Advertisement

Decomposition of Reduction

  • Ernst Zimmermann
Chapter
Part of the Trends in Logic book series (TREN, volume 39)

Abstract

The method of decomposition of reduction is presented for proofs of confluence and termination in rewriting and natural deduction. It is shown that any subclass of reductions, which is locally confluent and terminating and commutes with the full reduction in a specific way, transfers confluence and termination to the full reduction. The method is introduced on a pure rewriting level and subsequently applied to natural deduction, i.e. its calculi for Intuitionistic Logic and Intuitionistic Linear Logic, where respective subclasses of reductions are shown to exist. A key observation is that local confluence holds for reductions in natural deduction.

References

  1. 1.
    Baader, F., & Nipkow, T. (1998). Term rewriting and all that. Cambridge: Cambridge University Press.Google Scholar
  2. 2.
    Dershowitz, N., & Manna, Z. (1979). Proving termination with multiset orderings. Communications of the ACM, 22(8), 465–476.CrossRefGoogle Scholar
  3. 3.
    Gandy, R. O. (1980). Proofs of strong normalisation. To H.B. Curry: Essays on combinatory logic, \(\lambda \)-calculus and formalism (pp. 457–477). Boston: Academic Press.Google Scholar
  4. 4.
    Gentzen, G. (1934/1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 176–210, 405–431.Google Scholar
  5. 5.
    Girard, J.-Y. (1986). Proof theory and logical complexity. Napoli: Bibliopolis.Google Scholar
  6. 6.
    Joachimsky, F., & Matthes, R. (2003). Short proofs of normalization for the simple typed \(\lambda \)-calculus, permutative conversions and Gödel’s T. Archive for Mathematical Logic, 42, 59–87.CrossRefGoogle Scholar
  7. 7.
    Prawitz, D. (1965). Natural deduction. Stockholm: Almqvist and Wiksell.Google Scholar
  8. 8.
    Prawitz, D. (1971). Ideas and results in proof theory. Proceedings of the 2-nd Scand. Logic Symposium, Amsterdam, pp. 235–307.Google Scholar
  9. 9.
    Schwichtenberg, H., & Troelstra, A. S. (2000). Basic proof theory, 2nd edn. Cambridge: Cambridge University Press.Google Scholar
  10. 10.
    Zimmermann, E. (2010). Full Lambek calculus in natural deduction. Mathematical Logic Quarterly, 56, 85–88.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.DFG and ANR Project Hypotheses, Wilhelm-Schickard-InstitutUniversität TübingenTübingenGermany

Personalised recommendations