Small Jump with Negation-UTM Trampoline

TL;DR

This paper explores the complexity class hierarchy using fixpoints of Universal Turing Machines and negation, demonstrating new separations such as P not equal to NP through diagonalization arguments.

Contribution

It introduces a novel approach to complexity class separation by leveraging fixpoint properties of UTMs and negation, providing new proofs for class distinctions like P versus NP.

Findings

L is not P proven via UTM diagonalization

P is not NP demonstrated using similar hierarchy arguments

UTM and negation create new complexity class hierarchies

Abstract

This paper divide some complexity class by using fixpoint and fixpointless area of Decidable Universal Turing Machine (UTM). Decidable Deterministic Turing Machine (DTM) have fixpointless combinator that add no extra resources (like Negation), but UTM makes some fixpoint in the combinator. This means that we can jump out of the fixpointless combinator system by making more complex problem from diagonalisation argument of UTM. As a concrete example, we proof L is not P . We can make Polynomial time UTM that emulate all Logarithm space DTM (LDTM). LDTM set close under Negation, therefore UTM does not close under LDTM set. (We can proof this theorem like halting problem and time/space hierarchy theorem, and also we can extend this proof to divide time/space limited DTM set.) In the same way, we proof P is not NP. These are new hierarchy that use UTM and Negation.