Tractable Set Constraints

Abstract

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tractable. We introduce a large class of set CSPs that can be solved in quadratic time. Our class, which we call E I, contains all previously known tractable set CSPs, but also some new ones that are of crucial importance for example in description logics. The class of E I set constraints has an elegant universal-algebraic characterization, which we use to show that every set constraint language that properly contains all E I set constraints already has a finite sublanguage with an NP-hard constraint satisfaction problem.An extended abstract of the content of this article appeared in the proceedings of IJCAI'11 [BHK11].

Algebraic and model theoretic methods in constraint satisfaction

“…The solution to the following would subsume a considerable amount of results in the literature, e.g., completely the papers [30], [6], and some results in [19], [35]. It would moreover require the extension of our methods to functional signatures, a venture interesting in itself.…”

Agreeing to disagree: Reconciling conflicting taxonomic views using a logic‐based approach

“…This fragment is also closely related to set constraints and monadic first-order logic (Bachmair, Ganzinger & Waldmann, 1993;Bodirsky & Hils, 2012). RCC-5 is the variant of the RCC family used in Euler/X, consisting of five pairwise disjoint and mutually exclusive relations (Figure 2), i.e.…”

“…RCC is used for qualitative (often, but not necessarily spatial) representation and reasoning and can be seen as a decidable fragment of first-order predicate logic (Cohn & Renz, 2008). This fragment is also closely related to set constraints and monadic first-order logic (Bachmair, Ganzinger & Waldmann, 1993;Bodirsky & Hils, 2012). RCC-5 is the variant of the RCC family used in Euler/X, consisting of five pairwise disjoint and mutually exclusive relations (Figure 2), i.e., congruence (equals or "==" in Euler/X-generated figures), proper inclusion (includes or ">" in Euler/X), inverse proper inclusion (is_included_in or "<"), overlap (overlaps or "><"), and disjointness (disjoint or "!")…”

An initial study of time complexity in infinite-domain constraint satisfaction

“…Early examples of disjunctive constraints over infinite-domains can be found in, for instance, temporal reasoning [44,38,59], reasoning about action and change [26], and deductive databases [42]. More recent examples include interactive graphics [49], rule-based reasoning [47], and set constraints (with applications in descriptive logics) [10]. There are also works studying disjunctive constraints from a general point of view [16,21] but they are only concerned with the separation of polynomial cases from NP-hard cases, and do not further investigate the time complexity of the hard cases.…”