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Archive for Mathematical Logic

, Volume 52, Issue 1–2, pp 173–212 | Cite as

The taming of recurrences in computability logic through cirquent calculus, Part I

  • Giorgi Japaridze
Article

Abstract

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation \({\neg}\), parallel conjunction \({\wedge}\), parallel disjunction \({\vee}\), branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.

Keywords

Computability logic Cirquent calculus Interactive computation Game semantics Resource semantics 

Mathematics Subject Classification

Primary 03B47 Secondary 03B70 68Q10 68T27 68T15 

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References

  1. 1.
    Avron A.: A constructive analysis of RM. J. Symb. Logic 52, 939–951 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Blass A.: A game semantics for linear logic. Ann. Pure Appl. Logic 56, 183–220 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Girard J.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Guglielmi A.: A system of interaction and structure. ACM Trans. Comput. Logic 8, 1–64 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Japaridze G.: Introduction to computability logic. Ann. Pure Appl. Logic 123, 1–99 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Japaridze G.: Propositional computability logic I. ACM Trans. Comput. Logic 7, 302–330 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Japaridze G.: Propositional computability logic II. ACM Trans. Comput. Logic 7, 331–362 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Japaridze G.: From truth to computability I. Theor. Comput. Sci. 357, 100–135 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Japaridze G.: Introduction to cirquent calculus and abstract resource semantics. J. Logic Comput. 16, 489–532 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Japaridze G.: The logic of interactive Turing reduction. J. Symb. Logic 72, 243–276 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Japaridze G.: From truth to computability II. Theor. Comput. Sci. 379, 20–52 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Japaridze G.: Intuitionistic computability logic. Acta Cybern. 18, 77–113 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Japaridze G.: The intuitionistic fragment of computability logic at the propositional level. Ann. Pure Appl. Logic 147, 187–227 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Japaridze G.: Cirquent calculus deepened. J. Logic Comput. 18, 983–1028 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Japaridze G.: Sequential operators in computability logic. Inf. Comput. 206, 1443–1475 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Japaridze G.: In the beginning was game semantics. In: Majer, O., Pietarinen, A.-V., Tulenheimo, T. (eds) Games: Unifying Logic, Language and Philosophy, pp. 249–350. Springer, Berlin (2009)CrossRefGoogle Scholar
  17. 17.
    Japaridze G.: Many concepts and two logics of algorithmic reduction. Studia Logica 91, 1–24 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Japaridze G.: Towards applied theories based on computability logic. J. Symb. Logic 75, 565–601 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Japaridze G.: Toggling operators in computability logic. Theor. Comput. Sci. 412, 971–1004 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Japaridze, G.: From formulas to cirquents in computability logic. Logical Methods Comput. Sci. 7(2), Paper 1, 1–55 (2011)Google Scholar
  21. 21.
    Japaridze G.: Introduction to clarithmetic I. Inf. Comput. 209, 1312–1354 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Japaridze G.: A new face of the branching recurrence of computability logic. Appl. Math. Lett. 25, 1585–1589 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Japaridze G.: Separating the basic logics of the basic recurrences. Ann. Pure Appl. Logic 163, 377–389 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Japaridze G.: A logical basis for constructive systems. J. Logic Comput. 22, 605–642 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Japaridze, G.: The taming of recurrences in computability logic through cirquent calculus, Part II. Arch. Math. Logic (to appear)Google Scholar
  26. 26.
    Japaridze, G.: Introduction to Clarithmetic II. Manuscript at http://arxiv.org/abs/1004.3236
  27. 27.
    Japaridze, G.: Introduction to Clarithmetic III. Manuscript at http://arxiv.org/abs/1008.0770
  28. 28.
    Kolmogorov A.N.: Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift 35, 58–65 (1932)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kwon H., Hur S.: Adding sequential conjunctions to Prolog. Int. J. Comput. Appl. Technol. 1, 1–3 (2010)Google Scholar
  30. 30.
    Kwon H.: Adding a loop construct to Prolog. Int. J. Comput. Appl. Technol. 2, 121–123 (2011)Google Scholar
  31. 31.
    Mezhirov I., Vereshchagin N.: On abstract resource semantics and computability logic. J. Comput. Syst. Sci. 76, 356–372 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Pottinger G.: Uniform, cut-free formulations of T, S4 and S5 (abstract). J. Symb. Logic 48, 900 (1983)Google Scholar
  33. 33.
    Xu W., Liu S.: Knowledge representation and reasoning based on computability logic. J. Jilin Univ.47, 1230–1236 (2009)zbMATHGoogle Scholar
  34. 34.
    Xu W., Liu S.: Deduction theorem for symmetric cirquent calculus. Adv. Intell. Soft Comput. 82, 121–126 (2010)CrossRefGoogle Scholar
  35. 35.
    Xu W., Liu S.: Soundness and completeness of the cirquent calculus system CL6 for computability logic. Logic J. IGPL 20, 317–330 (2012)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyShandong UniversityJinanChina
  2. 2.Department of Computing SciencesVillanova UniversityVillanovaUSA

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