International Conference on Rewriting Techniques and Applications

RTA 2014: Rewriting and Typed Lambda Calculi pp 303-318 | Cite as

Proof Terms for Infinitary Rewriting

  • Carlos Lombardi
  • Alejandro Ríos
  • Roel de Vrijer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8560)

Abstract

We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. Transfinite reductions can be faithfully represented as infinitary proof terms, unique up to infinitary associativity. We use proof terms to define equivalence of transfinite reductions on the basis of permutation equations. A proof of the compression property via proof terms is presented, which establishes permutation equivalence between the original and the compressed reductions.

Keywords

infinitary rewriting proof terms permutation equivalence 

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References

  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Courcelle, B.: Fundamental properties of infinite trees. Theor. Comput. Sci. 25, 95–169 (1983)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Endrullis, J., Hansen, H.H., Hendriks, D., Polonsky, A., Silva, A.: A coinductive treatment of infinitary rewriting. Presented at WIR 2013, First International Workshop on Infinitary Rewriting, Eindhoven, Netherlands (June 2013)Google Scholar
  4. 4.
    Huet, G.P., Lévy, J.J.: Computations in orthogonal rewriting systems, i and ii. In: Computational Logic - Essays in Honor of Alan Robinson, pp. 395–414 (1991)Google Scholar
  5. 5.
    Kennaway, R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Transfinite reductions in orthogonal term rewriting systems. Inf. Comput. 119(1), 18–38 (1995)CrossRefMATHGoogle Scholar
  6. 6.
    Ketema, J.: Reinterpreting compression in infinitary rewriting. In: 23rd International Conference on Rewriting Techniques and Applications, RTA. LIPIcs, vol. 15, pp. 209–224. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)Google Scholar
  7. 7.
    Klop, J.W., de Vrijer, R.C.: Infinitary normalization. In: We Will Show Them: Essays in Honour of Dov Gabbay, vol. 2, pp. 169–192. College Publications (2005)Google Scholar
  8. 8.
    Lombardi, C., Ríos, A., de Vrijer, R.: Proof terms for infinitary rewriting, progress report (2014), http://arxiv.org/abs/1402.2245
  9. 9.
    van Oostrom, V., de Vrijer, R.: Four equivalent equivalences of reductions. Electronic Notes in Theoretical Computer Science 70(6), 21–61 (2002)CrossRefGoogle Scholar
  10. 10.
    Suppes, P.: Axiomatic Set Theory. D. van Nostrand, Princeton, USA (1960)Google Scholar
  11. 11.
    Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Carlos Lombardi
    • 1
    • 2
    • 3
  • Alejandro Ríos
    • 1
    • 3
  • Roel de Vrijer
    • 4
  1. 1.Universidad Nacional de QuilmesArgentina
  2. 2.PPS (Université Paris-Diderot and CNRS)France
  3. 3.Universidad de Buenos AiresArgentina
  4. 4.VU University AmsterdamThe Netherlands

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