Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

TL;DR

This paper reviews the complex computational properties of multi-head finite automata, highlighting their undecidable problems and exploring subclasses and characterizations to understand their computational limits.

Contribution

It provides a survey of the literature on multi-head finite automata, focusing on their complexity, undecidability issues, and open problems in the field.

Findings

Undecidability of emptiness, finiteness, universality, and equivalence.

Existence of subclasses with better decidability properties.

Open problems related to characterizations of multi-head automata.

Abstract

Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature.