F. Ajili and E. Contejean. Complete solving of linear Diophantine equations and inequations without adding variables. In U. Montanari and F. Rossi, editors, Proceedings of the First International Conference on Principles and Practice of Constraint Programming, volume 1949.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of all solutions of linear Diophantine equations and inequalities. For example, there are algorithms which focus on finding minimal solutions (e.g. 8] algorithms which focus on finding nonambiguous solutions (e.g. 1] and algorithms which focus on avoiding introducing slack variables (e.g. [2]) All of these algorithms resort to complete enumeration when the solution space is finite, which is ine#cient if it is particularly large. Whether this matters, and whether the integer hull would be a better representation, depends on the end use for the algorithm. 3. Preliminaries. In the ....

F. Ajili and E. Contejean, Complete solving of linear diophantine equations and inequations without adding variables, in Principles and Practice of Constraint Programming---CP '95, U. Montanari and F. Rossi, eds., Lecture Notes in Comput. Sci. 976, Springer-Verlag, Berlin, 1995, pp. 1--17.

....area of theorem proving. Again, our research in this area has opened doors for the coming years. 2.1 Diophantine Constraints Four sites are collaborating on related problems, COSYTEC, INRIA Lorraine, MPI and UPS. The design of algorithms for integer constraint solving has yielded to joint papers [1, 25]. Join work on sequencing constraints is going on between COSYTEC and UPS. The corresponding algorithms have been made available in CHIP, in the new commercial release CHIP V5. Logic based methods for pseudo Boolean constraint solving represent a very promising new approach for handling 0 1 ....

....Contejean and Devie are still satisfied by the new algorithm: it is complete, i.e. provides a (finite) description of the set of solutions, it can be implemented with a bounded stack, and it admits an incremental version. All of these characteristics enable its easy integration in the CLP paradigm [1]. Theorem proving and constraints We have further investigated the concept of deduction with constraints. Previous results already described in previous reports have also been publihed [57, 63] We have shown the completeness of an extension of SLD resolution to the equational setting [64] ....

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F. Ajili and E. Contejean. Complete solving of linear diophantine equations and inequations without adding variables. In Proc. First International Conference on Principles and Practice of Constraint Programming, Marseille, September 1995.

....equation systems is presented in the literature. The first algorithms [Hue78, For83] were limited to the case where a single equation has to be solved. More recent approaches can directly handle systems of equations [GH85, CF89, CD94] or even general linear constraints involving inequations [AC95, CD94] and disequations [DT95] Therefore we did not go into further details of the equation solving aspects. In our first application, the concept language T F , however, it turned out that all reasoning problems can be reduced to consistency checks for equation systems and thus no explicit ....

F. Ajili and E. Contejean. Complete solving of linear diophantine equations and inequations without adding variables. In U. Montanari and F. Rossi, editors, Proceedings of the 1st International Conference on Principles and Practice of Constraint Programming (CP-95). Springer, 1995.

....equation systems is presented in the literature. The first algorithms [Hue78, For83] were limited to the case where a single equation has to be solved. More recent approaches can handle systems of equations directly [GH85, CF89, CD94] or even general linear constraints involving inequations [AC95, CD94] and disequations [DT95] Therefore we have not gone into further details of the equation solving aspects. However, in our first application, the concept language T F , it turned out that all reasoning problems can be reduced to consistency checks for equation systems and thus no explicit ....

F. Ajili and E. Contejean. Complete solving of linear diophantine equations and inequations without adding variables. In U. Montanari and F. Rossi, editors, Proceedings of the 1st International Conference on Principles and Practice of Constraint Programming (CP-95). Springer, 1995.

....solution can be written as an N linear combination of elements of such a basis. Algorithms which compute such a basis are said complete. Complete solving methods, based generally on algebraic and automata techniques, do not make use of constraint propagation. The recent method of Ajili Contejean [1, 2] searches in N q for minimal solutions starting from the canonical vectors. A tuple is incremented if a successor generation criteria, which is based on a geometric interpretation, successes. A tuple is pruned iff the sub tree rooted at it can not contain minimal solutions. A strong motivation ....

....decisions by testing whether its local constraint store entails some pertinent constraints for the search. We show how the introduction of the notion of local constraint store allows to a tuple to exploit locally stronger pruning opportunities. In addition, we show how the depth first version of [1, 2], which is based on a freezing mechanism, can be encoded thanks to constraints. We capture the description of the solution set by a constrained parametric expression and where local consistency techniques is used to find admissible coefficients to have a solution. ....

Farid Ajili and Evelyne Contejean. Complete solving of linear diophantine equations and inequations without adding variables. Theoretical Computer Science, 1997. To appear. 2

....together with equations, extension to the heterogeneous case, incrementality) this new solver has a wide range of potential applications: it can be integrated in the CLP paradigm thanks to its ability to test the satis ability and to check constraint entailment. This work was rstly presented in [2] and will appear in [1] 2 Basic Notions 2.1 Notations As usually N denotes the set of non negative numbers and e j denotes the j th canonical tuple of N q , that is e j = 0; 0 z (j Gamma1)times ; 1; 0; 0 z (q Gammaj)times ) Delta denotes the scalar ....

Farid Ajili and Contejean Evelyne. Complete solving of linear diophantine equations and inequations without adding slack variables. In Nieuwenhuis R., editor, Proceedings of the ninth UNIF Workshop, Barclona, Spain, April 1995.

....solutions such that any solution can be written as a N linear combination of these non decomposable solutions. Most of the various mathematical techniques behind these methods originated in the areas of linear algebra (e.g. 10] automata theory (e.g. 5,6,15] and geometric interpretation (e.g. [1,2,6,8]) An examination of the solving machineries underlying those methods shows that they do not dynamically exploit constraints during the search. Compared to approaches based on constraint logic programming (CLP) we observe a lack of strong pruning criterion driven by constraint based reasoning. In ....

....preserves the advantages of each of its parts. This paper concentrates precisely on how to integrate constraint propagation and heuristics into the family of complete methods based on geometric interpretation. As a representative of this family we choose the algorithm of F. Ajili E. Contejean [1,2] (named here ACalg) since it is a generalisation of the algorithms [6,8] and solves systems of both equations and inequations. The ACalg algorithm searches for solutions starting from canonical tuples. A tuple is incremented if a successor generation criterion succeeds. This criterion, which is ....

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F. Ajili and E. Contejean. Complete solving of linear diophantine equations and inequations without adding variables. In Montanari and Rossi [14], pages 1--17.

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F. Ajili and E. Contejean. Complete solving of linear Diophantine equations and inequations without adding variables. In U. Montanari and F. Rossi, editors, Proceedings of the First International Conference on Principles and Practice of Constraint Programming, volume 1949.

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