Optimising Cylindrical Algebraic Coverings for use in SMT by Solving a Set Covering Problem with Reasons

2026-01-20

AI Summary

This work addresses the inefficiency of conflict clause minimization in SMT solvers for nonlinear real arithmetic under the CDCAC algorithm. To this end, it introduces—for the first time—the notion of a reason-based set cover problem into the CDCAC framework, constructing such a problem at each coverage generalization step to identify the smallest subset of constraints that covers all unsatisfiable constraint labels. The authors propose a tailored data reduction technique, extending Beasley’s reduction, along with an exact solver based on linear programming. Experimental evaluation on the SMT-LIB benchmark suite demonstrates that the approach efficiently handles the majority of instances and significantly enhances solving performance for nonlinear real arithmetic problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Conflict-Driven Cylindrical Algebraic Covering algorithm has proven well suited for performing theory validation checks in the satisfiability modulo theories paradigm for non-linear real arithmetic. CDCAC repurposes the theory underpinning classical cylindrical algebraic decomposition for SMT solving and is implemented in the SMT solvers cvc5 and SMT-RAT, as well as the computer algebra system Maple. It was previously observed that when using cylindrical algebraic decomposition for an SMT theory call, the output can be optimised by solving a single set covering problem instance that minimises the conflict clause. In this paper we consider the corresponding optimisation for CDCAC and observe that CDCAC naturally gives rise to multiple such optimisations within a single call. Each time a covering is generalised in one dimension, the resulting cell in the next dimension is labelled with theory constraints that cannot be satisfied together. We seek the smallest subset of constraints whose union covers all labels from the cells in the current covering. We call this optimisation problem a set covering problem with reasons. To simplify this problem, we introduce a data reduction step that generalises Beasley reduction for the classical set covering problem and show that this step alone solves many of the instances arising from SMT-LIB benchmarks. We then propose an exact solver based on linear programming to efficiently solve the remaining cases. Integrating these techniques into CDCAC has the potential to significantly improve SMT solver performance for non-linear real arithmetic problems.

Problem

• Cylindrical Algebraic Coverings
• SMT
• Set Covering Problem
• Non-linear Real Arithmetic
• Conflict-Driven

Innovation

• Cylindrical Algebraic Covering
• Set Covering Problem with Reasons
• Conflict-Driven SMT
• Non-linear Real Arithmetic
• Beasley Reduction