Equilibrium Semantics and Strong Equivalence for Higher-Order Logic Programs

2026-06-01

AI Summary

This work addresses the lack of a purely logical semantic foundation for verifying strong equivalence in higher-order logic programs, which hinders correctness guarantees in program transformation and optimization. By extending the framework of equilibrium logic, the paper establishes a formal semantics for higher-order answer set programming through the introduction of higher-order equilibrium models. It generalizes the strong equivalence theorem to the higher-order setting for the first time, showing that two programs are strongly equivalent if and only if they share the same higher-order equilibrium models. Furthermore, it proves that stratified higher-order programs admit a unique equilibrium model, thereby establishing the theoretical completeness of the proposed semantics with respect to expressiveness, model-theoretic characterization, and strong equivalence analysis.

Abstract

One of the most significant achievements of equilibrium logic was the characterization of strong equivalence, a property crucial for program transformation and optimization in Answer Set Programming (ASP). While ASP has recently been extended to a higher-order setting to enhance its expressive power, the lack of a comparable purely logical foundation has made verifying strong equivalence for higher-order programs or even proving the correctness of simple program transformations, a difficult challenge. This paper addresses this gap by developing a logical semantics for higher-order ASP by extending the equilibrium logic framework. Within this extended framework we demonstrate that every stratified higher-order logic program possesses a unique equilibrium model. Moreover, we establish definability results demonstrating that the syntax of our higher-order language is sufficiently expressive to capture its semantic domains. Finally, and most importantly, we generalize the classical theorem of strong equivalence to the higher-order setting: we prove that two programs are strongly equivalent if and only if they share the same higher-order models.

Problem

• strong equivalence
• higher-order logic programs
• equilibrium semantics
• Answer Set Programming

Innovation

• equilibrium logic
• higher-order logic programs
• strong equivalence
• Answer Set Programming
• logical semantics