DOCSLIB.ORG

Definability and Synthesis of Transductions Nathan Lhote

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Definability and synthesis of transductions Nathan Lhote To cite this version: Nathan Lhote. Definability and synthesis of transductions. Other [cs.OH]. Université de Bordeaux; Université libre de Bruxelles (1970-..), 2018. English. NNT : 2018BORD0185. tel-01960958 HAL Id: tel-01960958 https://tel.archives-ouvertes.fr/tel-01960958 Submitted on 19 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` en cotutelle present´ ee´ pour obtenir le grade de Docteur de l'Universite´ de Bordeaux et de l'Universite´ Libre de Bruxelles Ecole´ doctorale mathematiques´ et informatique specialit´ e´ informatique Par Nathan Lhote Definissabilit´ e´ et Synthese` de Transductions Definability and Synthesis of Transductions Sous la direction d'Emmanuel Filiot et d'Olivier Gauwin Soutenue le 12 octobre 2018 Membres du Jury: Miko laj Boja´nczyk Professeur, Uniwersytet Warszawski Examinateur Thomas Colcombet Directeur de recherche CNRS, Universit´eParis-Diderot Rapporteur Emmanuel Filiot Chercheur qualifi´eFNRS, Universit´eLibre de Bruxelles Directeur Olivier Gauwin Ma^ıtrede conf´erences,Universit´ede Bordeaux Directeur Christof L¨oding Chercheur associ´e,RWTH Aachen Rapporteur Anca Muscholl Professeure, Universit´ede Bordeaux Examinatrice Jean-Fran¸coisRaskin Professeur, Universit´eLibre de Bruxelles Examinateur Helmut Seidl Professeur, Technische Universit¨atM¨unchen Pr´esident Definissabilit´ e´ et Synthese` de Transductions Titre D´efinissabilit´eet Synth`esede Transductions R´esum´e Dans la premi`erepartie de ce manuscrit nous ´etudions les fonctions rationnelles, c’est-`a-dired´efiniespar des transducteurs unidirectionnels. Notre objectif est d'´etendreaux transductions les nombreuses correspondances logique-alg`ebrequi ont ´et´e´etablies concernant les langages, notamment le c´el`ebreth´eor`emede Sch¨utzenberger-McNaughton-Papert. Dans le cadre des fonctions rationnelles sur les mots finis, nous obtenons une caract´erisation`ala Myhill-Nerode en termes de congruences d'indice fini. Cette caract´erisation nous permet d'obtenir un r´esultat de transfert, `apartir d'´equivalences logique-alg`ebrepour les langages vers des ´equivalences pour les transductions. En particulier nous montrons comment d´ecidersi une fonction rationnelle est d´efinissableen logique du premier ordre. Sur les mots infinis, nous pouvons ´egalement d´ecider la d´efinissabilit´een logique du premier ordre, mais avec des r´esultatsmoins g´en´eraux. Dans la seconde partie nous introduisons une logique pour les transductions et nous r´esolvons le probl`emede synth`eser´eguli`ere: ´etant donn´eeune formule de la logique, peut-on obtenir un transducteur bidirectionnel d´eterministesatisfaisant la formule ? Plus pr´ecis´ement nous fournissons un algorithme qui produit toujours une fonction r´eguli`eresatisfaisant une sp´ecification donn´eeen entr´ee.Nous exposons ´egalement un lien int´eressant entre les transductions et les mots avec donn´ees.Par cons´equent nous obtenons une logique expressive pour les mots avec donn´ees, pour laquelle le probl`emede satisfiabilit´eest d´ecidable. Mots-clefs Transductions, minimisation, congruence syntaxique, aperiodicit´e,logique du sec- ond ordre monadique, logique du premier ordre, origine, v´erification,synth`ese,data words Title Definability and Synthesis of Transductions Abstract In the first part of this manuscript we focus on the study of rational functions, functions defined by one-way transducers. Our goal is to extend to transductions the many logic-algebra correspondences that have been established for languages, such as the celebrated Sch¨utzenberger-McNaughton-Papert Theorem. In the case of rational functions over finite words, we obtain a Myhill-Nerode-like characterization in terms of congruences of finite index. This characterization allows us to obtain a transfer result from logic-algebra equivalences for languages to logic-algebra equivalences for transductions. In particular, we show that one can decide if a rational function can be defined in first-order logic. Over infinite words, we obtain weaker results but are still able to decide first-order definability. In the second part we introduce a logic for transductions and solve the regular synthesis problem: given a formula in the logic, can we obtain a two-way deterministic transducer satisfying the formula? More precisely, we give an algorithm that always produces a regular function satisfying a given specification. We also exhibit an interesting link between transductions and words with ordered data. Thus we obtain as a side result an expressive logic