Intuitionistic computability logic

TL;DR

This paper establishes a soundness proof for Heyting's intuitionistic calculus within computability logic, providing a rigorous semantic foundation that aligns constructivistic logic with computational problem-solving.

Contribution

It offers the first formal semantic justification of INT using computability logic, bridging constructivistic logic and computational semantics.

Findings

Proof of soundness of INT in CL semantics

CL provides a rigorous semantic basis for intuitionistic logic

Supports Kolmogorov's thesis linking INT and problem logic

Abstract

Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and "truth" is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting's intuitionistic calculus INT, whose language can be seen as a special fragment of that of CL. The constructivistic philosophy of INT, however, has never really found an intuitively convincing and mathematically strict semantical justification. CL has good claims to provide such a justification and hence a materialization of Kolmogorov's known thesis "INT = logic of problems". The present paper contains a soundness proof for INT with respect to the CL semantics. A comprehensive online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.html