Learning a SAT Solver from Single-Bit Supervision (1802.03685v4)

Abstract: We present NeuroSAT, a message passing neural network that learns to solve SAT problems after only being trained as a classifier to predict satisfiability. Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve problems that are substantially larger and more difficult than it ever saw during training by simply running for more iterations. Moreover, NeuroSAT generalizes to novel distributions; after training only on random SAT problems, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs.

• The paper introduces NeuroSAT, an MPNN that learns SAT solving from single-bit supervision by predicting problem satisfiability.
• It achieves an overall test accuracy of 85% and decodes satisfying assignments from internal activations for 70% of satisfiable instances.
• NeuroSAT generalizes to larger and complex SAT instances, including SAT encodings of combinatorial graph problems, by extending its message passing iterations.

NeuroSAT: A Message Passing Neural Network for Solving SAT Problems

The paper "Learning a SAT Solver from Single-Bit Supervision", presented at ICLR 2019 by Daniel Selsam et al., introduces NeuroSAT, a novel Message Passing Neural Network (MPNN) designed to solve SAT problems. The approach employs a neural network framework to predict the satisfiability of SAT problems, trained solely with single-bit supervision.

Core Contributions and Methodology

The central contribution of this work is the development of NeuroSAT, an MPNN which serves remarkably as a SAT solver when trained to classify random SAT instances as satisfiable or unsatisfiable. Unlike traditional SAT solvers that rely on backtracking search, NeuroSAT exemplifies how neural networks can learn to navigate the solution space heuristically. This neural network-based approach accomplishes problem-solving even for instances significantly larger than those encountered during its training phase.

The architecture of NeuroSAT incorporates permutation and negation invariance crucial for working on SAT problems. SAT, being NP-complete, often involves encoding search problems where certifiable solutions require checking within polynomial time boundaries. NeuroSAT demonstrates adaptability beyond its training distribution, solving SAT transformations of combinatorial graph problems such as graph coloring and vertex cover.

Performance Evaluation

NeuroSAT was rigorously evaluated and demonstrated a test accuracy of 85% in predicting the satisfiability of SAT instances, with a noteworthy 96% accuracy on unsatisfiable instances. It was observed through experimentation that NeuroSAT effectively decodes satisfying assignments from its internal activations for about 70% of satisfiable problems, indicating that it has learned a form of heuristic search.

In an impressive extrapolation, NeuroSAT, despite being trained on small-scale problems from a specific distribution SR(U(10;40)), succeeds in solving larger and more complex instances from SR(n) by merely increasing the iterations of the message passing process. For example, for instances from SR(200), it can solve 25% of the cases, notable given their relative complexity compared to the training set.

Extensions and Generalization

The potential for generalization is further illustrated by NeuroSAT's capability to solve SAT encodings of various combinatorial graph problems generated from diverse random graph distributions. This cross-domain applicability underscores the abstract procedural knowledge NeuroSAT harnesses.

Additionally, the paper explores the construction of a variant model, NeuroUNSAT, tailored using a dataset where every unsatisfiable problem contains a pre-determined unsatisfiable core. NeuroUNSAT learns to identify such cores, providing insights into developing proof structures for unsatisfiability—though this methodology might not generalize beyond memorized subgraph patterns.

Implications and Future Work

The implications of this work are significant both theoretically and practically. It challenges the notion that neural networks inherently lack the precision required for logical reasoning tasks, showcasing an autonomous learning of search algorithms. Theoretically, NeuroSAT adds to our understanding of neural networks’ capability to handle complex decision problems.

Practically, while NeuroSAT itself is not benchmark-invoking when compared to state-of-the-art SAT solvers, its innovative approach points to potential enhancements in hybrid solver architectures, where neural networks augment traditional methods. Future work might focus on integrating such learned models with classical heuristics to improve robustness and efficiency.

This paper contributes to the evolving narrative of neural networks’ applicability in procedural problem solving and sets the stage for exploring broader spectrum cognitive tasks in artificial intelligence, promising intriguing directions for subsequent research.