for data words with decidable satisfiability. Keywords Transductions, minimization, syntactic congruence, aperiodicity, monadic second- order logic, first-order logic, origin, verification, synthesis, data words Laboratoires d'accueil Laboratoire Bordelais de Recherche en Informatique, 351 Cours de la Lib´eration,33405 Talence D´epartement d'informatique de l'Universit´eLibre de Bruxelles, B^atiment NO 8`eme´etage, Campus de la Plaine, ULB CP212, boulevard du Triomphe, 1050 Bruxelles ii Remerciements Ces ann´eesde th`eseont ´et´eune ´etape clef de ma vie pendant laquelle j'ai beaucoup appris et beaucoup chang´e. Une constante durant ces ann´eesest le soutient que j'ai re¸cude la part des personnes qui m'entourent et qui ont contribu´e`arendre ces ann´eesheureuses. Je souhaite commencer par remercier mes encadrants de th`ese,Manu, Olivier et Anca bien s^ur.Je me sens extr^emement privil´egi´ed'avoir eu trois encadrants d'une telle qualit´eet j'ai ´enorm´ement appris `aleur contact pendant ces trois virgule cinq ann´ees. Merci aux membres de mon jury et en particulier `aThomas et Christof d'avoir accept´ede relire mon manuscrit et merci pour vos remarques pertinentes. Merci aux membres du LaBRI qui contribuent `aune ambiance de travail tr`esagr´eable.Merci `aMohammed pour des discussions endiabl´eessur le cours du mitigeur, `aLouis-Marie pour ses calembours interstellaires, `aLuis pour les parties de ping-pong acrobatique, `aFilip pour son amour des cheese nan kebabs, `aNath pour ses le¸consd'escalade. Merci aux nombreux membres du bureau 123 pour une bonne ambiance: Edon, F´elix,Adithya, Varun, Jason, Paul, Karim, Kinda, Rohan, etc. Merci ´egalement aux membres de l'ULB, merci `aLuc de m'avoir support´e dans son bureau, `aIsma¨elpour ses discussions matinales, `aGuillermo pour la fondue, `aMarie pour les sabl´es,`aNicolas pour ses tenues ray´ees,`aL´eopour les soir´eesmusicales. Je souhaite ´egalement remercier les gestionnaires du LaBRI et de l'ULB qui nous rendent la vie infiniment plus facile, en particulier Emmanuelle, V´eronique,Pascaline et Maryka. Merci aux matheux de cette annexe du LaBRI qu'est L'IMB. Merci `aThibaut 'Magic' Kritter pour les parties de basket, merci `aElsa, Nicolas, Jonathan et les autres pour les verres en terrasse et les nombreuses et inoubliables soir´ees nlambda. Merci Elizabeth pour la d´egustationde fruits de mer et merci Cannelle pour les verres lu- mineux. Merci aux potes de Rennes, Vincent meilleur perdant que gagnant, Sarah pour ces supers ann´eescoloc', Lucile pour le voyage au bout du monde, Robert pour les soir´eesLudo & Robert , Ludo pour les soir´eesLudo & Robert, H´elianthe pour avoir fait de moi la fashionista que je suis aujourd'hui. Un merci tout particulier `aAm´eliesans qui je n'aurais sans doute pas fait cette th`ese. Ensuite je souhaite remercier mes parents Dominique et Thierry que j'aime qui m'ont apport´e un soutient inconditionnel pendant des ´etudeslongues et pas toujours lin´eaires. Et puis pour le reste, merci aussi ! Merci `ames sœurs Rachel et Clara, mes deux personnes pr´ef´er´eesex æquo que j'aime et que j'admire. Merci `aLo¨ıcet Nicolas et merci aussi `aleurs enfants (respectifs) Ya¨el,Marek, C^omeet Talia, que j'adore voir grandir et apprendre `aconna^ıtre. Merci `ames grand-parents, Jacqueline, Genevi`eve aka GM, Armand (et Andr´eque j'ai peu connu), pour l'amour et le soutient qu'ils m'ont apport´e. Enfin si j'ai beaucoup chang´e,beaucoup appris et si ces trois ann´eesont ´et´eheureuses, c'est gr^ace`atoi C´ecilia.Merci. iii Definissabilit´ e´ et Synthese` de Transductions iv R´esum´een fran¸cais Des langages aux transductions L'´etudedes langages formels est l'un des piliers de l'informatique th´eoriqueet a permis de d´evelopper de nombreux outils th´eoriqueset pratiques dans diff´erents domaines. Certaines classes de langages se d´emarquent des autres car elles b´en´eficient de plusieurs descriptions diff´erentes, par exemple les langages r´ecursivement ´enum´erablespeuvent ^etrecaract´eris´esen termes de machines de Turing ou en termes de grammaires de type 0 dans la hi´erarchie de Chomsky. Un exemple qui nous int´eresseparticuli`erement est celui de la classe des langages rationnels1 qui se situe au plus bas niveau de la hi´erarchie de Chomsky puisqu'elle est caract´eris´eepar les grammaires r´eguli`eres. Les langages rationnels sont ´egalement caract´eris´espar les expressions rationnelles, les formules de la logique monadique du second-ordre (MSO), les mono¨ıdesfinis (ou de mani`ere
Recommended publications
  • 1. Introduction This Paper Introduces a New Class of Languages of Infinite Words, Which Are Called Max-Regular Languages, and Include All Ω-Regular Languages

