Expansions, free inverse semigroups, and Schützenberger product

AbstractIn this paper we shall present a new construction of the free inverse monoid on a set X. Contrary to the previous constructions of [9, 11], our construction is symmetric and originates from classical ideas of language theory. The ingredients of this construction are the free group on X and the relation that associates to a word w of the free monoid on X, the set of all pairs (u, v) such that uv = w. It follows at once from our construction that the free inverse monoid on X can be naturally embedded into the Schützenberger product of two free groups of basis X. We shall also give some connections with the theory of expansions as developed by Rhodes and Birget [2, 3]

Towards a Higher-Dimensional String Theory for the Modeling of Computerized Systems

International audienceRecent modeling experiments conducted in computational music give evidence that a number of concepts, methods and tools belonging to inverse semigroup theory can be attuned towards the concrete modeling of time-sensitive interactive systems. Further theoretical developments show that some related notions of higher-dimensional strings can be used as a unifying theme across word or tree automata theory. In this invited paper, we will provide a guided tour of this emerging theory both as an abstract theory and with a view to concrete applications

Two-way automata and regular languages of overlapping tiles

International audienceWe 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 au- tomata (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 two-way automata recognizing languages of tiles, but the hierarchy induced by the number of allowed pebbles collapses to level one

On labeled birooted tree languages: algebras, automata and logic

International audienceWith 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

Syntactic semigroups

This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology