Two-way automata and regular languages of overlapping tiles
Résumé
We consider classes of languages of overlapping tiles, i.e. subsets of the McAlister monoid: the class REG of languages definable by Kleene's regular expressions, the class MSO of languages definable by formulas of monadic second-order logic, and the class REC of lan- guages definable by morphisms into finite monoids. By extending the semantics of finite-state two-way automata (possibly with pebbles) from languages of words to languages of tiles, we obtain a complete characterization of these classes. We show that adding pebbles strictly increases the expressive power of two-way automata recognizing languages of tiles, but the hierarchy induced by the number of allowed pebbles collapses to level one. Our study yields, as an immediate corollary, Shepherdson's result (and its extension to pebble automata) that, for languages of words, every finite-state two-way automaton is equivalent to a finite-state one-way automaton.
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