On the Equivalence of Two Systems of Affine Recurrence Equations (Research Note)

TLDR: It is proved that in the general case, the problem is undecidable, but there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations.

Abstract: This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.