Considerations on P vs NP

TL;DR

The paper argues that P does not equal NP by demonstrating that satisfiability problems require non-polynomial time for any search algorithm, due to the exponential size of hard instances.

Contribution

It introduces a proof that satisfiability problems inherently require non-polynomial time, supporting the separation of P and NP classes.

Findings

Any search algorithm faces exponential hard instances.

Hard instances require non-polynomial time to solve.

Satisfiability problem lacks polynomial complexity.

Abstract

In order to prove that the P of problems is different to the NP class, we consider the satisfability problem of propositional calculus formulae, which is an NP-complete problem. It is shown that, for every search algorithm A, there is a set E(A) containing propositional calculus formulae, each of which requires the algorithm A to take non-polynomial time to find the truth-values of its propositional letters satisfying it. Moreover, E(A)'s size is an exponential function of n, which makes it impossible to detect such formulae in a polynomial time. Hence, the satisfability problem does not have a polynomial complexity