A correspondence between the time and space complexity

TL;DR

This paper explores the relationship between time and space complexity in language recognition, proving that deterministic exponential time and polynomial space classes are equivalent, and showing polynomial time recognition can be achieved in nearly logarithmic space.

Contribution

It introduces a novel coding of Turing machine computations with short Boolean formulas, establishing key equivalences in complexity classes and improving lower bounds for decidable theories.

Findings

Deterministic exponential time and polynomial space classes coincide.

Languages recognized in polynomial time can be recognized in nearly logarithmic space.

Improved lower bounds for the complexity of decidable theories.

Abstract

We investigate the correspondence between the time and space recognition complexity of languages. For this purpose, we will code the long-continued computations of deterministic two-tape Turing machines by the relatively short-length quantified Boolean formulae. The modified Meyer and Stockmeyer method will appreciably be used for this simulation. It will be proved using this modeling that the complexity classes Deterministic Exponential Time and Deterministic Polynomial Space coincide. It will also be proven that any language recognized in polynomial time can be recognized in almost logarithmic space. Furthermore, this allows us slightly to improve the early founded lower complexity bound of decidable theories that are nontrivial relative to some equivalence relation (this relation may be equality) -- each of these theories is consistent with the formula, which asserts that there are two non-equivalent elements. Keywords: computational complexity, the coding of computations through formulae, exponential time, polynomial space, the lower complexity bound of the language recognition