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A correspondence between the time and space complexity

Ivan V. Latkin

TL;DR

The paper develops a framework to encode long deterministic two-tape Turing-machine computations as quantified Boolean formulae using a generalized Meyer–Stockmeyer approach. By constructing the modeling formula $\Omega^{(m)}(X,P)$, it shows that $ extbf{EXP} = extbf{PSPACE}$ and derives tight bounds relating input length, time, and space for these encodings, including the extension to $^{(k+1)}EXP = ^kEXPSPACE$ via padding. A key finding is that languages in $ extbf{P}$ can be recognized with almost logarithmic space, achieved by a careful, locality-focused construction that encodes only a small portion of tape content per step. These results have implications for lower bounds on decidable theories and demonstrate a deep link between time-bounded and space-bounded complexity via formula-based simulations, with discussion of potential extensions and limitations for nondeterministic or relativized contexts.

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

A correspondence between the time and space complexity

TL;DR

The paper develops a framework to encode long deterministic two-tape Turing-machine computations as quantified Boolean formulae using a generalized Meyer–Stockmeyer approach. By constructing the modeling formula , it shows that and derives tight bounds relating input length, time, and space for these encodings, including the extension to via padding. A key finding is that languages in can be recognized with almost logarithmic space, achieved by a careful, locality-focused construction that encodes only a small portion of tape content per step. These results have implications for lower bounds on decidable theories and demonstrate a deep link between time-bounded and space-bounded complexity via formula-based simulations, with discussion of potential extensions and limitations for nondeterministic or relativized contexts.

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
Paper Structure (27 sections, 27 theorems, 43 equations)

This paper contains 27 sections, 27 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Is the equality of the $\mathbf{EXP}$ and $\mathbf{PSPACE}$ classes possible?
  3. Main goals and objectives; the paper's structure
  4. Used methods and the main idea
  5. The main results
  6. The computational complexity of decidable theories
  7. Complexity classes and complete problems
  8. Necessary preparations for the proof of the main theorem
  9. On the Turing machines
  10. On the recording of Boolean formulae
  11. The beginning of the construction of modeling formulae
  12. The primary (basic) and auxiliary variables
  13. The description of the instruction actions
  14. The description of the running steps and configurations
  15. One step.
  16. ...and 12 more sections

Key Result

Theorem 1.1

For each one-tape deterministic Turing machine $P$ and every input string $X$, one can write a closed formula (sentence) $\Omega(X,P)$ of the signature of the two-element Boolean algebra $\mathcal{B}$ with the following properties: (i) $\Omega(X,P)$ can be written within polynomial time of $|X|$ an

Theorems & Definitions (53)

  • Theorem 1.1: Lat22
  • Corollary 1.2
  • Remark 1.3
  • Theorem 2.1: on modeling
  • Corollary 2.3: of the main theorem
  • proof
  • Proposition 2.4: Lat22, the generalized Rabin and Fischer method
  • Corollary 2.5
  • proof
  • Corollary 2.6: Lat22
  • ...and 43 more