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A Methodological View of Constraint Solving
, 1996
"... Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real nu ..."
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Cited by 6 (2 self)
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Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real numbers, finite domains, and term domains. We classify the different methods used for solving constraints, syntactic methods based on transformations, semantic methods based on adequate representations of constraints, hybrid methods combining transformations and enumerations. Examples are used throughout the paper to illustrate the concepts and methods. We also discuss applications of constraints to various fields, such as programming, operations research, and theorem proving. y CNRS and LRI, Bat. 490, Universit'e de Paris Sud, 91405 ORSAY Cedex, France fcomon, jouannaudg@lri.lri.fr z COSYTEC, Parc Club Orsay Universit'e, 4 Rue Jean Rostand, 91893 Orsay Cedex, France dincbas@cosytec.fr x ...
Solving linear Diophantine equations using the geometric structure of the solution space
 PROC. RTA'97, LNCS 1232
, 1997
"... In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we s ..."
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Cited by 3 (2 self)
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In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solutions and in having a qualitative evaluation of the difficulty in solving a given system.
Hilbert bases of cones related to simultaneous Diophantine approximations and linear Diophantine equations
, 1997
"... This paper investigates properties of the minimal integral solutions of a linear diophantine equation. We present best possible inequalities that must be satisfied by these elements which improves on former results. We also show that the elements of the minimal Hilbert basis of the dual cone of all ..."
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Cited by 2 (1 self)
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This paper investigates properties of the minimal integral solutions of a linear diophantine equation. We present best possible inequalities that must be satisfied by these elements which improves on former results. We also show that the elements of the minimal Hilbert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector "from above". Relations between these cones are applied to the knapsack problem.
An Algorithm for Solving Systems of Linear Diophantine . . .
 IN NATURALS. PROC. EPIA'97, LNAI 1323
, 1997
"... A new algorithm for finding the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which m ..."
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A new algorithm for finding the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which may be of use for anyone wanting to implement it.
On minimal solutions of linear Diophantine equations
"... This paper investigates the region in which all the minimal solutions of a linear diophantine equation lie. We present best possible inequalities which must be satisfied by these solutions and thereby improve earlier results. ..."
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This paper investigates the region in which all the minimal solutions of a linear diophantine equation lie. We present best possible inequalities which must be satisfied by these solutions and thereby improve earlier results.
The Computational Algorithm for Supported Solutions Set of Linear Diophantine Equations Systems in a Ring of Integer Numbers
"... ABSTRACT:The algorithm for computation of minimal supported set of solutions and base solutions of linear Diophantine equations systems in a ring of integer numbers is proposed. This algorithm is founded on the modified TSSmethod. Keywords:ring of integer numbers, linear Diophantine equations, suppo ..."
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ABSTRACT:The algorithm for computation of minimal supported set of solutions and base solutions of linear Diophantine equations systems in a ring of integer numbers is proposed. This algorithm is founded on the modified TSSmethod. Keywords:ring of integer numbers, linear Diophantine equations, supported set, supported set of solutions I.
Avoiding Infinite Loops in the Solving of Equations Involving Sequence Variables and Terms with Flexible Arity Function Symbols
"... Solving equations involving terms with variables that can be instantiated to sequences of terms (sequence variables) and terms with function symbols of arbitrary arity may lead to infinite loops. This is particularly relevant when the implementations traverse the solution tree using a depthfirst st ..."
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Solving equations involving terms with variables that can be instantiated to sequences of terms (sequence variables) and terms with function symbols of arbitrary arity may lead to infinite loops. This is particularly relevant when the implementations traverse the solution tree using a depthfirst strategy, because some solutions become unreachable when they appear after a branch that loops. In this paper we present a simple method for checking if a branch will lead to a loop. This makes it possible to go on with the unification algorithm without trying to explore those branches. The technique we use is based on an abstraction of the original unification by a Diophantine equation on the sizes of the terms involved. We present this abstraction and show that it is correct with respect to the original unification. 1