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The contraction rule and decision problems for logics without structural rules

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Abstract

This paper shows a role of the contraction rule in decision problems for the logics weaker than the intuitionistic logic that are obtained by deleting some or all of structural rules. It is well-known that for such a predicate logic L, if L does not have the contraction rule then it is decidable. In this paper, it will be shown first that the predicate logic FLec with the contraction and exchange rules, but without the weakening rule, is undecidable while the propositional fragment of FLec is decidable. On the other hand, it will be remarked that logics without the contraction rule are still decidable, if our language contains function symbols.

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References

  1. A. R. Anderson and N. D. Belnap, Jr., Modalities in Ackermann's “rigorous implication”, Journal of Symbolic Logic 24 (1959). pp. 107–111.

    Google Scholar 

  2. A. R. Anderson and N. D. Belnap, Jr., Entailment, The Logic of Relevance and Necessity, I, Princeton University Press, 1975.

  3. N. D. Belnap, Jr., A decision procedure for the system E 1 of entailment with negation, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 11 (1965), pp. 277–289.

    Google Scholar 

  4. C. L. Chang and R. C. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, 1973.

  5. D. van Dalen, Intuitionistic logic, in Handbook of Philosophical Logic III, D. Gabbay and F. Guenthner (eds.). D. Reidel, 1986, pp. 225–339.

  6. J. M. Dunn. Relevance and entailment, in Handbook of Philosophical Logic III, D. Gabbay and F. Guenthner (eds.), D. Reidel, 1986, pp. 117–224.

  7. J. Y. Glrard and Y. Lafont, Linear logic and lazy computation, Proceedings of TAPSOFT'87 vol. 2, Lecture Notes in Computer Science 250 (1987), pp. 52–66.

    Google Scholar 

  8. Y. Komori. Predicate logics without the structure rules, Studia Logica 45 (1986), pp. 393–404.

    Google Scholar 

  9. S. A. Kripke, The problem of entailment (abstract), Journal of Symbolic Logic 24 (1959). p. 325.

    Google Scholar 

  10. P. Lincoln, J. Mitchell, A. Scedrov and N. Shankar, Decision problems for propositional linear logic, to appear in Proceedings of 31st Annual IEEE Symposium on Foundations of Computer Science, St. Louis, Missouri, 1990.

  11. D. Mey, A predicate calculus with control of derivations, CSL'89. Lecture Notes in Computer Science 440 (1990), pp. 254–266.

    Google Scholar 

  12. R. K. Meyer, Intuitionism, entailment, negation, in Truth, Syntax and Modality, H. Leblanc (ed.), North-Holland, 1973, pp. 168–198.

  13. H. Ono, Structural rules and a logical hierarchy, in Mathematical Logic, Proceedings of the Summer School and Conference on Mathematical Logic, Heyting'88, P. P. Petkov (ed.), Plenum Press, 1990, pp. 95–104.

  14. G. Takeuti, Proof Theory 2nd ed., North-Holland, 1987.

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Authors and Affiliations

  1. Department of Information Systems, Kyushu University, Fukuoka 812, Japan

    Eiji Kiriyama & Hlroakira Ono

  2. Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 724, Japan

    Eiji Kiriyama & Hlroakira Ono

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  1. Eiji Kiriyama
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  2. Hlroakira Ono
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Kiriyama, E., Ono, H. The contraction rule and decision problems for logics without structural rules. Stud Logica 50, 299–319 (1991). https://doi.org/10.1007/BF00370189

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  • DOI: https://doi.org/10.1007/BF00370189

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