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

Inverse monoids of higher-dimensional strings

International audienceHalfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph mor-phism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Last, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory

Walking automata in free inverse monoids

International audienceWalking automata, be they running over words, trees or even graphs, possibly extended with pebbles that can be dropped and lifted on vertices, have long been defined and studied in Computer Science. However, questions concerning walking automata are surprisingly complex to solve. In this paper, we study a generic notion of walking automata over graphs whose semantics naturally lays within inverse semigroup theory. Then, from the simplest notion of walking automata on birooted trees, that is, elements of free inverse monoids, to the more general cases of walking automata on birooted finite subgraphs of Cayley's graphs of groups, that is, elements of free E-unitary inverse monoids, we provide a robust algebraic framework in which various classes of recognizable or regular languages of birooted graphs can uniformly be defined and related one with the other

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

Algebraic tools for the overlapping tile product

International audienceOverlapping tile automata and the associated notion of recognizability by means of (adequate) premorphisms in finite ordered monoids have recently been defined for coping with the collapse of classical recognizability in inverse monoids. In this paper, we investigate more in depth the associated algebraic tools that allows for a better understanding of the underlying mathematical theory. In particular, addressing the surprisingly difficult problem of language product and star, we eventually found some deep links with classical notions of inverse semigroup theory such as the notion of restricted product

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

A robust algebraic framework for high-level music writing and programming

International audienceIn this paper, we present a new algebraic model for music programming : tiled musical graphs. It is based on the idea that the definition of musical objects : what they are, and the synchronization of these objects : when they should be played, are two orthogonal aspects of music programming that should be kept separate although handled in a combined way. This leads to the definition of an algebra of music objects : tiled music graphs, which can be combined by a single operator : the tiled product, that is neither sequential nor parallel but both. From a mathematical point of view, this algebra is known to be especially robust since it is an inverse monoid. Various operators such as the reset and the coreset projections derive from these algebra and turned out to be fairly useful for music modeling. From a programming point of view, it provide a high level domain specific language (DSL) that is both hierarchical and modular. This language is currently under implementation in the functional programming language Haskell. From an applicative point of view, various music modeling examples are provided to show how notes, chords, melodies, musical meters and various kind of interpretation aspects can easily and robustly be encoded in this formalism

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

Programmer avec des tuiles musicales: le T-calcul en Euterpea

International audienceEuterpea est un langage de programmation dédié à la création et à la manipulation de contenus media temporisés - son, musique, animations, vidéo, etc... Il est enchassé dans un langage de programmation fonctionnelle avec typage polymorphe: Haskell. Il hérite ainsi de toute la souplesse et la robustesse d'un langage de programmation moderne. Le T-calcul est une proposition abstraite de modélisation temporelle qui, à travers une seule opération de composition: le produit tuilé, permet tout à la fois la composition séquentielle et la composition parallèle de contenus temporisés. En présentant ici une intégration du T-calcul dans le language Euterpea, nous réalisons un outil qui devrait permettre d'évaluer la puissance métaphorique du tuilage temporel combinée avec la puissance programmatique du langage Euterpea