    1. Introduction This Paper Introduces a New Class of Languages of Infinite Words, Which Are Called Max-Regular Languages, and Include All Ω-Regular Languages

    Submitted to the Symposium on Theoretical Aspects of Computer Science www.stacs-conf.org WEAK MSO WITH THE UNBOUNDING QUANTIFIER MIKOLAJBOJA NCZYK´ University of Warsaw E-mail address: [email protected] URL: www.mimuw.edu.pl/∼bojan Abstract. A new class of languages of infinite words is introduced, called the max- regular languages, extending the class of !-regular languages. The class has two equivalent descriptions: in terms of automata (a type of deterministic counter automaton), and in terms of logic (weak monadic second-order logic with a bounding quantifier). Effective translations between the logic and automata are given. 1. Introduction This paper introduces a new class of languages of infinite words, which are called max-regular languages, and include all !-regular languages. Max-regular languages can be described in terms of automata, and also in terms of a logic. A typical language in the class is the property \the distance between consecutive b's is unbounded", i.e. the language n1 n2 n3 L = fa ba ba ::: : 8m 9i ni > mg : (1.1) A practical motivation can be given for considering properties that speak of bounded distance; e.g. a formula of the logic in this paper could specify that a system responds to requests with bounded delay. We will begin, however, with a more fundamental motivation, which is the question: what is a regular language of infinite words? There is little doubt as to what is a regular language of finite words. For instance, the requirement that the Myhill-Nerode equivalence relation has finitely many equivalence classes uniquely determines which languages of finite words should be regular.
  • Memory and Delay in Regular Infinite Games

