Logic and Automata

Logic and Automata: History and Perspectives

Copyright Date: 2008
Pages: 736
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  • Book Info
    Logic and Automata
    Book Description:

    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. This title is available in the OAPEN Library - http://www.oapen.org.

    eISBN: 978-90-485-0128-1
    Subjects: Technology

Table of Contents

  1. Front Matter
    (pp. 1-4)
  2. Table of Contents
    (pp. 5-6)
  3. Preface
    (pp. 7-8)
    J. F., E. G. and T. W.
  4. On the topological complexity of tree languages
    (pp. 9-28)
    André Arnold, Jacques Duparc, Filip Murlak and Damian Niwiński

    Since the discovery of irrational numbers, the issue of impossibility has been one of the driving forces in mathematics. Computer science brings forward a related problem, that of difficulty. The mathematical expression of difficulty is complexity, the concept which affects virtually all subjects in computing science, taking on various contents in various contexts.

    In this paper we focus on infinite computations, and more specifically on finite-state recognition of infinite trees. It is clearly not a topic of classical complexity theory which confines itself to computable functions and relations over integers or words, and measures their complexity by the—supposedly finite...

  5. Nondeterministic controllers of nondeterministic processes
    (pp. 29-52)
    André Arnold and Igor Walukiewicz

    At the end of the eighties, Ramadge and Wonham introduced the theory of control of discrete event systems (see the survey [13] and the books [6] and [3]). In this theory a process (also called aplant) is a deterministic non-complete finite state automaton over an alphabetAof events, which defines all possible sequential behaviours of the process. The goal is to find for a given plant another process, calledcontroller, such that a synchronous product of the plant and the controller satisfies desired properties. The usual properties considered are for instance, that some dangerous states are never reachable,...

  6. Reachability in continuous-time Markov reward decision processes
    (pp. 53-72)
    Christel Baier, Boudewijn R. Haverkort, Holger Hermanns and Joost-Pieter Katoen

    Having their roots in economics, Markov decision processes (MDPs, for short) in computer science are used in application areas such as randomised distributed algorithms and security protocols. The discrete probabilities are used to model random phenomena in such algorithms, like flipping a coin or choosing an identity from a fixed range according to a uniform distribution, whereas the nondeterminism in MDPs is used to specify unknown or underspecified behaviour, e.g., concurrency (interleaving) or the unknown malicious behavior of an attacker.

    MDPs—also considered as turn-based 1\frac{1}{2}-player stochastic games—consist of decision epochs, states, actions, and transition probabilities. On entering...

  7. Logical theories and compatible operations
    (pp. 73-106)
    Achim Blumensath, Thomas Colcombet and Christof Löding

    The aim of this article is to give a survey of operations that can be performed on relational structures while preserving decidability of theories. We mainly consider first-order logic (FO), monadic second-order logic (MSO), and guarded second-order logic (GSO, also called MS₂ by Courcelle). For example, we might be interested in an operationfthat takes a single structureaand produces a new structuref(a) such that the FO-theory off(a) can be effectively computed from the MSO-theory ofa(we call such operations (MSO, FO)-compatible), i.e., for each FO-formulaφoverf(a) we can construct an MSO-formulaφ*...

  8. Forest algebras
    (pp. 107-132)
    Mikołaj Bojańczyk and Igor Walukiewicz

    There is a well-known decision problem in formal language theory:

    Decide if a given a regular language of finite binary trees can be defined by a formula of first-order logic with three relations: ancestor, left and right successor.

    If the language is a word language (there is only one successor relation in this case) the problem is known to be decidable thanks to fundamental results of Schützenberger [14] and McNaughton and Papert [11]. The problem is also decidable for words when only the successor relation is available [18, 1]. However, no algorithm is known for the case of tree languages,...

  9. Automata and semigroups recognizing infinite words
    (pp. 133-168)
    Olivier Carton, Dominique Perrin and Jean-Éric Pin

    Among the many research contributions of Wolfgang Thomas, those regarding automata on infinite words and more generally, on infinite objects, have been highly inspiring to the authors. In particular, we should like to emphasize the historical importance of his early papers [33, 34, 35], his illuminating surveys [36, 37] and the Lecture Notes volume on games and automata [15].

    Besides being a source of inspiration, Wolfgang always had nice words for our own research on thealgebraic approachto automata theory. This survey, which presents this theory for infinite words, owes much to his encouragement.

