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On Effective Representations of Well Quasi-Orderings Simon Halfon On Effective Representations of Well Quasi-Orderings Simon Halfon To cite this version: Simon Halfon. On Effective Representations of Well Quasi-Orderings. Other [cs.OH]. Université Paris-Saclay, 2018. English. NNT : 2018SACLN021. tel-01945232 HAL Id: tel-01945232 https://tel.archives-ouvertes.fr/tel-01945232 Submitted on 5 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. On Eective Representations of Well asi-Orderings ese` de doctorat de l’Universite´ Paris-Saclay prepar´ ee´ a` l’Ecole´ Normale Superieure´ de Cachan au sein du Laboratoire Specication´ & Verication´ Present´ ee´ et soutenue a` Cachan, le 29 juin 2018, par Simon Halfon Composition du jury : Dietrich Kuske Rapporteur Professeur, Technische Universitat¨ Ilmenau Peter Habermehl Rapporteur Maˆıtre de Conferences,´ Universite´ Paris-Diderot Mirna Dzamonja Examinatrice Professeure, University of East Anglia Gilles Geeraerts Examinateur Associate Professor, Universite´ Libre de Bruxelles Sylvain Conchon President´ du Jury Professeur, Universite´ Paris-Sud Philippe Schnoebelen Directeur de these` Directeur de Recherche, CNRS Sylvain Schmitz Co-encadrant de these` Maˆıtre de Conferences,´ ENS Paris-Saclay ` ese de doctorat ED STIC, NNT 2018SACLN021 Acknowledgements I would like to thank the reviewers of this thesis for their careful proofreading and pre- cious comment. You brought to my attention several flaws, and solving them helped me improving many parts, and gave me a deeper understanding of my results. In par- ticular, this made me found some connections I was not aware of, which gave me a great guideline for my defense. I also thank members of the jury for doing me the honor of attending to my PhD defense. In particular, special thanks for my PhD advisors for taking me on this three years long journey throughout the world of research. They have always been available for my questions and valued my work. It has been a pleasure working with you two: thank you a million. I would also like to thank the co-author (and main author) of my first paper: Christoph Haase. You taught me a lot as we were writing this article, and your advice became ground rules for writing this thesis and other papers. It has been a pleasure working with you. It has also been a pleasure working with Dmitry Chistikov who joined us on the journal version of this article, as he had solved our main open problem. I also thank all the people with whom I had the chance to share a publication: Piotr, thank you for inviting me to help you solve your problem. Stefan, and Piotr, thank you for all your precious comments on my talks. Georg: it was a pleasure working with you. I admire your efficiency at solving the problems we were working on, and your efficiency to write a very good paper in a rush (although for you, it was your “most relaxed LICS submission”). Most generally, thank you to all members of LSV, you all contribute in making this lab a terrific place, and it has truly been a pleasure being around you for the past three years. I remember Alain telling us at a MPRI lecture that LSV has the policy of only hiring nice people: “we would not hire the next Einstein if he was not a friendly person”. Well I have to believe you now Alain. And of all LSV members, I especially thank those that take care of us everyday, and take care of our silly mistakes: Cather- ine, Virginie, Thida, Imane, Francis, Hugues. Whenever one of us (no name) booked a flight for NYC when the conference only was in the state of New York, tried to print without being connected to the network, deleted files accidentely, came back from con- ferences without the receipts, or went to summer schools without train tickets: thank you for helping us then. More generally, I have met a lot of kind and fascinating people during my three-year journey in the world of research. There is a general feeling of being part of the same family, because we are inhabited of the same passion in mathematics and computer 1 science, and it is quite an enjoyable feeling. I particularly remember the joy I felt being part of event as the 20th anniversary of LSV, the Dagstuhl Seminar on WQOs (thanks to Jean and the organizers for their invitation on such a short notice), or the Bravas kickoff meeting. There were great experiences during which I have learned a lot, and have enjoyed