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

Janin, David

doi:10.1007/978-3-319-04298-5_2

Cited by 8 publications

(7 citation statements)

References 33 publications

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“…Beyond these applications, it is expected that inverse semigroup theory can be developed much further towards application in computer science and engineering [34]. Thanks to Stephen's representation theorem [61], Munn's graphical representation theorem can be generalized to arbitrary inverse semigroup S generated by a distinguished subset A ⊆ S. It follows that, via quotient by adequate equations, the language theory that is developed here may go far beyond the study of languages of birooted trees themselves, possibly in link with recognizable languages of graphs [10].…”

Section: Resultsmentioning

confidence: 99%

On labeled birooted tree languages: Algebras, automata and logic

Janin

2015

Information and Computation

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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.

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Section: Resultsmentioning

confidence: 99%

On labeled birooted tree languages: Algebras, automata and logic

Janin

2015

Information and Computation

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“…It has already been observed that an adequate algebraic theory for inverse monoid morphisms can be developed by means of certain kind of premorphisms instead of morphisms [9,10,12,14]. As a matter of fact, transition monoids of walking automata induce a different type of premorphisms that could also be investigated as new language recognizers.…”

Section: Resultsmentioning

confidence: 99%

Walking Automata in Free Inverse Monoids

Janin

2016

Lecture Notes in Computer Science

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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...

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“…More recently inverse semigroups have attracted also the attention of physics and computer science scholars because several notions and tools of inverse semigroup theory are attuned towards questions in solid-state physics (particularly those concerning quasi-crystals) and the concrete modeling of time-sensitive interactive systems (see for instance [8,9, 10]). Besides, inverse semigroup theory gives rise to interesting algorithmic problems, and in the cases where such problems are decidable, the analysis of the computational complexity is a quite natural issue.…”

Section: Introductionmentioning

confidence: 99%

Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

Ayyash

Rodaro

2016

Publ Inst Math (Belgr)

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Abstract. We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems.

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

Cited by 8 publications

References 33 publications

On labeled birooted tree languages: Algebras, automata and logic

On labeled birooted tree languages: Algebras, automata and logic

Walking Automata in Free Inverse Monoids

Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

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