#01Nov 12, 2026
cs.AI
Classical Energies Learn to Reason 1,000x Faster than Looped Models
Benjamin Hoover, Hugh Blayney, James C. R. Whittington
Looped models have emerged as a prominent paradigm for reasoning, with successes on simple reasoning puzzles such as Sudoku-Extreme and Maze-Unique. However, these models can be unstable, slow to train, and use many parameters and FLOPs to solve simple puzzles. Descent on an energy not only mitigates these issues but also offers advantages: built-in early stopping, compute that adapts to problem difficulty, fast training without differentiating through a solver, and interpretable weights that learn the constraints of the puzzle. Here we demonstrate this with a learned version of Hopfield & Tank's energy from the 80s, which we term CRAM, solves 99.996% of Sudoku-Extreme and 100% of Maze-Unique test puzzles after 42 ms and 132 ms of training on one GPU, at least 1,000x faster than looped models on these tasks. CRAM also matches baseline looped models at their highest accuracy on Sudoku-Extreme using at least 282x fewer inference FLOPs. Our work suggests two things. First, that energies are an extremely promising direction for training reasoning from data. Second, a caution against using trivially solvable benchmarks like Sudoku and Maze to test the scalability of looped reasoning techniques.
#02Nov 12, 2026
cs.IR
Sheaf-Inspired Structured Scoring Extensions of LightGCN for Collaborative Filtering
Ksenia Belkovskaya, Timur Lidzhiev, Maria Stebliy and 1 more
Graph-based collaborative filtering learns user and item representations by propagating embeddings over a user--item interaction graph. We investigate a modular, sheaf-inspired scoring extension placed on top of a three-layer LightGCN backbone. The motivation comes from cellular sheaves, where local vector spaces are related by linear restriction maps; here, this idea motivates structured, pair-dependent transformations of user and item embeddings rather than a change to graph propagation. We compare the identity inner-product scorer, diagonal, rank-one low-rank, and full matrix residual scorers, and a parameter-matched residual MLP baseline (\texttt{mlp\_full}) on Amazon-Book, Gowalla, MovieLens-1M, and Yelp2018. We report ranking quality and catalog coverage at cutoffs 10 and 20. The selected structured variants show modest, dataset-dependent changes in ranking quality and sometimes substantial changes in coverage. Our method is inspired by sheaf geometry but does not construct a cellular sheaf or implement sheaf-Laplacian diffusion; the experiments therefore do not establish a causal benefit from sheaf geometry itself.
#03Nov 12, 2026
cs.CL
Goal Encoding in LLMs
Arnab Sen Sharma, Eric Todd, Sheridan Feucht and 7 more
We investigate whether and how LLMs encode goals: internal neural representations that precede and guide action, and that are flexible enough to be realized through many different execution paths. We construct a coding setting with multiple sub-tasks that allows us to evaluate goal-seeking behavior and the selection of sub-goals. In this setting, we localize an encoding of the next sub-goal in the outputs of a sparse set of attention heads. The same heads can be used to decode the LM's future actions in natural language (e.g., "I'm checking if a word is a palindrome"). We verify that this encoding is causal for next action and is plastic: transplanting it into a different environment, e.g., another programming language, triggers the same operation realized with a different sequence of tokens. The LM also sustains the encoding throughout the action's execution, which is puzzling because a transformer LM without recurrence cannot carry a state forward for more tokens than it has layers left. We propose that it does not need to. Under a non-recurrent goal-seeking (NRGS) account, the LM frequently re-reads the goal from the task specification, compares it with the progress visible in the growing context, and pursues what remains. We test the predictions of this account with targeted causal interventions.
#04Nov 12, 2026
cs.AI
Al-Ghasali: A Computational Framework for Simulating Islamic Jurisprudential Rulings via Finite State Automata and Large Language Models
Elnaserledinellah Mahmoud Abdelwahab
This paper presents Al Ghasali, a computational framework designed to formalize and simulate Islamic jurisprudential (Fiqh) rulings. Building upon recent set theoretic formalizations of al sabr wa al Taqsim that extract minimal operational rules as simple Horn clause logical programs, this work proves that the semantics of closed Fiqh chapters collapse completely to deterministic Finite State Automata (FSA). We establish a formal computational hierarchy, demonstrating that restricting formalization to finite propositional Horn theories guarantees decidability and constant time (O(1)) verdict extraction after offline compilation. The framework formally proves that hierarchical compositions of sub-chapter FSAs preserve this O(1) complexity, and that temporal juristic requirements, such as Tartib (order), are deterministically flattened into static ontological states, preserving the memoryless property of the automaton. To generate human-readable juristic summaries, Al Ghasali incorporates Large Language Models (LLMs) via an architecture where the deterministic FSA strictly controls the probabilistic LLM. Crucially, the paper formulates sufficient design safety conditions for this FSA LLM integration, ensuring strict adherence to the epistemological boundaries of Usul al Fiqh. To neutralize the risk of LLM hallucination, the system employs a Grammar Guided Constrained Decoding mechanism. We formally define the mapping function that dynamically constructs unambiguous right linear grammars from the FSA execution history. The paper mathematically proves that this architecture, augmented by a Bounded Neutral Lexicon and Valence Locking, guarantees the absolute absence of Decisional Hallucination, Juristic Factual Hallucination, and Semantic Inversion reducing the LLM to a pure linguistic transformer of deterministic juristic facts
#05Nov 12, 2026
cs.LG
Towards a Theory of Learnability via Debate
Tosca Lechner, Vinayak Pathak, Sivan Sabato and 1 more
Debate has been recently proposed as a training protocol that is hoped to produce more aligned models even as model capabilities scale. The idea is that the human, instead of providing supervision in the form of labels, judges a debate between two models that compete to prove the correctness of opposite labels. It can be shown that functions of a much higher computational complexity can be correctly judged in this way, compared to the complexity of functions that can be directly evaluated using a computationally weak evaluator. This suggests that perhaps debate can be used to teach very complicated functions to models. However, previous work has largely ignored the fact that this is a training protocol, and simply analyzes the game being played between the two debaters for a known judge. We take the first steps towards understanding the learning problem implicit within the protocol, by formulating it as an online learning problem. We show that once we formulate the learning problem precisely, there are cases where the best way to learn given access to a judge is different from running an adversarial debate. This motivates us to define a cooperative version of the protocol, whose learnability we characterize using a combinatorial dimension. We also consider the possibility of the human judge being fallible and occasionally judging the debate incorrectly. We show that a small number of errors in judgement can lead to a large error in learning.