k-Hyperarc Consistency for Soft Constraints over Divisible Residuated Lattices

TL;DR

This paper explores divisible residuated lattices as a unifying algebraic framework for soft constraint satisfaction problems, introduces a polynomial-time algorithm for enforcing k-hyperarc consistency within this framework, and generalizes existing algorithms.

Contribution

It demonstrates that DRLs encompass key valuation structures for soft constraints and provides a novel, generalized polynomial-time algorithm for enforcing k-hyperarc consistency over DRLs.

Findings

DRLs include important valuation structures like semirings and fuzzy logics.

A polynomial-time algorithm for k-hyperarc consistency over DRLs is proposed.

The algorithm generalizes previous methods by handling non-idempotent and non-totally ordered DRLs.

Abstract

We investigate the applicability of divisible residuated lattices (DRLs) as a general evaluation framework for soft constraint satisfaction problems (soft CSPs). DRLs are in fact natural candidates for this role, since they form the algebraic semantics of a large family of substructural and fuzzy logics. We present the following results. (i) We show that DRLs subsume important valuation structures for soft constraints, such as commutative idempotent semirings and fair valuation structures, in the sense that the last two are members of certain subvarieties of DRLs (namely, Heyting algebras and BL-algebras respectively). (ii) In the spirit of previous work of J. Larrosa and T. Schiex [2004], and S. Bistarelli and F. Gadducci [2006] we describe a polynomial-time algorithm that enforces k-hyperarc consistency on soft CSPs evaluated over DRLs. Observed that, in general, DRLs are neither idempotent nor totally ordered, this algorithm amounts to a generalization of the available algorithms that enforce k-hyperarc consistency.