Is Space a Stronger Resource than Time? Positive Answer for the Nondeterministic at-Least-Quadratic Time Case

TL;DR

This paper demonstrates that for nondeterministic Turing machines running in at least quadratic time, the space complexity can be significantly reduced to the square root of the time bound, challenging the resource hierarchy.

Contribution

It introduces a simulation method showing space can be substantially less than time for certain nondeterministic computations, a novel insight in complexity theory.

Findings

Languages in time n^2 can be accepted in space O(n)

Space complexity can be reduced to the square root of time for these languages

The method is specific to certain Turing machine models and cannot be generalized easily.

Abstract

We show that all languages accepted in time f(n) >= n^2 can be accepted in space O(f(n)^{1/2})_and_ in time O(f(n)). The proof is carried out by simulation, based on the idea of guessing the sequences of internal states of the simulated TM when entering certain critical cells, whose location is also guessed. Our method cannot be generalised easily to many-tapes TMs, and in no case can it be relativised.