Narrowing the gap between Set-Constraints and CLP (SET)-Constraints

Abstract We compare two (apparently) rather di erent set-based constraint languages, and we show that, in spite of their di erent origins and aims, there are large classes of constraint formulae for which both proposals provide suitable procedures for testing constraint satisability with respect to a given privileged interpretation.

2011

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.

2018

The language {log} is a Constraint Logic Programming language that natively supports finite sets and constraints such as (non) equality and (non) membership. The set constraints resolution process is mathematically formalised by Dovier et al in [5] using rewriting rules. In this paper we present a formalisation in the Coq proof assistant of the term and constraint algebra, the rewriting rules and check all the examples given in the reference paper by applying the rewriting rules manually with the help of some tailored tactics. The main problem we encountered is the non-determinism captured by the rewriting rules, which prevents us from automating their application in Coq. However the rules for nonmembership and set checking are deterministic. So we propose a function that iteratively applies the latter rules. We prove its correctness with respect to the corresponding rewriting rules. This work is a first step of a larger project whose objective is to provide a formally verified reso...

Constraints, 2000

Set constraints (SC) are logical formulae in which atoms are inclusions between set expressions. Those set expressions are built over a signature , variables and various set operators. On a semantical point of view, the set constraints are interpreted over sets of trees built from and the inclusion symbol is interpreted as the subset relation over those sets. By restricting the syntax of those formulae and/or the set of operators that can occur in set expressions, different classes of set constraints are obtained. Several classes have been proposed and studied for some problems such as satisfiability and entailment. Among those classes, we focus in this article on the class of definite SC's introduced by Heintze and Jaffar, and the class of co-definite SC's studied by Charatonik and Podelski. In spite of their name, those two classes are not dual from each other, neither through inclusion inversion nor through complementation. In this article, we propose an extension for each of those two classes by means of an intentional set construction, so called membership expression. A membership expression is an expression {x | (x)}. The formula (x) is a positive first-order formula built from membership atoms t ∈ S in which S is a set expression. We name those two classes respectively generalized definite and generalized co-definite set constraints. One of the main point concerning those so-extended classes is that the two generalized classes turn out to be dual through complementation. First, we prove in this article that generalized definite set constraints is a proper extension of the definite class, as it is more expressive in terms of sets of solutions. But we show also that those extensions preserve some main properties of the definite and co-definite class. Hence for instance, as definite set constraints, generalized definite SC's have a least solution whereas the generalized co-definite SC's have a greatest solution, just as co-definite ones. Furthermore, we devise an algorithm based on tree automata that solves the satisfiability problem for generalized definite set constraints. Due to the dualization, the algorithm solves the satisfiability problem for generalized co-definite set constraints as well. This algorithm proves first that for those generalized classes, the satisfiability problem remains DEXPTIME-complete. It provides also a proof for regularity of the least solution of generalized definite constraints and so, by dualization for the greatest solution for the generalized co-definite SC's.

Journal of Logic Programming, 1998

The Constraint Logic Programming (CLP) Scheme was introduced by Jaar and Lassez. The scheme gave a formal framework, based on constraints, for the basic operational, logical and algebraic semantics of an extended class of logic programs. This paper presents for the ®rst time the semantic foundations of CLP in a self-contained and complete package. The main contributions are threefold. First, we extend the original conference paper by presenting de®nitions and basic semantic constructs from ®rst principles, giving new and complete proofs for the main lemmas. Importantly, we clarify which theorems depend on conditions such as solution compactness, satisfaction completeness and independence of constraints. Second, we generalize the original results to allow for incompleteness of the constraint solver. This is important since almost all CLP systems use an incomplete solver. Third, we give conditions on the (possibly incomplete) solver which ensure that the operational semantics is con¯uent, that is, has independence of literal scheduling. Ó 1998 Elsevier Science Inc. All rights reserved.

Proceedings of the 13th international ACM SIGPLAN symposium on Principles and practices of declarative programming - PPDP '11, 2011

This paper introduces and studies the notion of CLP projection for Constraint Handling Rules (CHR). The CLP projection consists of a naive translation of CHR programs into Constraint Logic Programs (CLP). We show that the CLP projection provides a safe operational and declarative approximation for CHR programs. We demónstrate moreover that a confluent CHR program has a least model, which is precisely equal to the least model of its CLP projection (closing henee a ten year-old conjecture by Abdenader et al.). Finally, we illustrate how the notion of CLP projection can be used in practice to apply CLP analyzers to CHR. In particular, we show results from applying AProVE to prove termination, and CiaoPP to infer both complexity upper bounds and types for CHR programs.

The Journal of Logic Programming, 1998

The Constraint Logic Programming (CLP) Scheme was introduced by Jaar and Lassez. The scheme gave a formal framework, based on constraints, for the basic operational, logical and algebraic semantics of an extended class of logic programs. This paper presents for the ®rst time the semantic foundations of CLP in a self-contained and complete package. The main contributions are threefold. First, we extend the original conference paper by presenting de®nitions and basic semantic constructs from ®rst principles, giving new and complete proofs for the main lemmas. Importantly, we clarify which theorems depend on conditions such as solution compactness, satisfaction completeness and independence of constraints. Second, we generalize the original results to allow for incompleteness of the constraint solver. This is important since almost all CLP systems use an incomplete solver. Third, we give conditions on the (possibly incomplete) solver which ensure that the operational semantics is con¯uent, that is, has independence of literal scheduling.

Lecture Notes in Computer Science, 2003

Lecture Notes in Computer Science, 2009

We develop a module-based framework for constraint modeling where it is possible to combine different constraint modeling languages and exploit their strengths in a flexible way. In the framework a constraint model consists of modules with clear input/output interfaces. When combining modules, apart from the interface, a module is a black box whose internals are invisible to the outside world. Inside a module a chosen constraint language (approaches such as CP, ASP, SAT, and MIP) can be used. This leads to a clear modular semantics where the overall semantics of the whole constraint model is obtained from the semantics of individual modules. The framework supports multi-language modeling without the need to develop a complicated joint semantics and enables the use of alternative semantical underpinnings such as default negation and classical negation in the same model. Furthermore, computational aspects of the framework are considered and, in particular, possibilities of benefiting from the known module structure in solving constraint models are studied.

Proceedings of the Ieee International Conference on Computer Systems and Applications, 2006

The past decade witnessed rapid development of constraint satisfaction technologies. More and more algorithms are now able to solve larger and harder problems. However, owing to the fact that constraints are inherently declarative, attention is quickly turning toward developing high-level programming languages within which such problems can be modeled and then solved. Along this direction, this paper presents DEPICT, the language. Its use is illustrated through modeling a couple of benchmark examples. The paper continues with a description of a prototype system within which such models may be interpreted. The paper concludes with a description of a sample run of the interpreter that shows how one such model is typically solved.