Incomplete sets in P for logspace-reduction

TL;DR

This paper explores the behavior of Turing machines with time and space limits, showing that certain incomplete sets in P can be constructed, which implies L is not equal to P.

Contribution

It demonstrates the existence of logspace-incomplete sets in P and establishes that L is not equal to P based on these constructions.

Findings

Logspace-incomplete sets in P can be constructed.

When both time and space limits go to infinity, the problem becomes undecidable.

L is not equal to P.

Abstract

In this article, we investigate the behaviour of TMs with time limit and tape space limit. This problem is in P when the time limit is unary coded. If both limits go to infinity, it is undecidable which limit is exceeded first. Thus logspace-incomplete sets in P can be constructed. This implies L $\not=$ P.