Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T05:10:00.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards applied theories based on computability logic

Published online by Cambridge University Press:  12 March 2014

Giorgi Japaridze
Giorgi Japaridze*
Affiliation:
Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, Pa 19085, USA. E-mail: giorgi.japaridze@villanova.edu, URL: http://www.csc.villanova.edu/~japaridz/
Get accessRights & Permissions [Opens in a new window]

Abstract

Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, “truth” means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logicbased Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings—an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010, pp. 565 - 601
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Japaridze, G., Introduction to computability logic, Annals of Pure and Applied Logic, vol. 123 (2003), no. 1–3, pp. 1–99.CrossRefGoogle Scholar
[2]Japaridze, G., Computability logic: a formal theory of interaction, Interactive computation: The new paradigm (Goldin, D., Smolka, S., and Wegner, P., editors), Springer, 2006, pp. 183–223.CrossRefGoogle Scholar
[3]Japaridze, G., From truth to computability I, Theoretical Computer Science, vol. 357 (2006), no. 1–3, pp. 100–135.CrossRefGoogle Scholar
[4]Japaridze, G., Introduction to cirquenl calculus and abstract resource semantics, Journal of Logic and Computation, vol. 16 (2006), no. 4, pp. 489–532.CrossRefGoogle Scholar
[5]Japaridze, G., Propositional computability logic I, ACM Transactions on Computational Logic, vol. 7 (2006), no. 2, pp. 302–330.CrossRefGoogle Scholar
[6]Japaridze, G., Propositional computability logic II, ACM Transactions on Computational Logic, vol. 7 (2006). no. 2. pp. 331–362.CrossRefGoogle Scholar
[7]Japaridze, G., From truth to computability II, Theoretical Computer Science, vol. 379 (2007), no. 1–2, pp. 20–52.CrossRefGoogle Scholar
[8]Japaridze, G., Intuitionistic computability logic, Acta Cybernetica, vol. 18 (2007), no. 1, pp. 77–113.Google Scholar
[9]Japaridze, G., The intuitionistic fragment of computability logic at the propositional level, Annals of Pure and Applied Logic, vol. 147 (2007), no. 3, pp. 187–227.CrossRefGoogle Scholar
[10]Japaridze, G., The logic of interactive Turing reduction, this Journal, vol. 72 (2007), no. 1, pp. 243–276.Google Scholar
[11]Japaridze, G., Cirquent calculus deepened, Journal of Logic and Computation, vol. 18 (2008), no. 6, pp. 983–1028.CrossRefGoogle Scholar
[12]Japaridze, G., Sequential operators in computability logic, Information and Computation, vol. 206 (2008), no. 12, pp. 1443–1475.CrossRefGoogle Scholar
[13]Japaridze, G., In the beginning was game semantics, Games: Unifying logic, language, and philosophy (Majer, O., Pietarinen, A.-V., and Tulenheimo, T., editors), Springer, 2009, pp. 249–350.CrossRefGoogle Scholar
[14]Japaridze, G., Many concepts and two logics of algorithmic reduction, Studia Logica, vol. 91 (2009). no. 1, pp. 1–24.CrossRefGoogle Scholar
[15]Kleene, S. C., Introduction to metamathematics, D. van Nostrand Company, New York/Toronto, 1952.Google Scholar