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

arXiv: https://arxiv.org/abs/2607.21201 · HTML: https://mlsanity.com/paper/2607.21201/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 17 references that are in the map.
- First posted on arXiv 23 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Clingo goes Linear Constraints over Reals and Integers](https://mlsanity.com/paper/1707.04053/) (2017)
- [Clingcon: The Next Generation](https://mlsanity.com/paper/1705.04569/) (2017)
- [Answer-Set Programming for Lexicographical Makespan Optimisation in Parallel Machine Scheduling](https://mlsanity.com/paper/2212.09077/) (2022)
- [Strong Equivalence in Answer Set Programming with Constraints](https://mlsanity.com/paper/2502.04302/) (2025)
- [Hybrid Metabolic Network Completion](https://mlsanity.com/paper/1808.04149/) (2018)

## Abstract (arXiv)

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.