discussion with talented people. This feeling of being part of a worldwide family will be one the aspect I will miss the most as I begin my career as a teacher. Speaking of teaching, my PhD offered me wonderful opportunities to teach at the ENS Cachan, under the direction of Jean Goubault-Larrecq. Thank you Jean for shar- ing your teaching experience with me and helping me provide relevant exercises to illustrate your lectures. Thank you to Claudine Picarrony for allowing me to intervene in her lectures for preparation to Agregation, and most of all for her many pieces of advice on what to do after my PhD. I would also like to thank all my fellow PhD students from LSV, from lunch to gouter, through coffe break and sometimes dinners, for their support throughout the difficult moments of a PhD. Jeremy: thank you for the introductionnary classes in category theory on the way home. Nathan thank you for the circuit complexity class walk through during lunches. Samy, thank you for bringing timed and robust systems to my attention. No I’m kiding, you know very well what I thank you for ;-) Many thanks to Antoine Dallon, whom I shared an office with, and that probably is the most quiet neighbour one could wish for. Now that I am done writing this manuscript, I can help you dig those moats you wanted ! Thanks to Anthony, my other neighbour, for your active participation to this manuscript: first the magnificient symbol (1 brownie point to whoever finds its occurence in the thesis), and later for brining to me a whole new view of the -WQOs. And I thank our older brothers: Marie, Guillaume, Rémi, Lucca: thank you for bringing to my attention that our diet is not healthy. I am writing this line at 9:30 pm at LSV (thank god this is not a ZRR !), just after a dinner out with other Cachannais, and we had a long discussion on where to get dinner, and had the drastic conclusion that there are no ways of eating healthy around that campus, hence the reputation of students from Cachan. I wish to all future students of the resplendent ENS Paris-Saclay that the landscape will be more diverse in Saclay. And finally I thank the youngsters: Adrien, for being my successor in the high concil, Charlie for being my heir to the Advenced Complexity course. Keep the traditions going. To all the Nadines I did not thank yet, and to those I thanked already, Nadine ! I nadine you with all my nadine: Presidente, Davy, Romaing, Samy, Sarah, Joris, Skippy, Seya, Jeremy, Sushi, Nala, Wasa, ... Thank you for your support during the B-weeks, thank you for keeping me from starvation, thank you for all your help around the house. Many thanks to my writing companions: Miette, after two bachelors and two mas- ters, you will soon be twice doctor in science ! Any typo in this manuscript can be attributed to Aloha, mistaking my keyboard for a DDR Pad (#Skippy). Thanks to Al- pha for sharing his deep understanding of -WQOs, and to Coquillette for biting my power plug. Thanks to Miss and Yang for their discrete presence once in a while. And of course, I want to thank my family, for always being there for me. There is a running joke in my family that each of my life’s “important” moment were shadowed by my older sisters greater achievment. My first day in middle school was my sister’s first day in medical studies. I graduated highschool the same year she became “Interne des Hopitaux” (Christian’s prounonciation). Some days before my oral exam for ENS, 2 I was driving my three-days-old niece from the hospital where she were born back to her home. Allison, maybe your constant crying at night is the reason I wasn’t accepted at Centrale Paris, hence in a sense, this thesis wouldn’t have happened without you ! Thank you ! And most recently, my sister was 8-months pregnant for my wedding, and almost couldn’t attend it. So sisters, what did you plan for June 2018 ? Or maybe this defense does not count as important: “So, as you know, me and Dad are in Paris on the 9th and 10th of June, could you maybe schedule your defense on the 8th so that your father can be there ?”. And of course, thank YOU, for being there and showing me how to write ac- knoledgements: thank you, for being you and making me whole. 3 Contents 1 Introduction 7 2 Basics of Order Theory and Well Quasi-Orderings 9 2.1 Preliminaries . 9 2.2 Order Theory . 11 2.3 Well Quasi Orderings . 13 2.4 Notes on Computability .
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