Abstract not found

When combining languages for symbolic constraints, one is typically faced with the problem of how to treat "mixed" constraints. The two main problems are (1) how to define a combined solution structure over which these constraints are to be solved, and (2) how to combine the constraint solving methods for pure constraints into one for mixed constraints. The paper introduces the notion of a "free amalgamated product" as a possible solution to the first problem. Subsequently, we define so-called simply-combinable structures (SC-structures). For SC-structures over disjoint signatures, a canonical amalgamation construction exists, which for the subclass of strong SC-structures yields the free amalgamated product. The combination technique of [BS92, BaS94a] can be used to combine constraint solvers for (strong) SC-structures over disjoint signatures into a solver for their (free) amalgamated product. In addition to term algebras modulo equational theories, the class of SC-stru...

We report on the extension of the concurrent constraint language Oz by constraints over finite sets of integers. Set constraints are an important addition to the constraint programming system Oz and are very employable in natural language processing and general problem solving. This extension profits much from its integration with the existing constraint systems over finite domains and feature trees, as well as from the availability of first-class procedures. This combination of features is unique to Oz. This paper focuses on the expressiveness gained by set constraints and on the benefits of the integration with finite domain constraints. A number of case studies demonstrates programming techniques exploring these advantages.

. The rst-order theories of lists, multisets, compact lists (i.e., lists where the number of contiguous occurrences of each element is immaterial), and sets are introduced via axioms. Such axiomatizations are shown to be very well-suited for the integration with free functor symbols governed by the classical Clark's axioms in the context of (Constraint) Logic Programming. Adaptations of the extensionality principle to the various theories taken into account is then exploited in the design of unication algorithms for the considered data structures. All the theories presented can be combined providing frameworks to deal with We acknowledge partial support from C.N.R. Grant 97.02426.CT12, C.N.R. project SETA, and from the MURST project \Tecniche formali per la specica, l'analisi, la verica, la sintesi e la trasformazione di sistemi software". 202 Dovier, Policriti, and Rossi / A uniform axiomatic view of lists, multisets, and sets several of the proposed data structures simultan...

Hereditarily finite sets and hypersets are characterized both as an algorithmic data structure and by means of a first-order axiomatization which, although rather weak, suffices to make the following two problems decidable: (1) Establishing whether a conjunction r of formulae of the form 8 y 1 \Delta \Delta \Delta 8 y m ((y 1 2 w 1 & \Delta \Delta \Delta & y m 2 wm ) ! q), with q quantifier-free and involving only the relators =; 2 and propositional connectives, and each y i distinct from all w j 's, is satisfiable. (2) Establishing whether a formula of the form 8 y q, q quantifier-free, is satisfiable. Concerning (1), an explicit decision algorithm is provided; moreover, significantly broad sub-problems of (1) are singled out in which a classification ---named the `syllogistic decomposition' of r--- of all possible ways of satisfying the input conjunction r can be obtained automatically. For one of these sub-problems, carrying out the decomposition results in producing a fi...

We review and compare the main techniques considered to represent finite sets in logic languages. We present a technique that combines the benefits of the previous techniques, avoiding their drawbacks. We show how to verify satisfiability of any conjunction of (positive and negative) literals based on =, ⊆, ∈, and ∪, ∩, \, and ||, viewed as predicate symbols, in a (hybrid) universe of finite sets. We also show that ∪ and | | (i.e., disjointness of two sets) are sufficient to represent all the above mentioned operations. 1

A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of most-general unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample problems that can be used as benchmarks for testing any set-unification algorithm. Based on this combinatorial study, a new Set-Unification Algorithm (named SUA) is also described and proved to be minimal for all the analyzed problems. Furthermore, an existing nave set-unification algorithm has also been tested to show its bad behavior for most of the sample problems.

Lists, multisets, and sets are well-known data structures whose usefulness is widely recognized in various areas of Computer Science. These data structures have been analyzed from an axiomatic point of view with a parametric approach in [12] where the relevant uni cation algorithms have been also parametrically developed. In this paper we extend these results considering more general constraints including not only equality but also membership constraints as well as their negative counterparts. This amounts to de ne the privileged structures for the considered axiomatic theories and to solve the relevant constraint satisfaction problems in each of the theories. We adopt a highly parametric approach which allows all the results obtained separately for each single theory to be easily combined so as to obtain a general framework where it is possible to deal with more than one data structure at a time.

The aim of this paper is to extend the Constructive Negation technique to the case of CLP (SET ), a Constraint Logic Programming (CLP ) language based on hereditarily (and hybrid) finite sets. The challenging aspects of the problem originate from the fact that the structure on which CLP (SET ) is based is not admissible closed, and this does not allow to reuse the results presented in the literature concerning the relationships between CLP and constructive negation. We propose a new constraint satisfaction algorithm, capable of correctly handling constructive negation for large classes of CLP (SET ) programs; we also provide a syntactic characterization of such classes of programs. The resulting algorithm provides a novel constraint simplification procedure to handle constructive negation, suitable to theories where unification admits multiple most general unifiers. We also show, using a general result, that it is impossible to construct an interpreter...

The goal of this paper is to provide a uniform overview of the unification problem in algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and it can find important applications in various research areas - e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The problem has been explored in depth, but the various solutions proposed are spread across a large literature, and some of the approaches have been ignored and/or rediscovered by different researchers. In this