Algorithmic Complexity of Well-Quasi-Orders Sylvain Schmitz To cite this version: Sylvain Schmitz. Algorithmic Complexity of Well-Quasi-Orders. Logic in Computer Science [cs.LO]. École normale supérieure Paris-Saclay, 2017. tel-01663266 HAL Id: tel-01663266 https://tel.archives-ouvertes.fr/tel-01663266 Submitted on 13 Dec 2017 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. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License Algorithmic Complexity of Well-Quasi-Orders Habilitation à diriger des recherches préparée à l’École Normale Supérieure Paris-Saclay au sein du Laboratoire Spécification & Vérification Présentée et soutenue à Cachan, le 27 novembre 2017, par Sylvain Schmitz Composition du jury : Mikołaj Bojańczyk Rapporteur Professeur, Universytet Warszawski Javier Esparza Rapporteur Professeur, Technische Universität München Alain Finkel Examinateur Professeur, École Normale Supérieure Paris-Saclay Géraud Sénizergues Rapporteur Professeur, Université de Bordeaux Andreas Weiermann Examinateur Professeur, Universiteit Gent James B. Worrell Examinateur Professeur, University of Oxford Habilitation à diriger des recherches Algorithmic Complexity of Well-Quasi-Orders Habilitation Thesis Sylvain Schmitz © Sylvain Schmitz licensed under Creative Commons License CC-BY-ND Acknowledgements * I am indebted to a very large number of people who supported me on insti- tutional, scientific, and personal levels while carrying this work. My first thanks go to all the people I have worked along with atthe Labora- toire Spécification et Vérification in Cachan over the nine years I have spent there since 2008: to the administrative and technical staff for their kindness and avail- ability, to my current students Simon Halfon and Anthony Lick and my former intern and visiting students for thought-provoking discussions, and to all my col- leagues for creating such a supportive and stimulating environment—it has been an incredibly formative experience, I have learned so much from all of you. Spe- cial thanks go to Alain Finkel, Jean Goubault-Larrecq, and Philippe Schnoebelen, who introduced me to the theory of well-quasi-orders when I arrived in Cachan. Let me thank more generally all my co-authors over the years: it has been a privilege to work by your side. My lasting collaborations with Ranko Lazić and Philippe Schnoebelen have been especially influential: little of the work described in this habilitation would have existed without you. I also thank the members of the jury for their excellent comments and cor- rections on the manuscript, and for their patience and goodwill during the entire process. I have also benefited from the fantastic opportunity to teach at École Normale Supérieure Paris-Saclay all over these nine years; the demands of tutoring and lecturing to such an exacting audience have had a deep influence. I thank my students and everyone I shared a course with, and in particular Paul Gastin for his guidance as head of the computer science department when I started on the job, at what was then called ENS Cachan. To end on a personal note, I give my heartful thanks to Marie Antczak, Claire David, Laure Daviaud, Mahsa Shirmohammadi, and everyone else who joined in the nudging and encouraged me to complete this habilitation thesis. Thank you! *I acknowledge the institutional and financial support of ENS Paris-Saclay, CNRS, Inria, ANR, the Leverhulme Trust, the University of Warwick, and the University of Warsaw. i Contents Chapter 1. Introduction 1 1.1. General Context 1 1.1.1. Verification of Infinite-state Systems 1 1.1.2. A Game Changer: Well-Structured Systems 1 1.1.3. Tools and Practical Applications 2 1.1.4. The Complexity Question 2 1.2. Contributions 3 1.2.1. Techniques for Upper Bounds 3 1.2.2. Techniques for Lower Bounds 4 1.2.3. Complexity Classes 5 1.2.4. A Challenge: the Complexity of VASS Reachability 5 1.2.5. Other Contributions 7 Chapter 2. Well-Quasi-Orders & Applications 9 2.1. Basic Definitions 10 2.1.1. Well-Quasi-Orders 10 2.1.2. Examples 11 2.2. Application: Program Termination 11 2.2.1. Ranking Functions 12 2.2.2. Quasi-Ranking Functions 13 2.3. Application: Well-Structured Transition Systems 14 2.3.1. Formal Definition 15 2.3.2. Example: Lossy Counter Machines 15 2.3.3. Verifying WSTS 16 Chapter 3. Length Function Theorems 19 3.1. Controlling Bad Sequences 20 3.1.1. Normed wqos 21 3.1.2. Controlled Sequences 21 3.1.3. Length Functions 22 3.2. Subrecursive Hierarchies 23 3.2.1. Fundamental Sequences and Predecessors 23 