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Studia Logica

, Volume 91, Issue 2, pp 273–293 | Cite as

Proof Systems for Reasoning about Computation Errors

  • Arnon AvronEmail author
  • Beata Konikowska
Article

Abstract

In the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options as a non-deterministic choice. If the two strategies are to be distinguished, Kleene and McCarthy logics are combined into a logic based on a 4-valued deterministic matrix featuring two kinds of computation errors which correspond to the two computation strategies described above. For the resulting logics, we provide sound and complete calculi of ordinary, two-valued sequents.

Keywords

three-valued logics four-valued logics parallel computation lazy sequential computation computation errors non-deterministic matrices sequent calculi 

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References

  1. AK05.
    Avron A., Konikowska B.: Proof Systems for Logics Based on Nondeterministic Multiple-valued Structures. Logic Journal of the IGPL 13, 365–387 (2005)CrossRefGoogle Scholar
  2. ABK06.
    Avron A., Ben-Naim J., Konikowska B.: Cut-free Ordinary Sequent Calculi for Logics Having Generalized Finite-Valued Semantics. Logica Universalis 1, 41–69 (2006)CrossRefGoogle Scholar
  3. AL01.
    Avron, A., and I. Lev, ‘Canonical Propositional Gentzen-Type Systems’, in R. Gor´e, A Leitsch, T. Nipkow (eds.), Proceedings of the 1st International Joint Conference on Automated Reasoning (IJCAR 2001), LNAI 2083, Springer Verlag, 2001, pp. 529--544.Google Scholar
  4. AL05.
    Avron A., Lev I.: Non-deterministic Multiple-valued Structures. Journal of Logic and Computation 15, 241–261 (2005)CrossRefGoogle Scholar
  5. BCJ84.
    Barringer H., Chang J.H., Jones C.B.: A logic covering undefinedness in program proofs. Acta Informatica 21, 251–269 (1984)CrossRefGoogle Scholar
  6. Bli91.
    Blikle A.: Three-valued predicates for software specification and validation. Fundamenta Informaticae 14, 387–410 (1991)Google Scholar
  7. Ho87.
    Hogevijs A.: Partial predicate logic in computer science. Acta Informatica 24, 381–393 (1987)CrossRefGoogle Scholar
  8. Kl52.
    Kleene, S.C., Introduction to Metamathematics, North-Holland, 1952.Google Scholar
  9. KTB91.
    Konikowska B., Tarlecki A., Blikle J.: A three-valued logic for software specification and validation. Fundamenta Informaticae 14(4), 411–453 (1991)Google Scholar
  10. Ko93.
    Konikowska B.: Two over three: a two-valued logic for software specification and validation over a three-valued predicate calculus. Journal for Applied Nonclassical Logic 3(1), 39–71 (1993)Google Scholar
  11. Ko08.
    Konikowska, B., A Four-Valued Logic for Reasoning about Finite and Infinite Computation Errors in Programs, accepted to: W. A. Carnielli, M. E. Coniglio, I. M. Loffredo D’ Ottaviano (eds.), The Many Sides of Logic, Series Studies in Logic, College Publications, London.Google Scholar
  12. MC67.
    McCarthy, J., A basis for a mathematical theory of computation, Computer Programming and Formal Systems, North-Holland, 1967.Google Scholar
  13. Owe85.
    Owe, O., An approach to program reasoning based on first order logic for partial functions, Res. Rep. Institute of Informatics, University of Oslo, no. 89, 1985.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of Computer ScienceTel-Aviv UniversityRamat AvivIsrael
  2. 2.Institute of Computer SciencePolish Academy of SciencesWarsawPoland

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