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Proof-theoretical investigation of temporal logic with time gaps

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Abstract

In the paper, the first-order intuitionistic temporal logic sequent calculus LBJ is considered. The invertibility of some of the LBJ rules, syntactic admissibility of the structural rules and the cut rule in LBJ, as well as Harrop and Craig's interpolation theorems for LBJ are proved. Gentzen's midsequent theorem is proved for the LBJ' calculus which is obtained from LBJ by removing the antecedent disjunction rule from it.

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References

  1. R. Alonderis, On intuitionistic branching tense logic with weak induction,Lith. Math. Society Scientific Works,2, 295–301 (1998).

    Google Scholar 

  2. H. Andreka, J. Nemeti, and J. Sain, On the strength of temporal proof,LNSC,379, 135–144 (1989).

    MATH  MathSciNet  Google Scholar 

  3. M. Baaz, A. Leitsch, and R. Zach, Completeness of a first-order temporal logic with time gaps,Theoret. Comput. Sci.,160, 241–270 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Dyckhoff, Contraction free sequent calculi for intuitionistic logic,JSL,51, 795–807 (1992).

    MathSciNet  Google Scholar 

  5. W. B. Ewalds, Intuitionistic tense and modal logic,JSL,51, 166–179 (1986).

    Google Scholar 

  6. K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme,Monatsh. Math. Phys.,38, 173–198 (1931).

    Article  MATH  Google Scholar 

  7. H. Kawai, Sequential calculus for a first-order infinitary temporal logic,Z. Math. Logik Grundlag. Math.,33, 423–452 (1987).

    MATH  MathSciNet  Google Scholar 

  8. F. Kröger,Temporal Logic of Programs, EATCS Monographs in Computer Science, vol. 8, Springer, Berlin (1987).

    MATH  Google Scholar 

  9. F. Kröger, On the interpretability of arithmetic in temporal logic,Theoret. Comput. Sci.,73, 47–60 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Leivant, Proof theoretic methodology for Propositional Dynamic Logic, Lecture Notes in Computer Science,107, 356–373 (1981).

    MATH  MathSciNet  Google Scholar 

  11. L. Maksimova, Temporal logics of “the next” do not have the Beth property,J. Appl. Non-Classical Logics,1(1), 73–76 (1991).

    MATH  MathSciNet  Google Scholar 

  12. H. Nishimura, Semantical analysis of constructive PDL,Publ. Res. Inst. Math. Sci.,18(2), 427–438 (1982).

    Google Scholar 

  13. H. Ono, On some intuitionistic modal logics,Publ. Res. Inst. Math. Sci.,13, 687–722 (1977).

    MATH  MathSciNet  Google Scholar 

  14. J. R. Shoenfield, Open sentences and the induction axiom,JSL 23(1), 7–12 (1958).

    MATH  MathSciNet  Google Scholar 

  15. J. C. Shepherdson, Non-standard models for fragments of number theory. Symp. Theor. Models, Amsterdam, pp. 137–170 (1965).

  16. J. C. Shepherdson, The rules of induction in the free variable arithmetic based on + and.Am. Fac. Univ. Clermont. Math.,4, 25–31 (1967).

    MathSciNet  Google Scholar 

  17. A. Szalas, A complete axiomatic characterization of first order temporal logic of linear time,Theoret. Comput. Sci.,54, 199–214 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Szalas, Concerning the semantic consequence relation in first-order temporal logic,Theoret. Comput. Sci.,47, 329–334 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Takeuti,Proof Theory, North-Holland, Amsterdam (1975).

    MATH  Google Scholar 

  20. B. Volozh, M. Matskin, B. G. Mints, and E. Tyugu,PRIZ System and the Propositional Calculus, Kibernetika and Software (1963).

  21. N. Vorob'ev, A new algorithm for derivability in a constructive propositional calculus,Amer. Math. Soc.,94, 37–71 (1970).

    MATH  Google Scholar 

  22. D. Wijesekera, Constructive Modal Logics I,Ann. Pure Appl. Logic,50, 271–301 (1990).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithuania

    R. Alonderis

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  1. R. Alonderis
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Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 255–276, July–September, 2000.

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Alonderis, R. Proof-theoretical investigation of temporal logic with time gaps. Lith Math J 40, 197–212 (2000). https://doi.org/10.1007/BF02465129

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  • DOI: https://doi.org/10.1007/BF02465129

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