On labeled birooted tree languages: algebras, automata and logic

International audience; With an aim to developing expressive language theoretical tools applicable to inverse semigroup languages, that is, subsets of inverse semigroups, this paper explores the language theory of finite labeled birooted trees: Munn's birooted trees extended with vertex labeling. To this purpose, we define a notion of finite state birooted tree automata that simply extends finite state word automata semantics. This notion is shown to capture the class of languages that are definable in Monadic Second Order Logic and upward closed with respect to the natural order defined in the inverse monoid structure induced by labeled birooted trees. Then, we derive from these automata the notion of quasi-recognizable languages, that is, languages recognizable by means of (adequate) premorphisms into finite (adequately) ordered monoids. This notion is shown to capture finite Boolean combinations of languages as above. Applied to a simple encoding of finite (mono-rooted) labeled tree languages in of labeled birooted trees, we show that classical regular languages of finite (mono-rooted) trees are quasi-recognizable in the above sense. The notion of quasi-recognizability thus appears as an adequate remedy to the known collapse of the expressive power of classical algebraic tools when applied to inverse semigroups. Illustrative examples, in relation to other known algebraic or automata theoretic frameworks for defining languages of finite trees, are provided throughout.