Computational Complexity on Signed Numbers

TL;DR

This paper explores a new way to represent natural numbers and examines how the presence or absence of the first Peano axiom affects computational complexity, suggesting that P equals NP with the axiom and P is a subset of NP without it.

Contribution

It introduces a novel representation of natural numbers and analyzes its implications for computability and complexity, challenging traditional assumptions about the Peano axioms.

Findings

With the Peano axiom, P equals NP.

Without the Peano axiom, P is a proper subset of NP.

Uncertainty in natural number existence affects computational complexity.

Abstract

This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of natural numbers is not essential. It claims that natural numbers remain usable in traditional ways without assuming the existence of at least one natural number. However, the uncertainty about the existence of natural numbers translates into every computation and introduces intrinsic uncertainty that cannot be avoided. The uncertainty in the output of a computation can be reduced, though, at the expense of a longer runtime and thus higher complexity. For the new representation of natural numbers, the paper claims that, with the first Peano axiom, P is equal to NP, and that without the first Peano axiom, P becomes a proper subset of NP.