    Büchi has extended the classical...

  10. Deterministic graph grammars
    (pp. 169-250)
    Didier Caucal

    Context-free grammars are one of the most classical and fundamental notions in computer science textbooks, in both theoretical and applied settings. As characterizations of the well-known class of context-free languages, they are a very prominent tool in the field of language theory. Since context-free grammars are powerful enough to express most programming languages, they also play an important role in compilation, where they form the basis of many efficient parsing algorithms.

    A similar notion can be adapted to the more general setting of grammars generating graphs instead of words. In this case, grammar rules no longer express the replacement of...

  11. Quantifier-free definable graph operations preserving recognizability
    (pp. 251-260)
    Bruno Courcelle

    Several algebras of graphs, and more generally of relational structures, can be defined in terms of disjoint union as unique binary operation and of several unary operations defined by quantifier-free formulas. These algebras are the basis of the extension to graphs and hypergraphs of thetheory of formal languagesin a universal algebra setting.

    In every algebra, one can define two families of subsets, the family ofequational setswhich generalizes the family of context-free languages, and the family ofrecognizable setswhich generalizes the family of recognizable languages. Equational sets are defined as least solutions of systems of recursive...

  12. First-order definable languages
    (pp. 261-306)
    Volker Diekert and Paul Gastin

    The study of regular languages is one of the most important areas in formal language theory. It relates logic, combinatorics, and algebra to automata theory; and it is widely applied in all branches of computer sciences. Moreover it is the core for generalizations, e.g., to tree automata [26] or to partially ordered structures such as Mazurkiewicz traces [6].

    In the present contribution we treat first-order languages over finite and infinite words. First-order definability leads to a subclass of regular languages and again: it relates logic, combinatorics, and algebra to automata theory; and it is also widely applied in all branches...

  13. Matrix-based complexity functions and recognizable picture languages
    (pp. 307-330)
    Dora Giammarresi and Antonio Restivo

    Picture (two-dimensional) languages were studied using different approaches and perspectives since the sixties as the natural counterpart in two dimensions of (one-dimensional) string languages. In 1991, a unifying point of view was presented in [6] where the family oftiling recognizable picture languagesis defined (see also [7]). The definition of recognizable picture language takes as starting point a well known characterization of recognizable string languages in terms of local languages and projections. Namely, any recognizable string language can be obtained as projection of a local string language defined over a larger alphabet. Such notion can be extended in a...

  14. Applying Blackwell optimality: priority mean-payoff games as limits of multi-discounted games
    (pp. 331-356)
    Hugo Gimbert and Wiesław Zielonka

    One of the major achievements of the theory of stochastic games is the result of Mertens and Neyman [15] showing that the values of mean-payoff games are the limits of the values of discounted games. Since the limit of the discounted payoff is related to Abel summability while the mean-payoff is related to Cesàro summability of infinite series, and classical abelian and tauberian theorems establish tight links between these two summability methods, the result of Mertens and Neyman, although technically very difficult, comes with no surprise.

    In computer science similar games appeared with the work of Gurevich and Harrington [12]...

  15. Logic, graphs, and algorithms
    (pp. 357-422)
    Martin Grohe

    In 1990, Courcelle [9] proved a fundamental theorem stating that graph properties definable in monadic second-order logic can be decided in linear time on graphs of bounded tree width. This is the first in a series ofalgorithmic meta theorems. More recent examples of such meta theorems state that all first-order definable properties of planar graphs can be decided in linear time [42] and that all first-order definable optimisation problems on classes of graphs with excluded minors can be approximated in polynomial time to any given approximation ratio [19]. The term “meta theorem” refers to the fact that these results...

  16. Non-regular fixed-point logics and games
    (pp. 423-456)
    Stephan Kreutzer and Martin Lange

    Modal and temporal logics. The most commonly used specification logics in the theory of computer aided verification are based on propositional modal logic augmented by temporal operators. Among those one can broadly distinguish between linear and branching time logics, depending on how they treat the temporal development of processes. The modalμ-calculus,L_{\mu}for short, provides a common generalization of most temporal logics. It is defined as the extension of basic propositional modal logic by rules to form the least and the greatest fixed point of definable monotone operators.

    L_{\mu}is aregularlogic in the sense that it can...

