Computability vs. Nondeterministic and P vs. NP

TL;DR

This paper explores the relativity of computability and nondeterminism, reinterprets foundational theories, and introduces new concepts like NP-algorithm, ultimately arguing that P does not equal NP.

Contribution

It redefines nondeterminism, challenges existing complexity class assumptions, and proposes NP-algorithm as an effective approximation method for NP problems.

Findings

NP is nondeterministic problem

NP-algorithm approximates NP effectively

P does not equal NP

Abstract

This paper demonstrates the relativity of Computability and Nondeterministic; the nondeterministic is just Turing's undecidable Decision rather than the Nondeterministic Polynomial time. Based on analysis about TM, UM, DTM, NTM, Turing Reducible, beta-reduction, P-reducible, isomorph, tautology, semi-decidable, checking relation, the oracle and NP-completeness, etc., it reinterprets The Church-Turing Thesis that is equivalent of the Polynomial time and actual time; it redefines the NTM based on its undecidable set of its internal state. It comes to the conclusions: The P-reducible is misdirected from the Turing Reducible with its oracle; The NP-completeness is a reversal to The Church-Turing Thesis; The Cook-Levin theorem is an equipollent of two uncertains. This paper brings forth new concepts: NP (nondeterministic problem) and NP-algorithm (defined as the optimal algorithm to get the best fit approximation value of NP). P versus NP is the relativity of Computability and Nondeterministic, P/=NP. The NP-algorithm is effective approximate way to NP by TM.