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Computability of recurrence equations. (English) Zbl 0780.65078
Summary: Systems of uniform recurrence equations were proposed by R. M. Karp et al. [J. Assoc. Comput. Mach. 14, 563–590 (1967; Zbl 0171.38305)] as a tool to derive programs for parallel architectures automatically. 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. [loc. cit.] on the computability of systems of uniform recurrence equations, by S. K. Rao [Regular iterative algorithms and their implementations on processor arrays. Ph.D. Thesis, Stanford Univ. (1985)] on regular iterative arrays and the undecidability result of B. Joinnault [Conception d’algorithmes et d’architectures systoliques. Thèse, Univ. de Rennes (1987)] on the computability of conditional systems of uniform recurrence equations with nonbounded domain.
Then we consider systems of parametrized affine recurrence equations, that is to say, systems of recurrence equations whose domains linearly depend on a size parameter, and establish that the computability of such systems is also undecidable.

MSC:
65Q05 Numerical methods for functional equations (MSC2000)
68W15 Distributed algorithms
68Q80 Cellular automata (computational aspects)
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