A Proof for P =? NP Problem

TL;DR

This paper claims to provide a theoretical proof that P equals NP by constructing a recursive sequence of Turing machines and defining related language classes, ultimately concluding P=NP.

Contribution

It introduces a novel recursive definition of Turing machines and a new language class to prove P equals NP, addressing a longstanding open problem.

Findings

Proves P=NP using a recursive Turing machine sequence.

Defines a language class Up as the union of decidable languages.

Establishes P=Up and Up=NP, leading to P=NP.

Abstract

The $\textbf{P}$ vs. $\textbf{NP}$ problem is an important problem in contemporary mathematics and theoretical computer science. Many proofs have been proposed to this problem. This paper proposes a theoretic proof for $\textbf{P}$ vs. $\textbf{NP}$ problem. The central idea of this proof is a recursive definition for Turing machine (shortly TM) that accepts the encoding strings of valid TMs. By the definition, an infinite sequence of TM is constructed, and it is proven that the sequence includes all valid TMs. Based on these TMs, the class $\textbf{D}$ that includes all decidable languages and the union and reduction operators are defined. By constructing a language $\textbf{Up}$ of the union of $\textbf{D}$, it is proved that $\textbf{P}=\textbf{Up}$ and $\textbf{Up}=\textbf{NP}$, and the result $\textbf{P}=\textbf{NP}$ is proven.