    Memory and Delay in Regular Infinite Games

    Memory and Delay in Regular Infinite Games Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Informatiker Michael Holtmann aus Düsseldorf Berichter: Universitätsprofessor Dr. Dr.h.c. Wolfgang Thomas Universitätsprofessor Dr. Erich Grädel Tag der mündlichen Prüfung: 9. Mai 2011 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Zusammenfassung Unendliche Zwei-Personen-Spiele sind ein ausdrucksstarkes und anpassungsfä- higes Werkzeug bei der Modellierung und Verifikation reaktiver Systeme. Es ist allgemein bekannt, dass beispielsweise die Konstruktion eines Controllers, der beliebig lange in der Umgebung eines Systems agiert, reduziert werden kann auf die Berechnung einer Gewinnstrategie in einem unendlichen Spiel (Pnueli und Rosner, 1989). Für die Klasse der Spiele mit regulärer Gewinnbedingung haben Büchi und Landweber 1969 gezeigt, dass einer der Spieler eine Gewinnstrategie hat, die durch einen endlichen Automaten realisierbar ist. Basierend auf diesem grundlegenden Resultat hat die Forschung viele Versuche unternommen, die Lö- sungsverfahren für unendliche Spiele weiterzuentwickeln. Dies betrifft sowohl den zeitlichen Aufwand, den man benötigt, um Gewinnstrategien zu berechnen, als auch den Speicherbedarf, um diese dann zu programmieren. In der vorliegen- den Arbeit geht es hauptsächlich um den zweiten Aspekt. Es werden zwei Proble- me betrachtet, die im Hinblick auf die Konstruktion kleiner Controllerspezifika- tionen von Bedeutung sind. Im ersten Teil der Arbeit beschäftigen wir uns mit dem klassischen Problem, endlich repräsentierbare Gewinnstrategien zu berechnen, die von Automaten mit möglichst wenig Speicher (das heißt mit möglichst wenig Zuständen) realisiert werden können. Auch wenn ein Resultat von Dziembowski et al.
  • First-Order Definable Languages

    First-Order Definable Languages

    First-order definable languages∗ Volker Diekert1 Paul Gastin2 1 Institut f¨ur Formale Methoden der Informatik Universit¨at Stuttgart Universit¨atsstraße 38 70569 Stuttgart, Germany [email protected] 2 Laboratoire Sp´ecification et V´erification Ecole´ Normale Sup´erieure de Cachan 61, avenue du Pr´esident Wilson 94235 Cachan Cedex, France [email protected] Abstract We give an essentially self-contained presentation of some princi- pal results for first-order definable languages over finite and infinite words. We introduce the notion of a counter-free B¨uchi automaton; and we relate counter-freeness to aperiodicity and to the notion of very weak alternation. We also show that aperiodicity of a regular ∞-language can be decided in polynomial space, if the language is specified by some B¨uchi automaton. 1 Introduction 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. More- over it is the core for generalizations, e.g., to tree automata [26] or to par- tially 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 lan- guages and again: it relates logic, combinatorics, and algebra to automata theory; and it is also widely applied in all branches of computer sciences. Let us mention that first-order definability for Mazurkiewicz traces leads essentially to the same picture as for words (see, e.g., [5]), but nice charac- tizations for first-order definable sets of trees are still missing.
  • Lic Pittou Maria 2017.Pdf

    Lic Pittou Maria 2017.Pdf

    Aalto University School of Science Department of Computer Science Contributions to multi-view modeling and the multi-view consistency problem for infinitary languages and discrete systems Licentiate thesis Maria Pittou Supervisor Full Professor Stavros Tripakis Espoo, Finland, 2017 Copyright © 2017 Maria Pittou Abstract of Licentiate Thesis Author Maria Pittou Title of Thesis Contributions to multi-view modeling and the multi-view consistency problem for infinitary languages and discrete systems Abstract The modeling of most large and complex systems, such as embedded, cyber-physical, or distributed systems, necessarily involves many designers. The multiple stakeholders carry their own perspectives of the system under development in order to meet a variety of objectives, and hence they derive their own models for the same system. This practice is known as multi- view modeling, where the distinct models of a system are called views. Inevitably, the separate views are related, and possible overlaps may give rise to inconsistencies. Checking for multi- view consistency is key to multi-view modeling approaches, especially when a global model for the system is absent, and can only be synthesized from the views. The present thesis provides an overview of the representative related work in multi-view modeling, and contributes to the formal study of multi-view modeling and the multi-view consistency problem for views and systems described as sets of behaviors. In particular, two distinct settings are investigated, namely, infinitary languages, and discrete systems. In the former research, a system and its views are described by mixed automata, which accept both finite and infinite words, and the corresponding infinitary languages.
  • Inclusion Testing of Büchi Automata Based on Well-Quasiorders

