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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.

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) |

### Keywords:

regular iterative arrays; systems of uniform recurrence equations; parallel architectures; systolic array synthesis; computability### Citations:

Zbl 0171.38305
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\textit{Y. Saouter} and \textit{P. Quinton}, Theor. Comput. Sci. 116, No. 2, 317--337 (1993; Zbl 0780.65078)

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### References:

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[2] | Delosme, J. M.; Ipsen, I. C.F., Efficient systolic arrays for the solution of Toeplitz systems: an illustration of a methodology for the construction of systolic architectures for VLS, (Moore, W.; McCabe, A.; Urquhart, R., Internat. Workshop on Systolic Arrays (1986), Adam Hilger, University of Oxford), 37-46 |

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[4] | Joinnault, B., Conception d’algorithmes et d’architectures systoliques, The“se de l”Universite´de Rennes I (1987) |

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[10] | Rao, S. K., Regular Iterative Algorithms and their Implementations on Processor Arrays, (Ph.D. Thesis (1985), Stanford University) |

[11] | Saouter, Y.; Quinton, P., Computability of recurrence equations, (Tech. Report 521 (1990), IRISA) · Zbl 0780.65078 |

[12] | Schrijver, A., Theory of Linear and Integer Programming, (Wiley-Interscience Series in Discrete Mathematics (1986), Wiley: Wiley New York) · Zbl 0665.90063 |

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