Introduction to clarithmetic II

TL;DR

This paper extends the axiomatic system of clarithmetic based on computability logic to include three new systems that characterize polynomial space, elementary recursive, and primitive recursive computability, demonstrating their soundness and completeness.

Contribution

It introduces three new sound and complete clarithmetic systems for different computational complexity classes, expanding the framework established in the previous work.

Findings

Established soundness and completeness for polynomial space computability system.

Developed systems for elementary recursive and primitive recursive computability.

Extended the axiomatic framework to broader computational classes.

Abstract

The earlier paper "Introduction to clarithmetic I" constructed an axiomatic system of arithmetic based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html), and proved its soundness and extensional completeness with respect to polynomial time computability. The present paper elaborates three additional sound and complete systems in the same style and sense: one for polynomial space computability, one for elementary recursive time (and/or space) computability, and one for primitive recursive time (and/or space) computability.