On the Equivalence of Two Systems of Affine Recurrence Equations
Résumé
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 first 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. The proof is by reducing any instance of Hilbert's tenth problem (the solution of Diophantine equations) to the equivalence of two SAREs. We then show that there neverthele- ss exists a semi-algorithm, in which the key ingredient is the computation of transitive closures of affine relations. This is again an undecidable problem 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.