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

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

Embedding finite and infinite words into overlapping tiles

International audienceIn 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) one- sorted 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

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Tiled Polymorphic Temporal Media

International audienceTiled Polymorphic Temporal Media (Tiled PTM) is an algebraic approach to specifying the composition of multimedia values having an inherent temporal quality --- for example sound clips, musical scores, computer animations, and video clips. Mathematically, one can think of a tiled PTM as a tiling in the one dimension of time. A tiled PTM value has two synchronization marks that specify, via an effective notion of tiled product, how the tiled PTMs are positioned in time relative to one another, possibly with overlaps. Together with a pseudo inverse operation, and the related reset and co-reset projection operators, the tiled product is shown to encompass both sequential and parallel products over temporal media. Up to observational equivalence, the resulting algebra of tiled PTM is shown to be an inverse monoid: the pseudo inverse being a semigroup inverse. These and other algebraic properties are explored in detail. In addition, recursively-defined infinite tiles are considered. Ultimately, in order for a tiled PTM to be \emph{renderable}, we must know its beginning, and how to compute its evolving value over time. Though undecidable in the general case, we define decidable special cases that still permit infinite tilings. Finally, we describe an elegant specification, implementation, and proof of key properties in Haskell, whose lazy evaluation is crucial for assuring the soundness of recursive tiles. Illustrative examples, within the Euterpea framework for musical temporal media, are provided throughout