  17. The universal automaton
    (pp. 457-504)
    Sylvain Lombardy and Jacques Sakarovitch

    With every language is canonically associated an automaton, calledthe universal automaton of the language, which is finite whenever the language is regular. It is large, it is complex, it is complicated to compute, but it contains, hopefully, many interesting informations on the language. In the last forty years, it has been described a number of times, more or less explicitly, more or less approximately, in relation with one or another property of the language. This is what we review here systematically.

    The origin of the universal automaton is not completely clear. A well-publicized note [1] credits Christian Carrez of...

  18. Deterministic top-down tree automata: past, present, and future
    (pp. 505-530)
    Wim Martens, Frank Neven and Thomas Schwentick

    The goal of this article is to survey some results concerning deterministic top-down tree automata motivated by purely formal language theoretic reasons (past) and by the advent of the data exchange format XML (present). Finally, we outline some new research directions (future).

    The Past. Regular tree languages have been studied in depth ever since their introduction in the late sixties [10]. Just as for regular string languages, regular tree languages form a robust class admitting many closure properties and many equivalent formulations, the most prominent one in the form of tree automata. A striking difference with the string case where...

  19. Expressive power of monadic logics on words, trees, pictures, and graphs
    (pp. 531-552)
    Oliver Matz and Nicole Schweikardt

    There is a close relationship between (generalized) automata theory and the expressive power of certain monadic logics. Already in 1960, Büchi and Elgot proved that a word-language is recognizable by a finite automaton if, and only if, it can be characterized by a monadic second-order formula. Since then, various analogous results, e.g., for labeled trees rather than words, and also for more general classes of labeled graphs, have been obtained. Alluding to the notion of “descriptive complexity theory”, in his survey article [39] for theHandbook of Formal Languages, Wolfgang Thomas called the branch of research that investigates the relationship...

  20. Structured strategies in games on graphs
    (pp. 553-574)
    R. Ramanujam and Sunil Simon

    We discuss strategies in non-zero sum games of perfect information on graphs. The study of non-zero sum games on graphs is motivated by the advent of computational tasks on the world-wide web and related security requirements which have thrown up many interesting areas of interaction between game theory and computer science. For example, signing contracts on the web requires interaction between principals who do not know each other and typically distrust each other. Protocols of this kind which involveselfish agentscan be easily viewed as strategic games of imperfect information. These are complex interactive processes which critically involve players...

  21. Counting in trees
    (pp. 575-612)
    Helmut Seidl, Thomas Schwentick and Anca Muscholl

    Tree automata and logics for finite trees have been considered since the seminal work of Thatcher and Wright [38] in the late sixties, with emphasis onrankedtrees. More recently, research on semi-structured data and XML in particular, raised new questions aboutunrankedtrees, i.e., trees where the number of children of a node is not fixeda priori, [8, 22]. Trees in XML are unranked, labeled, and may occur in two versions, ordered or unordered, depending on whether the sequence of children of a node is ordered or not.

    In XML schema description languages like DTDs and XML Schema,...

  22. Modular quantifiers
    (pp. 613-628)
    Howard Straubing and Denis Thérien

    In the late nineteen-eighties much of our research concerned the application of semigroup-theoretic methods to automata and regular languages, and the connection between computational complexity and this algebraic theory of automata. It was during this period that we became aware of the work of Wolfgang Thomas. Thomas had undertaken the study of concatenation hierarchies of star-free regular languages—a subject close to our hearts—by model-theoretic methods. He showed that the levels of the dot-depth hierarchy corresponded precisely to levels of the quantifier alternation hierarchy within first-order logic [26], and applied Ehrenfeucht-Fraïssé games to prove that the dot-depth hierarchy was...

  23. Automata: from logics to algorithms
    (pp. 629-736)
    Moshe Y. Vardi and Thomas Wilke

    In his seminal 1962 paper [17], Büchi states: “Our results [. . .] may therefore be viewed as an application of the theory of finite automata to logic.” He was referring to the fact that he had proved the decidability of the monadic-second order theory of the natural numbers with successor function by translating formulas into finite automata, following earlier work by himself [16], Elgot [35], and Trakthenbrot [122]. Ever since, the approach these pioneers were following has been applied successfully in many different contexts and emerged as a major paradigm. It has not only brought about a number of...