Complexity of multi-head finite automata: Origins and directions

Multi-head finite automata were introduced and first investigated by Rabin and Scott in 1964 and Rosenberg in 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. Additionally the conversions between different types of multi-head finite automata induce in most cases size bounds that cannot be bounded by any recursive function, so-called non-recursive trade-offs. 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.