Symmetry and Uncountability of Computation

TL;DR

This paper explores the relationship between symmetry, order structure, and computational complexity, establishing that orderly problems are P-complete and chaotic problems are NP-complete, thereby supporting P != NP.

Contribution

It introduces the concepts of orderly and chaotic problems, linking symmetry and order structure to complexity classes, and provides new conditions for P and NP completeness.

Findings

Orderly problems are P-complete.

Chaotic problems are NP-complete.

P is not equal to NP.

Abstract

This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named "testing problem". And I get some condition of the P complete and NP complete by using testing problem. Second, I make two problem "orderly problem" and "chaotic problem". Orderly problem have some order structure. And DTM can limit some possible symbol effectly by using symmetry of each symbol. But chaotic problem must treat some symbol as a set of symbol, so DTM cannot limit some possible symbol. Orderly problem is P complete, and chaotic problem is NP complete. Finally, I clear the computation time of orderly problem and chaotic problem. And P != NP.