    Inclusion Testing of Büchi Automata Based on Well-Quasiorders

    Inclusion Testing of Büchi Automata based on Well-Quasiorders Kyveli Doveri ! IMDEA Software Institute and Universidad Politécnica de Madrid, Spain Pierre Ganty ! IMDEA Software Institute, Spain Francesco Parolini ! Sorbonne Université, France Francesco Ranzato ! University of Padova, Italy Abstract We introduce an algorithmic framework to decide whether inclusion holds between languages of infinite words over a finite alphabet. Our approach falls within the class of Ramsey-based methods and relies on a least fixpoint characterization of ω-languages leveraging ultimately periodic infinite words of type uvω, with u a finite prefix and v a finite period of an infinite word. We put forward an inclusion checking algorithm between Büchi automata, called BAInc, designed as a complete abstract interpretation using a pair of well-quasiorders on finite words. BAInc is quite simple: it consists of two least fixpoint computations (one for prefixes and the other for periods) manipulating finite sets (of pairs) of states compared by set inclusion, so that language inclusion holdswhenthe sets (of pairs) of states of the fixpoints satisfy some basic conditions. We implemented BAIncin a tool called BAIT that we experimentally evaluated against the state-of-the-art. We gathered, in addition to existing benchmarks, a large number of new case studies stemming from program verification and word combinatorics, thereby significantly expanding both the scope and sizeofthe available benchmark set. Our experimental results show that BAIT advances the state-of-the-art on an overwhelming majority of these benchmarks. Finally, we demonstrate the generality of our algorithmic framework by instantiating it to the inclusion problem of Büchi pushdown automata into Büchi automata.
  • Recognisable Languages Over Monads

    Recognisable Languages Over Monads

    Recognisable Languages over Monads Miko laj Boja´nczyk February 18, 2015 Contents I Introduction 3 1 Introduction 4 2 Monads and their algebras 6 3 Syntactic morphisms 8 3.1 Proof of the Syntactic Morphism Theorem . 10 4 Pseudovarieties 17 4.1 The Syntactic Pseudovariety Theorem. 20 4.2 The Polynomial Pseudovariety Theorem . 22 5 Representing an algebra 27 6 Monadic second-order logic 30 6.1 Language theoretic definition of mso ................ 30 6.2 Deciding satisfiability of mso .................... 32 II Example Monads 34 7 Monads for chains 35 arXiv:1502.04898v1 [cs.LO] 17 Feb 2015 8 Pointed words 36 8.1 Unary queries definable in two-variable first-order logic . 37 9 Monads for trees 46 9.1 Ranked trees over a fixed alphabet . 46 9.2 Clones . 49 9.3 Forests of unranked trees . 53 9.4 A monad for infinite unranked forests. 57 1 10 Future work 58 III Profinite Monads 59 11 Stone duals and topology on an algebra 60 11.1 Stone duals of Boolean algebras . 60 11.2 Stone duals of T-algebras . 62 11.3 Uniform continuity . 67 12 Profinite monads 72 12.1 Definition of the profinite monad . 72 12.2 From a T-algebra to a T-algebra. 79 12.3 From a T-algebra to a T-algebra. 82 12.4 Clopen languages and Stone algebras . 84 13 Profinite words 86 13.1 The unboundedness language . 86 2 Part I Introduction In this part, we introduce monads and their algebras. Section 2 contains the basic definitions: first illustrated on examples of finite words and 1-words, and then formally defined.
  • Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular Omega-Language Olivier Finkel

    Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular Omega-Language Olivier Finkel

    Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language Olivier Finkel To cite this version: Olivier Finkel. Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language. International Journal of Foundations of Com- puter Science, World Scientific Publishing, 2012, 23 (7), p. 1481-1497. 10.1142/S0129054112400606. hal-00543309v2 HAL Id: hal-00543309 https://hal.archives-ouvertes.fr/hal-00543309v2 Submitted on 28 Jul 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular ω-Language Olivier Finkel Equipe de Logique Math´ematique CNRS et Universit´eParis Diderot Paris 7 UFR de Math´ematiques case 7012, site Chevaleret, 75205 Paris Cedex 13, France. fi[email protected] Abstract 1 It was noticed by Harel in [Har86] that “one can define Σ1-complete versions of the well-known Post Correspondence Problem”. We first give a complete proof of this result, showing that the infinite Post 1 Correspondence Problem in a regular ω-language is Σ1-complete, hence located beyond the arithmetical hierarchy and highly undecidable.
  • A Novel Learning Algorithm for Büchi Automata Based on Family of Dfas

    A Novel Learning Algorithm for Büchi Automata Based on Family of Dfas

    A Novel Learning Algorithm for Buchi¨ Automata based on Family of DFAs and Classification Trees a,b c a,d,b, a,b Yong Li , Yu-Fang Chen , Lijun Zhang ∗, Depeng Liu aState Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190 bUniversity of Chinese Academy of Sciences, Beijing 100049 cInstitute of Information Science, Academia Sinica, Taipei 11529 dInstitute of Intelligent Software, Guangzhou, Beijing 511400 Abstract In this paper, we propose a novel algorithm to learn a Buchi¨ automaton from a teacher who knows an !-regular language. The learned Buchi¨ automaton can be a nondeterministic Buchi¨ automaton or a limit deterministic Buchi¨ automaton. The learning algorithm is based on learning a formalism called family of DFAs (FDFAs) recently proposed by Angluin and Fisman [1]. The main catch is that we use a classification tree structure instead of the standard observation table structure. The worst case storage space required by our algorithm is quadratically better than that required by the table-based algorithm proposed in [1]. We implement the proposed learning al- gorithms in the learning library ROLL (Regular Omega Language Learning), which also consists of other complete !-regular learning algorithms available in the literature. Experimental results show that our tree-based learning algorithms have the best performance among others regarding the number of solved learning tasks. Keywords: Buchi¨ Automata, Learning Algorithm, Automata Learning, Observation Table, Family of DFAs, Classification Tree, L∗ 1. Introduction In the past two decades, learning-based automata inference techniques [2–5] have received significant attention from the community of formal verification.
  • The Greatest Fixed-Points and Rational Omega-Tree Languages

    The Greatest Fixed-Points and Rational Omega-Tree Languages

    Theoretical Computer Science 44 (1986) 259-274 259 North-Holland THE GREATEST FIXED-POINTS AND RATIONAL OMEGA-TREE LANGUAGES Masako TAKAHASHI* LITP, Universitd de Paris VII, 75005 Paris, France Communicated by D. Pert'in Received November 1985 Abstract. Rational omega-tree languages are defined by extending the notion of rational (or regular) expressions for languages, and their properties are investigated. They are characterized by means of greatest fixed-points of certain functions, and are shown to be precisely the omega-tree languages recognized by Rabin's 'special automata'. A simple normal form for rational omega-tree languages is also presented. Introduction It is well known that the regular (or rational) languages can be characterized by means of the least fixed-points of right-linear functions; that is, the class of regular languages is equal to the class of components of the least solutions of equation systems of the form n X,=ULr~uL, (i=1,2,..., n) j=l where L o and Li are given finite languages and Xi are unknown languages. Park [7] has studied the same equation systems in a different domain, i.e., in the domain of infinitary languages, and characterized the class of regular infinitary languages as the class of components of the 'max-min' solutions of such equation systems. This paper can be considered as a continuation of his work. First we note that his result can be reformulated as follows; the class of regular infinitary languages is equal to the class of components of the greatest solutions of equation systems of the same form as above, but with regular languages L o and L, and unknown infinitary languages X~.
  • On the Length of the Wadge Hierarchy of Omega Context Free Languages Olivier Finkel

    On the Length of the Wadge Hierarchy of Omega Context Free Languages Olivier Finkel

