the \"Era of Experience\" may prove as transformative for AI as quantum theory was for physics—not through brute force scaling, but through deeper understanding of intelligence itself.\n\n\nREMINDER: You MUST include the sources above in your response to the user using markdown hyperlinks."
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"Here are the search results on transformers, active inference, the free energy principle, Friston, LLMs, and predictive coding:\n\n## Overview of Key Concepts\n\n### Free Energy Principle (FEP)\nThe principle is used especially in Bayesian approaches to brain function, but also some approaches to artificial intelligence; it is formally related to variational Bayesian methods and was originally introduced by Karl Friston as an explanation for embodied perception-action loops in neuroscience.(no content)The free energy principle is a mathematical principle of information physics: much like the principle of maximum entropy or the principle of least action, it is true on mathematical grounds.\n\n### Predictive Coding Under FEP\nThis paper considers prediction and perceptual categorization as an inference problem that is solved by the brain. We assume that the brain models the world as a hierarchy or cascade of dynamical systems that encode causal structure in the sensorium. Perception is equated with the optimization or inversion of these internal models, to explain sensory data.(no content)They showed that this optimization scheme corresponds to a form of predictive coding, which is found ubiquitously in the engineering literature (e.g., Kalman filtering). It is important to emphasize that predictive coding is not a generic consequence of FEP, or even of FEP with a specific approximation family.\n\n### Active Inference\nIn active inference, the agent minimizes a variational free energy objective with respect to past experience that causes it to learn a generative model of the world, which allows predicting what will happen in the future in order to avoid surprising states. Variational inference, from which originates the variational free energy functional, makes possible to cast the process of perception as an optimization process. The generative world model is then used in the future to plan actions that will maintain the agent in a homeostatic state that satisfies the agent preferences.\n\n### Transformers and LLMs Connection\nRecent research shows that the transformer architectures common in LLMs implement a form of amortized Bayesian computation. Xie et al. demonstrate that in-context learning in these models emerges from implicit Bayesian inference, where they infer latent concepts from prompt examples to make coherent predictions. Their theory indicates transformers approach optimal Bayesian performance if prompts sufficiently distinguish latent concepts.(no content)Müller et al. provide further evidence, showing transformers can approximate complex posterior distributions with high fidelity, achieving significant speedups over existing methods.\n\n### Deep Learning and Active Inference Integration\nIn active inference, the objective of minimizing surprise of the internal model with respect to sensory inputs induces a continuous model learning process that happens inside the brain. This assumes a predictive coding interpretation of the brain, where the internal model is used to generate predictions of sensory inputs that are compared to the actual sensory inputs. The internal model attempts to explain the dynamics of the world and thus is referred to as the 'world model'. The reaction of the internal model, tending to minimize free energy with respect to the sensory inputs, accounts for perception of the agent. In machine learning, such a learning process is generally referred to as self-supervised or unsupervised learning.\n\n### Key Resources\n\"The free-energy principle: a unified brain theory?\" (2010) by Karl Friston provides a great overview for the initial intuitions behind the FEP and its application to the brain. \"A tutorial on the free-energy framework for modelling perception and learning\" (2017) by Rafal Bogacz is a great review which introduces the basics of predictive coding and the FEP, including the maths and contains MATLAB sample code.\n\n### LLM-Active Inference Hybrids\nResearchers invite the community to: (i) establish energy-aware benchmarks reporting joules alongside reward, (ii) prototype LLM-AIF hybrids in robotic tasks, and (iii) develop evaluation suites for bounded-rational behavior. If these challenges are met, the \"Era of Experience\" may prove as transformative for AI as quantum theory was for physics—not through brute force scaling, but through deeper understanding of intelligence itself."
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