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 languages 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 the classes REG and MSO. In particular, we show that adding pebbles strictly increases the expressive power of twoway automata recognizing languages of tiles, but the hierarchy induced by the number of allowed pebbles collapses to level one. We now prove several closure properties of the class MSO of MSO-definable languages of tiles with an associated corollary. Theorem 4.3 (Robustness). For all MSO-definable languages of tiles M, N ⊆ T (A), the languages

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In this paper, we study languages of finite and infinite birooted words. We show how the embedding of free ω-semigroups of finite and infinite words into the monoid of birooted words can be generalized to the embedding of two-sorted ω-semigroups into (some notion of) onesorted ordered ω-monoids. This leads to an algebraic characterization of regular languages of finite and infinite birooted words that generalizes and unifies the known algebraic characterizations of regular languages of finite and infinite words 1 .