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On the Computational Content of Termination Proofs

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Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

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Abstract

Given that a program has been shown to terminate using a particular proof, it is natural to ask what we can infer about its complexity. In this paper we outline a new approach to tackling this question in the context of term rewrite systems and recursive path orders. From an inductive proof that recursive path orders are well-founded, we extract an explicit realiser which bounds the derivational complexity of rewrite systems compatible with these orders. We demonstrate that by analysing our realiser we are able to derive, in a completely uniform manner, a number of results on the relationship between the strength of path orders and the bounds they induce on complexity.

This work is supported by FWF (Austrian Science Fund) project P-25781.

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References

  1. Hofbauer, D.: Termination proofs by multiset path orderings imply primitive recursive derivation lengths. TCS 105, 129–140 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Weiermann, A.: Termination proofs for term rewriting systems with lexicographic path ordering imply multiply recursive derivation lengths. TCS 139, 355–362 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Buchholz, W.: Proof-theoretic analysis of termination proofs. APAL 75, 57–65 (1995)

    MATH  MathSciNet  Google Scholar 

  4. Parsons, C.: On a number theoretic choice schema and its relation to induction. In: Proceedings of the Intuitionism and Proof Theory, pp. 459–473 (1970)

    Google Scholar 

  5. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  6. Moser, G., Weiermann, A.: Relating derivation lengths with the slow-growing hierarchy directly. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 296–310. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Cichon, E.A.: Termination orderings and complexity characterisations. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory, pp. 171–193. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  8. Touzet, H.: Encoding the hydra battle as a rewrite system. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, p. 267. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  9. Moser, G.: KBOs, ordinals, subrecursive hierarchies and all that. JLC (2015, to appear)

    Google Scholar 

  10. Figueira, D., Figueira, S., Schmitz, S., Schnoebelen, P.: Ackermannian and primitive-recursive bounds with dickson’s lemma. In: Proceedings of the 26th LICS, pp. 269–278. IEEE (2011)

    Google Scholar 

  11. Berardi, S., Oliva, P., Steila, S.: Proving termination with transition invariants of height \(\omega \). In: Proceedings of the 15th ICTCS, pp. 237–240 (2014)

    Google Scholar 

  12. Terese, : Term Rewriting Systems. Cambridge Tracks in Theoretical Computer Science. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  13. Ferreira, M.C.F., Zantema, H.: Well-foundedness of term orderings. In: Lindenstrauss, N., Dershowitz, N. (eds.) CTRS 1994. LNCS, vol. 968, pp. 106–123. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  14. Goubault-Larrecq, J.: Well-founded recursive relations. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, p. 484. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  15. Weiermann, A.: How is it that infinitary methods can be applied to finitary mathematics? Gödel’s T: a case study. JSL 63, 1348–1370 (1998)

    MATH  MathSciNet  Google Scholar 

  16. Arai, T.: Some results on cut-elimination, provable well-orderings, induction, and reflection. APAL 95, 93–184 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Cichon, E.A., Weiermann, A.: Term rewriting theory for the primitive recursive functions. APAL 83, 199–223 (1997)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, University of Innsbruck, Innsbruck, Austria

    Georg Moser & Thomas Powell

Authors
  1. Georg Moser
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  2. Thomas Powell
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Corresponding author

Correspondence to Thomas Powell .

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Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

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Moser, G., Powell, T. (2015). On the Computational Content of Termination Proofs. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_28

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  • DOI: https://doi.org/10.1007/978-3-319-20028-6_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20027-9

  • Online ISBN: 978-3-319-20028-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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