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The Structure of Orders in the Pushdown Hierarchy Laurent Braud

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The Structure of Orders in the Pushdown Hierarchy Laurent Braud The structure of orders in the pushdown hierarchy Laurent Braud To cite this version: Laurent Braud. The structure of orders in the pushdown hierarchy. Modeling and Simulation. Uni- versité Paris-Est, 2010. English. NNT : 2010PEST1009. tel-00587409 HAL Id: tel-00587409 https://pastel.archives-ouvertes.fr/tel-00587409 Submitted on 20 Apr 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. THESE` pour obtenir le grade de DOCTEUR DE L’UNIVERSITE´ PARIS-EST The structure of orders in the pushdown hierarchy Les structures d’ordre dans la hi´erarchie `apile Sp´ecialit´e informatique Ecole´ doctorale MSTIC Soutenue publiquement par Laurent Braud le 10 d´ecembre 2010 JURY : Zolt´an Esik´ , rapporteur, Wolfgang Thomas, rapporteur, Arnaud Carayol, examinateur, Damian Niwinski´ , examinateur, Dominique Perrin, examinateur, Didier Caucal, directeur de th`ese. 2 Contents 1 Introduction 7 1.1 (enfran¸cais)................................... 7 1.2 (inEnglish) ................................... 13 2 Preliminaries 19 2.1 Notationsandfirststructures . .. 19 2.1.1 Finitewords............................... 19 2.1.2 Structures ................................ 20 2.1.3 Graphs.................................. 21 2.1.4 Deterministictrees ........................... 22 2.2 Linearorderings................................. 22 2.2.1 Ordinals................................. 24 2.2.2 ScatteredorderingsandHausdorffrank . .. 26 2.2.3 Ordersinadeterministictree . 29 2.3 Logic....................................... 30 2.3.1 First-orderlogic............................. 30 2.3.2 Monadicsecond-orderlogic. 31 2.3.3 Decidability ............................... 31 2.4 Graphtransformations . 33 2.4.1 Graphinterpretations. 33 2.4.2 Graphexpansions............................ 35 2.5 Thepushdownhierarchy . 37 2.5.1 Definition ................................ 37 2.5.2 Someproperties............................. 38 3 Linear order construction 43 3.1 Ordinalsinthepushdownhierarchy . .... 43 3.2 Powers of ζ ................................... 45 3.3 n-regularpresentation ............................. 48 3.3.1 Prefix-recognizablegraphs . 49 3.3.2 Configuration graphs of n-hopdas ................... 49 3.3.3 Encodingordinals............................ 50 3 4 CONTENTS 3.4 Coveringgraphs................................. 53 3.4.1 Fundamentalsequence . 54 3.4.2 Coveringgraphs............................. 56 3.4.3 Otherpropertiesofcoveringgraphs . ... 60 3.4.4 Strictness of covering graphs in the hierarchy . ....... 62 3.4.5 The case of Gε0 ............................. 65 4 The structure of tree frontiers 67 4.1 Tree-walkingautomaton . 67 4.2 Fromgraphstofrontiers . 69 4.3 Ordinals..................................... 73 4.4 Scatteredlinearorders . .. 78 4.4.1 Treeswithscatteredfrontiers . .. 78 4.4.2 Permutationofsubtrees . 79 4.4.3 Cantor-Bendixson rank of deterministic trees . ....... 81 4.4.4 Hausdorff rank of scattered orders in Graphn ............. 86 4.5 Finitecombs................................... 87 5 Schemes and morphic words 91 5.1 Recursionschemes ............................... 92 5.1.1 Definition ................................ 92 5.1.2 Schemesinthepushdownhierarchy . 94 5.2 Morphicwords ................................. 97 5.2.1 Definitionandproperties. 97 5.2.2 Constructioninthepushdownhierarchy . ... 99 5.2.3 Words in Graph2 aremorphic,directproof . 99 5.2.4 Words in Graph2 are morphic, by recursion schemes . 102 5.3 Secondorder ..................................106 5.3.1 Second-ordermorphicwords . 106 5.3.2 Second-order scheme ω-frontiers . .109 5.3.3 Liouvilleword..............................117 List of notations 119 Index 121 Bibliography 123 List of Figures 1.1 Un graphe fini et une propri´et´ede ce graphe. ...... 7 1.2 L’arbrebinairecomplet. .. 8 1.3 Afinitegraphandoneofitsproperties.. ..... 13 1.4 Thecompletebinarytree. 14 2.1 Exampleofagraph:theladder. 21 2.2 The graph representation of the ordinal ω +2................. 24 2.3 Theladderanditsunfolding. .. 36 2.4 Treegraphofthecompletebinarytree. ..... 36 2.5 Thepushdownhierarchy.. 38 2.6 Exemple of graph constructions in the hierarchy. ......... 39 3 3.1 Finite graph G3 which unfolding has frontier ω ................ 44 3.2 Folded graphs of trees of frontier ζ, η and ω(1 + η). ............. 45 3.3 Folded regular graph of a tree of frontier ζω.................. 46 3.4 {0, 1}-treegraph of ω............................... 48 3.5 The operation inc(ωω). ............................. 52 3.6 Covering graph of ωω. ............................. 56 3.7 Exponentiation of the covering graph of ω................... 62 4.1 Order in a finite tree t and “arranged” tree s(t)................ 72 4.2 Non-full tree of frontier ω having a branch with infinitely many 0’s. 74 4.3 A finite graph G, “spanning tree” T , completed tree T¯, and unfolding.. 