3.2.2. Hardy and Cichoń Hierarchies 23 3.3. Length Functions for Ordinals Below "0 24 3.3.1. Residuals and a Descent Equation 25 3.3.2. Upper Bounds 25 3.4. Length Functions for Dickson’s Lemma 26 3.4.1. Polynomial Normed wqos 27 3.4.2. Reflecting Normed wqos 27 3.4.3. Maximal Order Types 29 iii iv CONTENTS 3.4.4. Upper Bounds 30 3.5. Related Work & Perspectives 31 3.5.1. Length Functions for Ordinals 31 3.5.2. Length Functions for Well-Quasi-Orders 32 3.5.3. Further Applications 32 3.5.4. Perspectives 33 Chapter 4. Fast-Growing Complexity 35 4.1. Complexity Classes Beyond ELEMENTARY 36 4.1.1. The Extended Grzegorczyk Hierarchy 36 4.1.2. The Fast-Growing Complexity Hierarchy 38 4.1.3. Example: LCM Coverability is in ACKERMANN 42 4.2. Lower Bounds Through Hardy Computations 43 4.2.1. Hardy-like Computations 44 4.2.2. Weak Computation in Lossy Counter Machines 45 4.2.3. Lower Bound 47 4.3. Related Work & Perspectives 49 4.3.1. Lower Bounds Through Counter Objects 49 4.3.2. Provability in Theories of Arithmetic 50 Chapter 5. Reachability in Vector Addition Systems 53 5.1. Basic Definitions 54 5.1.1. Vector Addition Systems with States 54 5.1.2. Closely Related Models 56 5.2. The Decomposition Algorithm 56 5.2.1. Marked Witness Graph Sequences 57 5.2.2. An Example of a KLMST Decomposition 58 5.2.3. Termination 60 5.3. Complexity Upper Bound 61 5.3.1. An Ordinal Ranking Function 61 5.3.2. Applying the Length Function Theorems 61 5.4. Related Work & Perspectives 63 5.4.1. Tightness 63 5.4.2. Restrictions 64 5.4.3. Extensions 67 Chapter 6. Perspectives 69 6.1. Complexity of VASS Reachability 69 6.2. Parameterised Bounds 69 6.3. Algorithmic Applications of Ideals 70 6.3.1. Ideals 70 6.3.2. A Dual Backward Coverability Algorithm 71 6.3.3. Beyond Well-Quasi-Orders 72 6.4. Reachability in VASS Extensions 72 6.4.1. Branching VASS Reachability 73 6.4.2. Ideal Decomposition of the Set of Executions 73 Appendix A. Technical Appendix 75 A.1. Ordinals 75 A.1.1. Ordinals Terms 75 CONTENTS v A.1.2. Maximal Order Types 76 A.2. Subrecursive Functions 76 A.2.1. Monotonicity Properties 76 References 79 List of Publications 93 CHAPTER 1 Introduction 1.1. General Context The main motivations for this thesis are rooted in formal verification, and in particular in model-checking techniques. This might come as a surprise, be- cause the title seems centred on well-quasi-orders, which are a rather abstract concept, whose invention can be tracked back to the early 1910s and whose theory was developed by mathematicians like Higman [1952], Kruskal [1972], de Jongh and Parikh [1977], and many others. Model-checking on the other hand, while certainly grounded on sound mathematical concepts [e.g. Vardi, 2009], is an emi- nently practical field of computer science with a widespread adoption in thehard- ware industry. At its core, it consists of a fully automated, algorithmic check that a model of a system—like a concrete sequential circuit—satisfies its specification— like a safety check that no bad configuration can ever be reached in the model. 1.1.1. Verification of Infinite-state Systems. Once the general underpin- nings of model-checking for finite-state systems were well-understood, the focus of the formal verification community turned in the 1990s to infinite-state systems. Indeed, many critical systems worth the time and effort required by formal ver- ification are infinite, due to unbounded variables: integer counters, numberof active threads in concurrent settings, real-time clocks, stacks, nonces, communi- cation channels, etc. In some specific instances, these infinite-state systems are ‘essentially finite-state’ and can be faithfully abstracted into finite onesby finite partitioning, i.e. by quotienting them using a bisimulation equivalence of finite index—this method was notably behind the success of timed automata [Alur and Dill, 1994]. But at that time this research area was lacking common methods and general principles to cope with cases where finite partitioning failed. 1.1.2. A Game Changer: Well-Structured Systems. This situation was remedied by Abdulla, Čerāns, Jonsson, and Tsay [2000] and Finkel and Schnoebe- len [2001], who popularised the notion of well-structured systems. Their approach can be understood as a generalisation of the finite partitioning method, using instead simulation quasi-orders. The finiteness condition on the equivalence’s in- dex was accordingly replaced by a much more flexible condition: this quasi-order should be a well-quasi-order (a wqo).