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Simple termination is difficult

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Abstract

A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termination. We show that simple termination is an undecidable property, even for one-rule systems. This contradicts a result by Jouannaud and Kirchner. The proof is based on the ingenious construction of Dauchet who showed the undecidability of termination for one-rule systems. Our results may be summarized as follows: being simply terminating, (non-)self-embedding, and (non-)looping are undecidable properties of orthogonal, variable preserving, one-rule constructor systems.

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Authors and Affiliations

  1. Institute of Information Sciences and Electronics, University of Tsukuba, 305, Tsukuba, Japan

    Aart Middeldorp

  2. Fachbereich Informatik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Bernhard Gramlich

Authors
  1. Aart Middeldorp
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  2. Bernhard Gramlich
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Additional information

A preliminary version of this paper appeared in the Proceedings of the 5th International Conference on Rewriting Techniques and Applications, Montreal, Lecture Notes in Computer Science 690, pp. 228–242, 1993.

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Middeldorp, A., Gramlich, B. Simple termination is difficult. AAECC 6, 115–128 (1995). https://doi.org/10.1007/BF01225647

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  • DOI: https://doi.org/10.1007/BF01225647

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