Topological approach to solve P versus NP

TL;DR

This paper explores the P versus NP problem using a topological perspective on resolution principles, introducing RCNF and TCNF to analyze complexity and reducibility within CNF formulas.

Contribution

It presents a novel topological framework for understanding resolution in CNF, defining RCNF and TCNF to distinguish P-Complete and NP-Complete classes.

Findings

RCNF is P-Complete and represents the topology of resolution.

TCNF is NP-Complete and irreducible, indicating complexity differences.

CCNF cannot be polynomially reduced to RCNF, supporting P vs NP separation.

Abstract

This paper talks about difference between P and NP by using topological space that mean resolution principle. I pay attention to restrictions of antecedent and consequent in resolution, and show what kind of influence the restrictions have for difference of structure between P and NP regarding relations of relation. First, I show the restrictions of antecedent and consequent in resolution principle. Antecedents connect each other, and consequent become a linkage between these antecedents. And we can make consequent as antecedents product by using some resolutions which have same joint variable. We can determine these consequents reducible and irreducible. Second, I introduce RCNF that mean topology of resolution principle in CNF. RCNF is HornCNF and that variable values are presence of restrictions of CNF formula clauses. RCNF is P-Complete. Last, I introduce TCNF that have 3CNF's character which relate 2 variables relations with 1 variable. I show CNF complexity by using CCNF that combine some TCNF. TCNF is NP-Complete and product irreducible. I introduce CCNF that connect TCNF like Moore graph. We cannot reduce CCNF to RCNF with polynomial size. Therefore, TCNF is not in P.