C. Gervet. Interval Propagation to Reason about Sets: De nition and Implementation of a Practical Language. Constraints, 1:191-246, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....language LISP, and list predicates, such as member and append, are among the rst predicates that are taught to students in logic programming language courses. Sets are the main data structure used in speci cation languages (e.g. in Z [26] and in high level declarative programming languages [5, 13, 19, 17]; but also imperative programming languages may take advantage from the set data abstraction (e.g. SETL [27] Multisets (often called bags in the literature) emerge as the most natural data structure in several interesting applications. For instance, solutions to the equation x 2x 1 = 0 ....

C. Gervet. Interval Propagation to Reason about Sets: De nition and Implementation of a Practical Language. Constraints, 1:191-246, 1997.

....theories describing the associative (A) commutative (C) and idempotent (I) nature of a function symbol. Constraints in the context of ACI theories (or similar theories for set like structures) have been shown to be very important from the theoretical as well as the practical point of view [18, 14, 13, 7]. The ultimate goal of our e ort is to develop a framework for handling ACI constraints which can be used in Constraint Logic Programming (CLP) The problem of handling positive and negative constraints under equational theories has been explored in the literature. In [8] a general solution to ....

....do not introduce disjunctions. Starting from a constraint made of disequations, the complexity of our approach seems to be more promising (e.g. O(n ) for the implicit solved form) and practical. In the context of CLP with sets, three major proposals have been presented in the literature. In [14], Gervet presents a language, called Conjunto, which incorporates a constraint solver over boolean lattices built from ( at) set intervals. The constraints can be more complex (e.g. boolean constraints) than those considered in this paper, but the domain isless general. In particular, the ....

C. Gervet. Interval Propagation to Reason about Sets: De nition and Implementation of a Practical Language. Constraints, 1:191-246, 1997.

....concerned with (di erent forms of) sets. Therefore, speci c CLP languages dealing with sets can be obtained by simply considering the instances of the general CLP scheme based on these constraint domains: CLP (SC) is the instance based on SC, CLP(SET ) that based on SET . CLPS [32] and Conjunto [23] are other two examples of CLP languages with sets based on quite di erent notions of set constraints. Being instances of the general CLP scheme, CLP (SC) and CLP(SET ) inherit from this scheme all its general features. In particular, the syntactic form of a program, as well as the semantics of ....

....are certain application areas in which the use of sets ts more naturally. These areas include database applications (see for instance [31] combinatorial problems, graph related applications and operational research in general (e.g. resource allocation problems) as pointed out for instance in [23, 32, 10]. A number of simple examples in some of these areas using CLP(SET ) can be found in [16] 9 Conclusions We have presented two di erent kinds of set based constraints: SI constraints (based on set inclusion and using the powerset of T as its domain) and SM constraints (based on membership and ....

C. Gervet. Interval Propagation to Reason about Sets : De nition and Implementation of a Practical Language. International Journal of Constraints 1, 191-246, 1997.

....distribution, which solve dominance constraints with set operators. We illustrate the power of the propagation rules and prove soundness, completeness, and termination in nondeterministic polynomial time. We then derive a concrete implementation in a constraint programming language with nite sets [11, 6] and prove its faithfulness to the abstract saturation rules. The resulting solver is not only well suited for formal reasoning but also improves in expressiveness on the saturation based solver for pure dominance constraints of [8] and produces smaller search trees than the earlier set based ....

....Constraint Programming with Finite Sets Current constraint programming technology provides no support for our Dsaturation algorithm. Instead, improving on [2] we reformulate the task of nding solutions of a tree description as a constraint satisfaction problem solvable by constraint programming [11, 6]. In this section, we de ne our target language. Its propagation rules are given in Fig 12 and are used in proving correctness of implementation. Distribution rules, however, are typically problem dependent and we assume that they can be programmatically stipulated by the application. Thus, the ....

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C. Gervet. Interval propagation to reason about sets: De nition and implementation of a practical language. Constraints, 1(3):191-244, 1997.

....to solve all instances. We are interested in breaking as much symmetries as possible, as well as combining and improving previously proposed techniques to nd all solutions of one speci c instance of the problem. The rst important choice concerns the model. We naturally choose a set model [9] which automatically removes one kind of symmetry. The next step is to statically remove symmetries by adding constraints. Additional redundant constraints may be added to detect failures as soon as possible. The crucial point is then to be able to nd an isomorphism relating two solutions ....

....their associated domain is a lattice of sets de ned by its greatest lower bound http: www.icparc.ic. ac.uk eclipse examples, which automatically removes symmetries inside groups is the one we chose for our experiments (the necessary elements) and its lowest upper bound (the possible elements) [9]. The G i;j s are subsets of the set of golfers. Each of them contains exactly s elements. All the groups of a week are disjoint and every pair of groups from di erent weeks share at most one element. All these properties are expressed with the following contraints: 1 i w; 1 j g G i;j ....

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Carmen Gervet. Interval propagation to reason about sets: De nition and implementation of a practical language. Constraints, 1(3):191-244, 1997. http://www.icparc.ic.ac.uk/~cg6.

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