Ptarithmetic

Abstract

The present article introduces ptarithmetic (short for “polynomial time arithmetic”) — a formal number theory similar to the well known Peano arithmetic, but based on the recently born computability logic instead of classical logic. The formulas of ptarithmetic represent interactive computational problems rather than just true/false statements, and their “truth” is understood as existence of a polynomial time solution. The system of ptarithmetic elaborated in this article 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 effectively 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 self-contained, and can be read without any prior familiarity with computability logic.