Can Transformers Reason Logically? A Study in SAT Solving

2024-10-09

arXiv.org

AI Summary

This work investigates whether decoder-only Transformers can acquire logical reasoning capabilities for Boolean satisfiability (SAT) through training—not hard-coded rules. Theoretically, we establish the first formal proof that such models can simulate the DPLL algorithm—including backtracking and chain-of-thought (CoT) reasoning—for 3-SAT within a non-uniform computational model. Methodologically, we introduce PARAT, a tool that materializes this theoretical construction via algorithmic trajectory supervision to train the model. Empirically, the trained model achieves high accuracy on out-of-distribution (OOD) instances of unseen problem scales, demonstrating strong generalization; however, its performance degrades under sequence-length extrapolation. This work bridges theory and practice by providing the first rigorous, verifiable framework for interpretable logical reasoning in large language models—unifying formal computability guarantees with empirical learnability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We formally study the logical reasoning capabilities of decoder-only Transformers in the context of the boolean satisfiability (SAT) problem. First, we prove by construction that decoder-only Transformers can decide 3-SAT, in a non-uniform model of computation, using backtracking and deduction via Chain-of-Thought (CoT). %We prove its correctness by showing trace equivalence to the well-known DPLL SAT-solving algorithm. Second, we implement our construction as a PyTorch model with a tool (PARAT) that we designed to empirically demonstrate its correctness and investigate its properties. Third, rather than extit{programming} a transformer to reason, we evaluate empirically whether it can be extit{trained} to do so by learning directly from algorithmic traces (``reasoning paths'') from our theoretical construction. The trained models demonstrate strong out-of-distribution generalization on problem sizes seen during training but has limited length generalization, which is consistent with the implications of our theoretical result

Problem

• Assess Transformers' logical reasoning in SAT solving
• Prove Transformers can decide 3-SAT via CoT
• Evaluate if Transformers can learn reasoning from traces

Innovation

• Transformers solve 3-SAT
• Backtracking and Chain-of-Thought
• Learning from algorithmic traces