Introduction to clarithmetic I

TL;DR

This paper introduces a new formal system called clarithmetic based on computability logic, which models interactive computational problems and their polynomial time solutions, providing a sound and complete framework for such problems.

Contribution

It presents a novel clarithmetic system for polynomial time computability, establishing its soundness and completeness in representing interactive problems.

Findings

The system accurately models polynomial time interactive problems.

Every theorem corresponds to a problem with a polynomial time solution.

Solutions can be efficiently extracted from proofs.

Abstract

"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics. Formulas of clarithmetical theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Imposing various complexity constraints on such solutions yields various versions of clarithmetic. The present paper introduces a system of clarithmetic for polynomial time computability, which is shown to be sound and complete. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a polynomial time solution and, furthermore, such a solution can be efficiently extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a polynomial time solution is represented by some theorem T of the system. The paper is written in a semitutorial style and targets readers with no prior familiarity with computability logic.