Abstract not found

In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is difficult, whereas the proofs of completeness and correctness are straightforward generalizations of Fortenbacher 's proof. 1 Introduction Linear Diophantine equations occur frequently in mathematics and computer science, namely in the decision procedure of the accessibility property for Petri-Nets, in associative-commutative unification [1], in constrained logic programming, in the vectorization of FORTRAN programs, etc. By definition, a Diophantine equation has the form Q = 0, where Q is a 1 polynomial with integer coefficients. The equation is homogeneous if Q has no constant and linear if all monomial have the form kx i where k ...

. We present a method for characterizing the least fixed-points of a certain class of Datalog programs in Presburger arithmetic. The method consists in applying a set of rules that transform general computation paths into "canonical" ones. We use the method for treating the problem of reachability in the field of Petri nets, thus relating some unconnected results and extending them in several directions. Keywords: decomposition, linear arithmetic, least fixed-point, Petri nets, reachability set 1. Introduction The problem of computing fixpoints for arithmetical programs has been investigated from the seventies in an imperative framework. A typical application was to check whether or not array bounds were violated. A pionneering work in this field is the work by Cousot-Halbwachs (Cousot, 78). The subject has known a renewal of interest with the development of logic programming and deductive databases with arithmetical constraints. Several new applications were then possible in these f...

We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satisfiability of an homogeneous system, as well as upper bounds on height and length of all minimal solutions of such a system. We also show how our results apply to inhomogeneous systems yielding necessary conditions for satisfiability and upper bounds on the minimal solutions. Solving linear diophantine equations is a problem which appears in many fields, from linear programming to resolution of equations in semigroups or constrained logic programming. In the framework of semigroups, one needs an efficient test for the satisfiability of equations as well as a way to generate a basis for the set of all solutions. The efficiency of the satisfiability test becomes crucial in constrained equational logic because constraints are...

In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie [9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm of Fortenbacher [6] for solving a single linear Diophantine equation. All the nice properties of the algorithm of 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.

In this paper, we show how to handle linear Diophantine constraints incrementally by using several variations of the algorithm by Contejean and Devie (hereafter called ABCD) for solving linear Diophantine systems [4, 5]. The basic algorithm is based on a certain enumeration of the potential solutions of a system, and termination is ensured by an adequate restriction on the search. This algorithm generalizes a previous algorithm due to Fortenbacher [2], which was restricted to the case of a single equation. Note that using Fortenbacher's algorithm for solving systems of Diophantine equations by repeatedly applying it to the successive equations is completely unrealistic: the tuple of variables in the solved equation must then be substituted in the rest of the system by a linear combination of the minimal solutions found in which the coefficients stand for new variables. Unfortunately, the number of these minimal solutions is actually exponential in both the number of variables and the v...

In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the meta-language needed to manipulate the constraints. This has the advantage to provide descriptions that are very close to the proof theoretical setting used now to describe constraint manipulations like unification or numerical constraint solving. We examplify the approach by presenting examples of constraint solvers descriptions and combinations written in the ELAN language. 1

Abstract. We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we are convinced that ours are the fastest known to date when the equation coefficients are not too small (ie., greater than 2 or 3). 1

. We describe a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural numbers. We derive our algorithm from one proposed by Elliott in 1903 for solving a single homogeneous equation. This algorithm was then extended to solve homogeneous systems of equations by MacMahon. We show how it further extends to an algorithm which solves general linear constraints in nonnegative integers and allows a parallel implementation. This algorithm provides a parametric representation of the solutions from which minimal solutions may be extracted immediately. Moreover, it may be easily implemented in parallel. It has however one drawback: it is redundant which means that the same minimal solution is usually generated many times. We show how this redundancy may be eliminated at the cost of an increase in the space complexity. keywords: integer programming, linear diophantine constraints Introduction The problem of solving linear equations with i...

We introduce a new technique that translates cardinality information about finite sets into simple arithmetic terms and thereby enables a system to reason about such set cardinalities by solving arithmetic equation problems. The atomic decomposition technique separates a collection of sets into mutually disjoint smallest components ("atoms") such that the cardinality of the sets are just the sum of the cardinalities of their atoms. With this idea it is possible to have languages combining arithmetic formulae with set terms, and to translate the formulae of this combined logic into pure arithmetical formulae. As a particular application we show how this technique yields new inference procedures for concept languages with so called number restriction operators. 1 Imperial College, Department of Computing, London SW7 2BZ, email: h.ohlbach@doc.ic.ac.uk. 2 On leave from DFKI, Stuhlsatzenhausweg 3, 66123 Saarbrucken, e-mail: koehler@dfki.uni-sb.de. Contents 1 Introduction 1 2 Atomic D...