Fourier's Elimination: Which to Choose?

1.1 Abstract This paper is concerned with variable elimination methods in linear in-equation systems, related to Fourier's elimination8]. Our aim is to make visible the links between the diierent contributions of S.N. Yap 14]. We show that the three methods proposed by Cernikov, Kolher and Imbert produce exactly the same output (without more or less redundant inequations), up to multiplying by a non-zero positive scalar. We present and discuss the improvements of Cernikov, Duun, Imbert and Jaaar et al., and propose a new improvement. We give a short analysis of the complexity of the main improvements and discuss the choice of the method in relation to the problem at hand. We propose a pattern algorithm. Finally, we conclude with a comparative assessment through a brief example and a few remarks. 1.2 Introduction Variable elimination is of major interest for Constraint Logic Programming Languages 12], and Constraint Query Languages 15], where we would like to eliminate auxiliary variables introduced during the execution of a program. This elimination is always suitable for nal results. It can also increase the eeciency of the intermediary processes. We focus on linear inequalities of the form ax b, where a denotes a n-real vector, x an n-vector of variables, b a real number, and the juxtaposition ax denotes the inner product. This type of constraint occurs in CLP languages such as CHIP 7, 22], CLP(<) 13], and Prolog III 4, 5]. A constraint system is a conjunction of constraints. Using matrix notation, an inequation system fa i x b i j i = 1;. . .; mg can be written Ax b, where A denotes an (m x n)-matrix, and b an m-real vector. The main problem we face during the variable elimination process in linear in-equation systems, is the size of the output. It is doubly exponential. Variable elimination in inequation systems has been extensively investigated. Among these investigations one can cite the C. and JL. Lassez