76 4.4 Ordersinatree.................................. 84 ω 5.1 Rules for S1, of frontier ω . .......................... 95 5.2 General rules for Sn, of frontier ω ↑↑ (n +1). ................ 95 5.3 Paperfoldingsequence. .. 98 5.4 A graph which unfolding yields the morphic word abaab . a2i b... ....100 5.5 Generalshapeofthefoldedgraph. 101 5.6 Rules of a scheme which frontier is a morphic word. .......105 5.7 Order-2 safe scheme which frontier is the Champernowne word. .107 5.8 Order-2 safe scheme which frontier is the Liouville word. ..........118 5 6 LIST OF FIGURES Chapter 1 Introduction 1.1 (en fran¸cais) En logique math´ematique, on distingue les objets math´ematiques, ou structures, et leurs propri´et´es, ou logique. Les structures `aleur tour sont constitu´ees d’´el´ements et de relations entre ces ´el´ements. Dans cette th`ese, nous travaillerons uniquement sur ce que l’on appelle des “graphes color´es dont les arcs sont ´etiquet´es”, c’est-`a-dire des structures dont les relations sont d’arit´eau plus 2. De plus, ces structures seront d´enombrables — on peut num´eroter chaque objet avec un entier — et de pr´esentation finie — il existe une quantit´e finie d’information permettant de repr´esenter la structure de fa¸con non ambig¨ue. “Un chemin entre deux sommets blancs est en pointill´es ou passe par un sommet noir.” Figure 1.1: Un graphe fini et une propri´et´ede ce graphe. La logique la plus simple est celle dite du premier ordre, car elle manipule simplement les ´el´ements proprement dits de la structure consid´er´ee. Elle permet d’exprimer des propri´et´es du type “tout sommet colori´een noir a un voisin blanc” ou “s’il existe un sommet blanc, alors il existe 3 arcs distincts en pointill´es”. Elle est cependant restreinte `ades propri´et´es ponctuelles et ne dit rien sur la structure dans sa globalit´e. Une solution est alors de consid´erer une logique plus forte, qui permet de manipuler directement des ensembles d’´el´ements. C’est la logique du second ordre monadique, qui permet d’exprimer des propri´et´es raisonnablement compliqu´ees. Des exemples classiques sont “il y a un chemin entre x et y” ou “le sous-arbre t est infini”. On la notera souvent par le sigle MSO. De fa¸con g´en´erale, la logique du second ordre d´esigne l’ensemble des formules qui parlent des relations entre les ´el´ements. Mais elle a le d´efaut d’ˆetre paradoxalement trop puissante; elle ´echappe ainsi aux outils usuels de l’informatique, selon la d´efinition donn´ee 7 8 CHAPTER 1. INTRODUCTION ci-apr`es. C’est pourquoi on se restreint aux relations qui ne prennent qu’une variable : ce sont bien les ensembles. Ceci explique l’adjectif monadique. Notre probl`eme n’est pas de connaˆıtre enti`erement une structure donn´ee, mais de savoir exprimer ses propri´et´es. Cette nuance autorise l’usage d’un automatisme. En effet, la question qui se pose est alors la suivante. Etant´ donn´ees une logique et une structure, existe-t-il un algorithme qui prenne en entr´ee une formule close de cette logique et renvoie oui ou non selon que la structure satisfait la formule ou non? C’est la question de la d´ecidabilit´ede du model-checking de la logique `alaquelle ap- partiennent les formules que l’on veut tester. Savoir construire un tel algorithme est un d´efi majeur de l’informatique d’aujourd’hui, car cette question est naturellement li´ee `ala notion de v´erification de programmes. En effet, savoir si un programme fait effectivement ce que l’on veut — par exemple, savoir s’il termine — est expressible par une formule, que doit v´erifier la structure des configurations du programme. Construction de structures Par exemple, un r´esultat fondemental de Rabin [Rab69] indique que l’arbre binaire com- plet, esquiss´eFigure 1.2, a une th´eorie monadique d´ecidable. Plus g´en´eralement, pour une logique donn´ee, peut-on esp´erer trouver une caract´erisation de toutes les structures qui jouissent de la mˆeme propri´et´e? Probablement pas, mais cela n’empˆeche pas les sci- entifiques de chercher `aen d´ecrire le plus possibles. En particulier, pour la logique du second ordre monadique, l’int´erˆet de la communaut´escientifique se porte sur la hi´erarchie `apile [Cau03], aussi appel´ee hi´erarchie de Caucal. Figure 1.2: L’arbre binaire complet. Pour d´ecrire historiquement cette hi´erarchie, il nous faut remonter `a[MS85], o`ules auteurs s’int´eressent aux graphes des configurations des automates `apiles en partant d’une configuration donn´ee. Il caract´erisent exactement ces graphes par un crit`ere g´eom´etrique : si on fixe un sommet, et que l’on retire successivement ce sommet, puis les sommets `a distance 1, puis `adistance 2 et ainsi de suite, on obtient une suite de graphes. Le r´esultat fondamental est que l’ensemble de ces graphes est fini `aisomorphisme pr`es : la suite se 1.1. (EN FRANC¸AIS) 9 r´ep`ete. Cette propri´et´eleur permet d’´etendre le r´esultat de Rabin `atout une classe de graphes.
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