SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver

TL;DR

This paper introduces a differentiable MAXSAT solver integrated into deep learning systems, enabling learning of logical structures like parity and Sudoku from minimal supervision, bridging logical reasoning and neural networks.

Contribution

The paper presents a novel differentiable MAXSAT solver based on SDP and coordinate descent, allowing end-to-end learning of logical reasoning tasks within deep networks.

Findings

Successfully learned parity function with single-bit supervision

Achieved Sudoku solving from examples using integrated MAXSAT solver

Mapped Sudoku images to solutions combining CNNs with logical reasoning

Abstract

Integrating logical reasoning within deep learning architectures has been a major goal of modern AI systems. In this paper, we propose a new direction toward this goal by introducing a differentiable (smoothed) maximum satisfiability (MAXSAT) solver that can be integrated into the loop of larger deep learning systems. Our (approximate) solver is based upon a fast coordinate descent approach to solving the semidefinite program (SDP) associated with the MAXSAT problem. We show how to analytically differentiate through the solution to this SDP and efficiently solve the associated backward pass. We demonstrate that by integrating this solver into end-to-end learning systems, we can learn the logical structure of challenging problems in a minimally supervised fashion. In particular, we show that we can learn the parity function using single-bit supervision (a traditionally hard task for deep networks) and learn how to play 9x9 Sudoku solely from examples. We also solve a "visual Sudok" problem that maps images of Sudoku puzzles to their associated logical solutions by combining our MAXSAT solver with a traditional convolutional architecture. Our approach thus shows promise in integrating logical structures within deep learning.