Towards applied theories based on computability logic

Abstract

Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) 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.

To appear in 2010 in the Journal of Symbolic Logic

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Cited by in corpus (9)

• The taming of recurrences in computability logic through cirquent calculus, Part I
• Toggling operators in computability logic
• From formulas to cirquents in computability logic
• Introduction to clarithmetic II
• A logical basis for constructive systems
• A new face of the branching recurrence of computability logic
• Introduction to clarithmetic I
• Separating the basic logics of the basic recurrences
• Ptarithmetic