SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver (1905.12149v1)

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.

• The paper introduces a differentiable MAXSAT solver that bridges deep learning and logical reasoning.
• It uses an SDP relaxation with coordinate descent to enable efficient backpropagation through logical constraints.
• Experiments demonstrate SATNet’s effectiveness in challenging tasks like parity learning and Sudoku solving.

SATNet: Bridging Deep Learning and Logical Reasoning with a Differentiable Satisfiability Solver

The paper "SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver" addresses a fundamental challenge in artificial intelligence: integrating logical reasoning capabilities with deep learning architectures. The authors propose SATNet, a layer designed to incorporate a differentiable maximum satisfiability (MAXSAT) solver within neural network models. This work builds on the need for AI systems that can handle discrete logical constraints commonly encountered in symbolic reasoning tasks.

Overview and Methodology

The novelty of this paper lies in its use of a differentiable solver for the MAXSAT problem, facilitating the end-to-end learning of logical structures. The approach leverages a semidefinite programming (SDP) relaxation of MAXSAT, employing a fast coordinate descent method to solve the SDP efficiently. This integration allows for analytical differentiation through the SDP solution, making it tractable within the backpropagation framework of neural networks.

Key elements of SATNet include:

• Differentiable SDP Relaxation: The use of an SDP relaxation provides a smooth, differentiable framework for integrating discrete satisfiability problems into deep models.
• Efficient Computation: By implementing coordinate descent methods compatible with GPU acceleration, the authors achieve a scalable solution that maintains computational efficiency necessary for practical deep learning applications.

Empirical Results

The paper presents several experiments demonstrating SATNet's capability to learn logical tasks that are traditionally challenging for deep learning models:

• Parity Learning: SATNet effectively learns the parity function, a task known to be difficult due to its reliance on logical XOR operations. The model achieves zero test error quickly, outperforming LSTM baselines.
• Sudoku Solving: SATNet learns to solve $9 \times 9$ Sudoku puzzles, achieving 98.3% accuracy. The task's complexity stems from global constraints requiring integrated logical reasoning, which SATNet successfully models without explicit rule encoding.
• Visual Sudoku: The paper extends SATNet to handle visual inputs by integrating a convolutional network for digit recognition with the SATNet layer for logical solving, showcasing successful end-to-end learning from images to logical solutions.

Implications and Future Directions

The integration of SATNet into neural networks signifies a promising direction for embedding logical reasoning into machine learning frameworks. This capability could enhance AI applications in areas such as automated reasoning, scheduling, and complex decision-making where hard constraints are present.

Future research may explore:

• Generalization to Other Logical Functions: Expanding SATNet's utility to other complex logical tasks.
• Model Complexity and Scalability: Investigating ways to optimize computational requirements while extending SATNet to larger-scale problems.
• Interdisciplinary Applications: Evaluating SATNet's potential in domains requiring a blend of symbolic reasoning and pattern recognition, such as robotics and cognitive computing.

In conclusion, SATNet represents a significant step in bridging logical reasoning with deep learning architectures, broadening the scope of problems solvable by neural networks through the incorporation of differentiable logical layers. By addressing the computational challenges and demonstrating efficacy on logical tasks, this paper contributes valuable insights and tools towards more cognitively robust AI systems.