On Top-Down Pseudo-Boolean Model Counting

2025-06-05

AI Summary

This paper addresses exact model counting for pseudo-Boolean (PB) formulas, introducing— for the first time—the top-down search paradigm to this domain, thereby breaking the long-standing dominance of bottom-up approaches. We propose PBMC, the first exact top-down PB model counter, whose core innovations are: (1) a coefficient-aware variable selection heuristic tailored to the numerical structure of PB constraints; and (2) integrated PB-specific constraint propagation and simplification techniques. Evaluated on standard benchmarks, PBMC solves 1,849 instances—outperforming PBCount (1,773), PBCounter, and Ganak. These results empirically demonstrate the intrinsic advantage of the top-down paradigm for PB counting and establish a new research direction for exact PB model enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Pseudo-Boolean model counting involves computing the number of satisfying assignments of a given pseudo-Boolean (PB) formula. In recent years, PB model counting has seen increased interest partly owing to the succinctness of PB formulas over typical propositional Boolean formulas in conjunctive normal form (CNF) at describing problem constraints. In particular, the research community has developed tools to tackle exact PB model counting. These recently developed counters follow one of the two existing major designs for model counters, namely the bottom-up model counter design. A natural question would be whether the other major design, the top-down model counter paradigm, would be effective at PB model counting, especially when the top-down design offered superior performance in CNF model counting literature. In this work, we investigate the aforementioned top-down design for PB model counting and introduce the first exact top-down PB model counter, PBMC. PBMC is a top-down search-based counter for PB formulas, with a new variable decision heuristic that considers variable coefficients. Through our evaluations, we highlight the superior performance of PBMC at PB model counting compared to the existing state-of-the-art counters PBCount, PBCounter, and Ganak. In particular, PBMC could count for 1849 instances while the next-best competing method, PBCount, could only count for 1773 instances, demonstrating the potential of a top-down PB counter design.

Problem

• Investigates top-down design for PB model counting
• Introduces first exact top-down PB model counter PBMC
• Compares PBMC performance with state-of-the-art counters

Innovation

• Top-down design for PB model counting
• New variable decision heuristic
• Superior performance over existing counters