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Simply terminating rewrite systems with long derivations

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Abstract.

A term rewrite system is called simply terminating if its termination can be shown by means of a simplification ordering. According to a result of Weiermann, the derivation length function of any simply terminating finite rewrite system is eventually dominated by a Hardy function of ordinal less than the small Veblen ordinal. This bound had appeared to be of rather theoretical nature, because all known examples had had multiple recursive complexities, until recently Touzet constructed simply (and even totally) terminating examples with complexities beyond multiple recursion. This was established by simulating the Hydra battle for all ordinal segments below the proof-theoretic ordinal of Peano arithmetic. By extending this result to the small Veblen ordinal we prove the huge bound of Weiermann to be sharp. As a spin-off we can show that total termination allows for complexities as high as those of simple termination.

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  1. Institut für Mathematische Logik und Grundlagenforschung der Westfälischen Wilhelms-Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Ingo Lepper

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  1. Ingo Lepper
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Correspondence to Ingo Lepper.

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This paper is part of the author’s doctoral dissertation project (under the supervision of Professor A. Weiermann at the University of Münster).

The work on this paper was supported by DFG grant WE 2178/2–1

Mathematics Subject Classification (2000): 03D20, 68Q15, 68Q42

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Lepper, I. Simply terminating rewrite systems with long derivations. Arch. Math. Logic 43, 1–18 (2004). https://doi.org/10.1007/s00153-003-0190-2

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  • DOI: https://doi.org/10.1007/s00153-003-0190-2

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