2-ASP(Q) programs with weak constraints: Complexity and efficient implementation

2026-05-26

AI Summary

This work investigates the computational complexity and efficient solving techniques for second-order Quantified Answer Set Programming with weak constraints (2-ASP(Q)ʷ). It provides the first complete characterization of the complexity of its primary reasoning tasks within the Δ₃^P level of the polynomial hierarchy. The paper introduces a Counterexample-Guided Abstraction Refinement (CEGAR) solving strategy specifically tailored to the semantics of ASP(Q), effectively addressing non-trivial cases not handled by prior approaches. Implemented in the Casper system, the proposed method demonstrates significant performance improvements over existing solvers on multiple challenging benchmark instances, thereby confirming its effectiveness and practical applicability.

Abstract

ASP(Q) extends Answer Set Programming (ASP) with Quantifiers over answer sets. In this paper we focus on the class of ASP(Q) programs with two quantifiers and weak constraints, denoted as 2-ASP(Q)^w. 2-ASP(Q)^w is a practically relevant fragment of ASP(Q) that is expressive enough to capture optimization problems up to the class Delta_3^P. On the theoretical side, we provide a complete complexity characterization of the main computational tasks for 2-ASP(Q)^w programs, including tight completeness results and the analysis of nontrivial cases that have not been addressed in previous works. On the practical side, we introduce novel strategies for computing (optimal) quantified answer sets in the Casper system, that rely on a Counterexample-Guided Abstraction Refinement (CEGAR) technique tailored to ASP(Q). An experimental evaluation on hard benchmarks from different application domains shows that the proposed techniques are effective in practice.

Problem

• ASP(Q)
• weak constraints
• complexity
• quantified answer sets
• optimization problems

Innovation

• ASP(Q)
• weak constraints
• CEGAR
• quantified answer sets
• Delta_3^P