An Exact Solver for Satisfiability Modulo Counting with Probabilistic Circuits

2025-03-02

AI Summary

Existing Satisfiability Modulo Counting (SMC) solvers suffer from either a lack of formal correctness guarantees (approximate solvers) or prohibitively low efficiency due to frequent interleaving of SAT solving and probabilistic inference (exact solvers). Method: We propose KOCO-SMC—the first tightly integrated, exact SMC solver enabling synergistic statistical and symbolic AI reasoning. It unifies SAT solving, probabilistic circuit (PC) inference, and interval propagation within a single framework. Contribution/Results: Its key innovations include (1) the first dynamic upper- and lower-bound tracking mechanism in probabilistic inference, enabling early probability estimation from partial variable assignments—breaking dependence on full assignments; and (2) deep integration of logical and statistical reasoning components. Experiments on large-scale benchmarks and real-world applications demonstrate that KOCO-SMC significantly outperforms state-of-the-art approximate and exact SMC solvers, achieving strict formal correctness while improving runtime efficiency by several orders of magnitude.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Satisfiability Modulo Counting (SMC) is a recently proposed general language to reason about problems integrating statistical and symbolic artificial intelligence. An SMC formula is an extended SAT formula in which the truth values of a few Boolean variables are determined by probabilistic inference. Existing approximate solvers optimize surrogate objectives, which lack formal guarantees. Current exact solvers directly integrate SAT solvers and probabilistic inference solvers resulting in slow performance because of many back-and-forth invocations of both solvers. We propose KOCO-SMC, an integrated exact SMC solver that efficiently tracks lower and upper bounds in the probabilistic inference process. It enhances computational efficiency by enabling early estimation of probabilistic inference using only partial variable assignments, whereas existing methods require full variable assignments. In the experiment, we compare KOCO-SMC with currently available approximate and exact SMC solvers on large-scale datasets and real-world applications. Our approach delivers high-quality solutions with high efficiency.

Problem

• Develops an exact solver for Satisfiability Modulo Counting (SMC).
• Improves efficiency by tracking probabilistic inference bounds.
• Compares KOCO-SMC with existing solvers on large datasets.

Innovation

• Integrated exact SMC solver KOCO-SMC
• Efficient probabilistic inference bounds tracking
• Early estimation with partial variable assignments