k-hyperarc consistency for soft constraints over divisible residuated lattices

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 [GJKO07, Háj98]. We present the following results. (i) We show that DRLs subsume important valuation structures for soft constraints, such as commutative idempotent semirings [BMR97] and fair valuation structures [CS04], 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 [LS04, BG06], 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.