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The logic of interactive turing reduction

Published online by Cambridge University Press:  12 March 2014

Giorgi Japaridze
Giorgi Japaridze*
Affiliation:
Villanova University, Department of Computing Sciences, 800 Lancaster Avenue, Villanova, PA 19085, USA. E-mail: [email protected] URL: http://www.csc.villanova.edu/&U0007E;japaridz/
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Abstract

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.

Keywords

Computability logicIntuitionistic logicInteractive computationGame semantics
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 243 - 276
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Blass, A., Degrees of indeterminacy of games, Fundamenta Mathematicae, vol. 77 (1972), pp. 151–166.CrossRefGoogle Scholar
[2]Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic, vol. 56 (1992), pp. 183–220.CrossRefGoogle Scholar
[3]Felscher, W., Dialogues, strategies, and intuitionistic provability, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 217–254.CrossRefGoogle Scholar
[4]Girard, J. Y., Linear logic, Theoretical Computer Scince, vol. 50 (1987), pp. 1–102.Google Scholar
[5]Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280–287.CrossRefGoogle Scholar
[6]Japaridze, G., Introduction to computability logic, Annals of Pure and Applied Logic, vol. 123 (2003), pp. 1–99.CrossRefGoogle Scholar
[7]Japaridze, G., Computability logic: A formal theory of interaction, Interactive Computation: The New Paradigm (Goldin, D., Smolka, S., and Wegner, P., editors), Springer-Verlag, Berlin, 2006, pp. 183–223.Google Scholar
[8]Japaridze, G., From truth to computability I, Theoretical Computer Science, vol. 357 (2006), pp. 100–135.CrossRefGoogle Scholar
[9]Japaridze, G., Propositional computability logic I, ACM Transactions on Computational Logic, vol. 7 (2006), no. 2, pp. 302–330.Google Scholar
[10]Japaridze, G., Propositional computability logic II, ACM Transactions on Computational Logic, vol. 7 (2006), no. 2, pp. 331–362.Google Scholar
[11]Japaridze, G., From truth to computability II, Theoretical Computer Science, to appear.Google Scholar
[12]Japaridze, G., Intuitionistic computability logic, Acta Cybernetica, to appear.Google Scholar
[13]Japaridze, G., In the beginning was game semantics, Logic and Games: Foundational Perspectives (Majer, O., Pietarinen, A.-V., and Tulenheimo, T., editors), Springer-Verlag, Berlin, to appear. Preprint is available at http://arxiv.org/abs/cs.LO/0507045.Google Scholar
[14]Kleene, S. C., Introduction to Metamathematics, D. van Nostrand Company, New York, Toronto, 1952.Google Scholar
[15]Kolmogorov, A. N., Zur Deutung der intuitionistischen Logik, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 35 (1932), pp. 58–65.Google Scholar
[16]Kripke, S., Semantical analysis of intuitionistic logic, Formal Systems and Recursive Functions (Crossley, J. and Dummet, M., editors), Amsterdam, 1965, pp. 92–130.Google Scholar
[17]Lorenzen, P., Ein dialogisches Konstruktivitätskriterium, Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, PWN, Warsaw, 1961, pp. 193–200.Google Scholar
[18]Medvedev, Y., Interpretation of logical formulas by means of finite problems and its relation to the realiability theory, Soviet Mathematics Doklady, vol. 4 (1963), pp. 180–183.Google Scholar