Two-way Automata and Regular Languages of Overlapping Tiles

In this paper, we show how the study of two-way automata on words may relevantly be extended to the study of two-way automata on one-dimensional overlapping tiles that generalize finite words. Indeed, over tiles, languages recognizable by finite two-way automata (resp. multi-pebble automata) coincide with languages definable by Kleene’s (resp. Kleene’s extended) regular expressions. As an immediate corollary, if we restrict our observations to words, we obtain a new proof of Shepherdson’s theorem which posits that every finite state two-way automaton is equivalent to a finite state one-way automaton. We also obtain a new proof that this is still true for two-way automata with pebbles. Concerning tiles, we show that adding pebbles strictly increases the expressive power of two way automata. The hierarchy induced by the number of allowed pebbles is however shown to collapse to level one. A single pebble is enough to reach maximal expressive power: the class of languages definable in monadic second order logic.