Journal of Symbolic Logic

Towards applied theories based on computability logic

Giorgi Japaridze

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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-logic-based 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.

Article information

Source
J. Symbolic Logic Volume 75, Issue 2 (2010), 565-601.

Dates
First available: 18 March 2010

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1268917495

Digital Object Identifier
doi:10.2178/jsl/1268917495

Zentralblatt MATH identifier
05725864

Mathematical Reviews number (MathSciNet)
MR2648156

Subjects
Primary: 03F50: Metamathematics of constructive systems
Secondary: secondary: 03F30 03D75: Abstract and axiomatic computability and recursion theory 03F50: Metamathematics of constructive systems 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85] 68T27: Logic in artificial intelligence 68T30: Knowledge representation

Keywords
Computability logic game semantics Peano arithmetic constructive logics

Citation

Japaridze, Giorgi. Towards applied theories based on computability logic. Journal of Symbolic Logic 75 (2010), no. 2, 565--601. doi:10.2178/jsl/1268917495. http://projecteuclid.org/euclid.jsl/1268917495.


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