An algebraic theory of real-time formal languages Catalin Dima To cite this version: Catalin Dima. An algebraic theory of real-time formal languages. Modeling and Simulation. Univer- sité Joseph-Fourier - Grenoble I, 2001. English. tel-00004672 HAL Id: tel-00004672 https://tel.archives-ouvertes.fr/tel-00004672 Submitted on 16 Feb 2004 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. UNIVERSITE´ JOSEPH FOURIER -GRENOBLE 1 SCIENCES ET GEOGRAPHIE THESE` pour obtenir le grade de DOCTEUR de l’UNIVERSITE´ JOSEPH FOURIER Specialit´ e:´ Informatique present´ ee´ et soutenue publiquement par M. Cat˘ alin˘ DIMA le 11 decembre´ 2001 THEORIE´ ALGEBRIQUE´ DES LANGAGES FORMELS TEMPS REEL´ Directeurs de these:` Prof. Eugene` Asarin, Dr. Oded Maler Composition du jury : Jean-Claude Fernandez President´ Paul Gastin Rapporteur Nils Klarlund Rapporteur P. S. Thiagarajan Examinateur Pascal Weil Examinateur Eugene` Asarin Directeur de These` Oded Maler Directeur de These` These´ preparee´ au sein du Laboratoire Verimag´ Lui Gabi si Iuliei, cu drag 4 Remerciements Je remercie d’abord a` Oded Maler et Eugene` Asarin pour m’avoir donne´ la chance de finir mes etudes´ sous leur direction et soutenir ma these` aV` erimag.´ Eugene` a patiemment resist´ e´ aux tous mes essais echou´ es´ et ses critiques m’ont fait apprendre un nouveau sens de la recherche. Je remercie a` Joseph Sifakis pour m’avoir accueili pendant plus d’une annee´ au sein du labora- toire Verimag,´ en me donnant ainsi la possibilite´ de faire la recherche dans un milieu effervescent, entretenu par des chercheurs de haute qualite.´ Je remercie aussi a` Alain Girault et aux membres du groupe BIP de l’INRIA Rhone-Alpes,ˆ ou` j’ai et´ e´ accueilli pendant une annee´ en 2000. C’est eux qui on fait mon accomodation plus facile. Graceˆ a eux, j’ai pu poursuivre mes recherches pour la these,` en parallele` avec mon travail dans le cadre du projet TOLERE.` Je remercie a` Gheorghe S¸tefanescu˘ qui m’a toujours pousse´ a` contacter diverses groupes de recherce et a` qui je dois en fait mon arrivee´ en France. Many thanks to the UNU/IIST, in particularly to Prof. Zhou Chaochen. It was there, during my fellowship at UNU/IIST from Febrary to August 1998, that my interest for the theory of real-time systems emerged. I also thank the TCS group at TIFR Mumbai for giving me the means to visit from September to December 1998. Xu Qiwen and Dang Van Hung, at UNU/IIST Macau, and Paritosh Pandya at TIFR Mumbai have helped me a lot in the early stages of the research for the thesis. Merci a` Liana et Marius pour leur chaleureuse amitie.´ Merci a` Yasmina et Moez, qui sont des bons copains de bureau, et a` Ana et Gerardo qui sont des bons copains tout court. Merci finalement a` tous les chercheurs du Verimag´ pour cet environnement de bonne qualite´ qu’ils entretiennent. 6 Table of Contents 1. Introduction 11 2. Signals and their basic properties 19 2.1 Basic notions. ........................................................... 20 2.1.1 Coproduct monoids . ................................................ 20 2.1.2 Kleene algebras . ................................................... 21 2.2 Signals . .............................................................. 22 2.2.1 Timed languages: basic properties . .................................... 24 2.3 Timed words ............................................................ 24 2.3.1 Relating the monoids of signals and of timed words ...................... 25 2.4 Timed regular languages defined by inverse monoid morphisms ................. 26 2.4.1 Essentially untimed regular languages ................................. 27 Sig 2.4.2 Syntactic monoids on are not interesting . .................. 28 3. Real-time automata 33 3.1 Real-time automata and their regular expressions ............................. 34 3.1.1 Real-time automata defined . ........................................ 34 3.1.2 Regular expressions and the Kleene theorem ............................ 35 3.1.3 The problem of complementation of real-time automata .................. 38 3.2 The Kleene algebra of sets of real numbers ................................... 40 3.2.1 Normal forms . ................................................... 41 3.2.2 A normal form theorem ............................................. 