    On the Length of the Wadge Hierarchy of Omega Context Free Languages Olivier Finkel To cite this version: Olivier Finkel. On the Length of the Wadge Hierarchy of Omega Context Free Languages. Journal of Automata Languages and Combinatorics, Otto-von-Guericke-Universität Magdeburg, 2005, 10 (4), pp.439-464. hal-00122154v2 HAL Id: hal-00122154 https://hal.archives-ouvertes.fr/hal-00122154v2 Submitted on 3 Jan 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the length of the Wadge hierarchy of ω-context free languages ∗ Olivier Finkel Equipe de Logique Math´ematique, U.F.R. de Math´ematiques, Universit´eParis 7 2 Place Jussieu 75251 Paris cedex 05, France. E Mail: fi[email protected] Abstract We prove in this paper that the length of the Wadge hierarchy of ω-context free th languages is greater than the Cantor ordinal εω, which is the ω fixed point of the 0 ordinal exponentiation of base ω. We show also that there exist some Σω-complete ω-context free languages, improving previous results on ω-context free languages and the Borel hierarchy. Keywords: ω-context free languages; infinitary context free languages; topological properties; Borel hierarchy; Wadge hierarchy; conciliating Wadge hierarchy.
  • Paths in Infinite Trees: Logics and Automata

    Paths in Infinite Trees: Logics and Automata

    PATHS IN INFINITE TREES: LOGICS AND AUTOMATA Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades einer Doktorin der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Informatikerin ALEXANDRA SPELTEN aus Koblenz, Rheinland-Pfalz Berichter: Universitätsprofessor Dr. Dr.h.c. Wolfgang Thomas Universitätsprofessor Dr. Erich Grädel Tag der mündlichen Prüfung: 28. Januar 2013 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Abstract In this thesis, several logical systems over infinite trees and infinite words are studied in their relation to finite automata. The first part addresses (over the infinite binary tree) “path logic” and “chain logic” as fragments of monadic second-order logic that allow quantification over paths, respectively subsets of paths of the binary tree. Many systems of bran- ching-time logic are subsumed by chain logic. We introduce ranked alternating tree automata as a computation model that characterizes chain logic. The main idea is to associate ranks to states such that in an automaton run, starting from the root, the ranks have to decrease and are allowed to remain unchanged only in one direction (either in existential or universal branching). The second part of the thesis is motivated by chain logic over infinite-bran- ching trees (where the successors of a node are indexed by natural numbers). A path through the N-branching tree is given by an ω-word over the infinite alpha- bet N. As a preparation for the study of path logics over such trees, we develop a theory of logics and automata over infinite alphabets, more precisely over alpha- bet frames (M, L), given by a relational structure M, supplying the alphabet, and a logic L that is used in specifying letter properties and automaton transitions.
  • Synthesis and Planning  Dagstuhl Seminar 

    Synthesis and Planning  Dagstuhl Seminar 

    05241 Abstracts Collection Synthesis and Planning Dagstuhl Seminar Henry Kautz1, Wolfgang Thomas2 and Moshe Y. Vardi3 1 Univ. of Washington, US [email protected] 2 RWTH Aachen, DE [email protected] 3 Rice University, US [email protected] Abstract. From 12.06.05 to 17.06.2005 the Dagstuhl Seminar 05241 Synthesis and Planning was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. Keywords. AI planning, controller synthesis, partially observed do- mains, reactive computation, program analysis, games, model checking, satisability, Markov decision processes 05241 Executive Summary Synthesis and Planning This seminar has brought together researchers working in two complementary elds: automatic synthesis of (control) programs, and methods for devising plan- ning algorithms in artical intelligence (AI). This combines a strong thread of current research in automata theory with an area of possible but so far unex- plored applications. Keywords: Synthesis, planning Joint work of: Kautz, Henry; Thomas, Wolfgang; Vardi, Moshe Full Paper: http://drops.dagstuhl.de/opus/volltexte/2006/452 Dagstuhl Seminar Proceedings 05241 Synthesis and Planning http://drops.dagstuhl.de/opus/volltexte/2006/453 2 H. Kautz, W. Thomas and M. Y. Vardi Planning for Partially Observable, Non-Deterministic Domains Piergiorgio Bertoli (IRST - Trento, I) The problem of planning for partially observable, nondeterministic domains has lately attracted considerable interest, due to its relevance and complexity.