Two-way Automata and Regular Languages of Overlapping Tiles

Information and Computation

2015

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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. 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, a class that contains classical regular languages of finite (mono-rooted) trees. This contrasts with the known collapse of classical algebraic tools when applied to inverse semigroups.

Section: Proof (⇒)mentioning

confidence: 93%

Section: Proof (⇒)mentioning

confidence: 97%

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On labeled birooted tree languages: Algebras, automata and logic

Information and Computation

2015

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“…The walking automata defined here are walking automata with unmarked and invisible pebbles in the sense of [6]. However, generalizing Pécuchet's study of two-way automata on strings [18] (see also [11,5]), we do not require that accepting runs starts and ends in the same vertex of the input structures. Moreover, our definition also differs from the definition proposed in [6,1] in the sense that, a priori, the absence of edges cannot be detected by the automata and the walking automaton is not required to traverse the entire structure.…”

Section: Remark 11mentioning

confidence: 99%

“…The more classical notion of accepting runs defined by complete traversals of the input structure can be related with ours as follows. In particular, when the birooted structures cannot be related by morphisms (as with end markers in two-way word automata [5]), studying strict recognizability just amounts to study recognizability.…”

Section: Then We Havementioning

confidence: 99%

Walking Automata in Free Inverse Monoids

Lecture Notes in Computer Science

2016

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Walking 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. 1 Introduction General context. Walking automata, be they running over words, trees or even graphs, possibly extended with pebbles, have long been defined and studied in Computer Science[7,8]. For instance, tree walking automata with pebbles have been an important subject of study the last decades since they are natural abstract models of machine for XML query languages such as XPATH, or XML transformation languages such as XSL [6]. Although based on well studied computation models: finite state machines or pushdown automata, questions about walking automata are often surprisingly complex to solve and, to a lesser extent, quite dependent on such or such details in automata's definition. For instance, in the case of tree languages, bounding the number of pebbles an automaton leads to defining classes of recognizable languages. Various logical characterizations of these classes have been obtained [8] and difficult separation results have also been proved [2,3,1]. However, for separation results, proof arguments apply to the case of pebbles that are marked and visible [2,3], leaving open the cases of unmarked and/or invisible pebbles. Even though walking automata are sequential machines much like string automata, the classical algebraic tools that have been developed to study word automata are not easily applicable to tree walking automata. Despite numerous results, little is known about the underlying mathematical framework, say in algebra, that walking automata may induce. Contribution of the paper. In this paper, we initiate the development of an algebraic framework, within inverse semigroup theory, for walking automata. We provide a generic notion of automata walking on edge-labeled graphs. They act as some kind of observers of their input graphs much in the same way observational semantics has been defined in concurrency theory by Hennessy and Milner [17]. Unlike most classical definitions, we do not require walking automata to start and end in the same vertex, neither do we require the complete traversal of input structures. Moreover, the capacity given to a walking automaton to check or not the absence of an (incoming or outgoing) edge labeled by a given lett...

Embedding Finite and Infinite Words into Overlapping Tiles

Dicky

Developments in Language Theory

2014

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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 .