Affine timings for systems of affine recurrence equations

Abstract

This paper is devoted to the problem of the existence of affine timings for problems defined by Systems of Affine Recurrence Equations. After a first analysis, such problems may have no affine timing not because the problem is uncomputable but only because of the initial system of equations. This system can induce dependencies organized in an inappropriate way. We give conditions for a dependency to be well-organized in such a way that an affine timing may exist. When a dependency does not satisfy these conditions, we describe how to transform it in order to meet the conditions. A problem defined by a system of equations is analyzed by a step-by-step examination of its dependencies. For each dependency organized in an inappropriate way, a transformation is applied. The whole transformation process yields to the determination of a new equivalent system of equations from which an affine timing can usually be computed. Many practical problems need such transformations. We illustrate this transformation process on the Algebraic Path Problem.