43 3.2.3 Matrices of normal forms ............................................ 45 3.3 Determinization and complementation of RTA . ............................. 46 3.4 The Pumping Lemma and expressiveness issues . ............................. 50 3.5 Stuttering-free concatenation .............................................. 51 3.5.1 Syntactic monoids for stuttering-free concatenation and real-time automata . 52 4. Timed automata 55 4.1 Clocks and clock constraints ............................................... 55 4.2 Timed automata and their clock valuation semantics ........................... 56 4.2.1 A Kleene theorem with indexed concatenation .......................... 61 8 Table of Contents 4.3 Reset time semantics for timed automata .................................... 65 5. Timed regular expressions 69 5.1 Basic properties of timed regular expressions . ............................. 70 5.1.1 Timed regular expressions without brackets . ............................ 70 5.2 Undecidability of the language emptiness problem for extended timed regular ex- pressions . .............................................................. 72 5.3 Relating timed regular expressions and timed automata ........................ 75 5.4 Colored parentheses: basic ideas and problems . ............................. 76 5.4.1 Changing the concatenation . ........................................ 77 5.4.2 The “overlapping” concatenation for timed automata . .................. 79 6. Matrices of signals 83 n 6.1 n-dominoes and -signals ................................................. 84 6.1.1 n-dominoes over a one-letter alphabet ................................. 85 6.2 Operations on n-dominoes ................................................ 86 6.2.1 Projection ......................................................... 86 6.2.2 Juxtaposition . ................................................... 88 6.2.3 Properties of juxtaposition . ........................................ 90 6.2.4 Concatenation . ................................................... 93 6.3 n-domino languages . ................................................... 96 6.4 Regminoes, regsignals, and regular expressions over them ...................... 97 6.4.1 Projection and juxtaposition on n-regsignals ............................ 98 n n 6.4.2 -domino regular expressions and -signal regular expressions . ....... 101 n 6.5 -signal regular expressions and timed automata ............................. 102 n 6.6 The emptiness problem for -signal regular expressions is undecidable . ....... 105 7. n-words and their automata 109 7.1 n-words . .............................................................. 110 7.2 n-automata ............................................................. 112 7.2.1 The emptiness problem for n-automata. ............................. 114 n 7.2.2 -transitions in -automata . ........................................ 116 7.2.3 Basic operations with n-automata ..................................... 118 7.2.4 Relationship with n-regwords ........................................ 120 7.3 Non-elasticity ........................................................... 121 7.4 The non-elastic star closure theorem ........................................ 125 8. Representing timing information with n-words 151 8.1 Difference bound matrices ................................................ 158 8.2 Regions . .............................................................. 164 8.2.1 Juxtaposition and concatenation on regions ............................. 166 Table of Contents 9 n 8.3 Representing DBMs with the aid of n-words and -relations . .................. 169 8.3.1 n-relations ........................................................ 170 8.3.2 Operations on n-relations ............................................ 171 8.3.3 n-word representations .............................................. 175 8.3.4 Operations on n-word representations. ................................. 178 8.4 n-region automata ....................................................... 181 8.4.1 Basic closure properties for n-region automaton ......................... 182 n 8.4.2 Non-elasticity for -DBMs . ........................................ 186 8.4.3 Closure under concatenation and star .................................. 188 9.