First-Order Definable Languages
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Definability and Synthesis of Transductions Nathan Lhote
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 -
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 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. -
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 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 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 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 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
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 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 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
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.