Computability of recurrence equations

Abstract

Systems of uniform recurrence equations were proposed by Karp et al. (1967) as a means to derive automatically programs for parallel architectures. Since then, extensions of this formalism were used by many authors, in particular, in the fields of systolic array synthesis. The computability of a system of recurrence equations is, therefore, of primary importance, and is considered as the first point to be examined when trying to implement an algorithm. This paper investigates the computability of recurrence equations. We first recall the results established by Karp et al. (1967) on the computability of systems of uniform recurrence equations, by Rao (1985) on regular iterative arrays, and Joinnault's (1987) undecidability result on the computability of conditional systems of uniform recurrence equations with nonbounded domain. Then we consider systems of parameterized affine recurrence equations, that is to say, systems of recurrence equations whose domains depend linearly on a size parameter, and establish that the computability of such system is also undecidable.