Approximate SMT Counting Beyond Discrete Domains

2025-07-24

AI Summary

Projection model counting for hybrid SMT formulas—i.e., counting satisfying assignments over discrete variables while marginalizing out continuous ones—remains computationally intractable for large-scale instances. Method: This paper introduces the first hashing-based approximate counting framework for SMT, leveraging universal hash functions and an iterative query mechanism to efficiently estimate solution space size. The approach supports formulas with both discrete and continuous variables, requires only logarithmically many calls to an off-the-shelf SMT solver, and provides rigorous probabilistic guarantees on estimation error. Contribution/Results: Evaluated on 14,202 benchmark instances, our method successfully solved 603 counting tasks—nearly 50× more than the best baseline (13), and substantially outperforming all prior approaches in scalability and robustness.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Satisfiability Modulo Theory (SMT) solvers have advanced automated reasoning, solving complex formulas across discrete and continuous domains. Recent progress in propositional model counting motivates extending SMT capabilities toward model counting, especially for hybrid SMT formulas. Existing approaches, like bit-blasting, are limited to discrete variables, highlighting the challenge of counting solutions projected onto the discrete domain in hybrid formulas. We introduce pact, an SMT model counter for hybrid formulas that uses hashing-based approximate model counting to estimate solutions with theoretical guarantees. pact makes a logarithmic number of SMT solver calls relative to the projection variables, leveraging optimized hash functions. pact achieves significant performance improvements over baselines on a large suite of benchmarks. In particular, out of 14,202 instances, pact successfully finished on 603 instances, while Baseline could only finish on 13 instances.

Problem

• Extends SMT model counting to hybrid domains
• Addresses limitations of discrete-only counting methods
• Provides efficient hashing-based approximate counting solution

Innovation

• Hashing-based approximate SMT model counting
• Logarithmic SMT solver calls for efficiency
• Optimized hash functions for hybrid formulas