Finite and infinite basis in P and NP

TL;DR

This paper introduces a novel approach to the P vs NP problem using the concept of bases in function spaces, suggesting NP-Complete problems cannot be expressed within P due to their infinite dimensionality.

Contribution

It proposes a new perspective on P vs NP by analyzing the cardinality of bases in function spaces, linking problem complexity to basis dimensionality and fixed point operators.

Findings

NP-Complete problems are in infinite-dimensional function spaces.

P-Complete problems have at most finite bases.

NP-Complete problems cannot be described in P.

Abstract

This article provide new approach to solve P vs NP problem by using cardinality of bases function. About NP-Complete problems, we can divide to infinite disjunction of P-Complete problems. These P-Complete problems are independent of each other in disjunction. That is, NP-Complete problem is in infinite dimension function space that bases are P-Complete. The other hand, any P-Complete problem have at most a finite number of P-Complete basis. The reason is that each P problems have at most finite number of Least fixed point operator. Therefore, we cannot describe NP-Complete problems in P. We can also prove this result from incompleteness of P.