Bound-Founded Semantics for Answer Set Programming with Difference Constraints: Preliminary Report

2026-07-23

AI Summary

This work addresses the lack of a unified logical semantic foundation in existing hybrid Answer Set Programming (ASP) solvers when handling linear constraints, which leads to inconsistent system behavior. It proposes a multi-sorted variant of Bound-founded Here-and-There logic (HTb), establishing for the first time a uniform semantic framework for ASP with difference constraints and formally defining the notion of foundedness for numeric variables. Grounded in equilibrium model theory and a semantic analysis of constraint atoms, the approach not only uncovers the root causes of semantic discrepancies among systems such as clingo[DL], clingcon, and flingo, but also provides a foundational platform capable of uniformly characterizing the semantics of diverse hybrid ASP systems. This framework further supports program simplification and facilitates the integration of future semantic principles.

Abstract

While the integration of linear constraints has significantly expanded the reach of Answer Set Programming (ASP), existing hybrid solvers often rely on disparate semantic underpinnings that lack a unified logical foundation. We address this gap by introducing a many-sorted variant of the Bound-founded Logic of Here-and-There (HTb), providing a versatile framework capable of characterizing equilibrium models across a wide spectrum of alternative semantics for extensions of ASP with linear constraints. We apply this framework to the setting of difference constraints, focusing on the semantic characterization of clingo[DL]. Central to our approach is the formalization of foundedness for numeric variables. By investigating how different hybrid systems - such as clingo[DL], clingcon, and flingo - justify constraint atoms, we uncover the semantic roots of their varying behaviors. This investigation results in a single, consistent framework that not only formalizes the foundations of current systems like clingo[DL] but also facilitates the rigorous study of program simplifications and the future integration of diverse semantic principles.

Problem

• Answer Set Programming
• Difference Constraints
• Semantic Foundation
• Hybrid Solvers
• Linear Constraints

Innovation

• Bound-founded semantics
• Answer Set Programming
• Difference constraints
• Here-and-There logic
• Foundedness