Reversible Computations of One-Way Counter Automata

TL;DR

This paper investigates the computational power and limitations of reversible deterministic one-way multi-counter automata, revealing hierarchies, separation results, and undecidability issues in their computational capabilities.

Contribution

It introduces the concept of reversible automata, establishes hierarchy results using Kolmogorov complexity, and compares reversible and ordinary counters in computational power.

Findings

Reversible counters do not outperform ordinary counters with the same number.

An infinite hierarchy of reversible automata exists for exponential time.

Most decidability questions are undecidable for reversible multi-counter automata.

Abstract

Deterministic one-way time-bounded multi-counter automata are studied with respect to their ability to perform reversible computations, which means that the automata are also backward deterministic and, thus, are able to uniquely step the computation back and forth. We study the computational capacity of such devices and obtain separation results between irreversible and reversible k-counter automata for superpolynomial time. For exponential time we obtain moreover an infinite and tight hierarchy with respect to the number of counters. This hierarchy is shown with Kolmogorov complexity and incompressibility arguments. In this way, on passing we can prove this hierarchy also for ordinary counter automata. This improves the known hierarchy for ordinary counter automata in the sense that here we consider a weaker acceptance condition. Then, it turns out that k+1 reversible counters are not better than k ordinary counters and vice versa. Finally, almost all usually studied decidability questions turn out to be undecidable and not even semidecidable for reversible multi-counter automata, if at least two counters are provided.