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 l ogic, and the class REC of languages definable by morphisms into finite monoids. By extending the s emantics of finite-state two-way automata (possibly with pebbles) from languages of words to languages of tiles, we obtain a com- plete characterization of the classes REG and MSO . In particular, 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.
['Anne Dicky', 'David Janin']
Two-way Automata and Regular Languages of Overlapping Tiles