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Quantifiers and Duality Luca Reggio

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Quantifiers and Duality Luca Reggio Quantifiers and duality Luca Reggio To cite this version: Luca Reggio. Quantifiers and duality. Logic in Computer Science [cs.LO]. Université Sorbonne Paris Cité, 2018. English. NNT : 2018USPCC210. tel-02459132 HAL Id: tel-02459132 https://tel.archives-ouvertes.fr/tel-02459132 Submitted on 29 Jan 2020 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. THÈSE DE DOCTORAT DE L’UNIVERSITÉ SORBONNE PARIS CITÉ PRÉPARÉE À L’UNIVERSITÉ PARIS DIDEROT ÉCOLE DOCTORALE DES SCIENCES MATHÉMATIQUES DE PARIS CENTRE ED 386 Quantifiers and duality Par: Dirigée par: Luca REGGIO Mai GEHRKE Thèse de doctorat de Mathématiques Présentée et soutenue publiquement à Paris le 10 septembre 2018 Président du jury: Samson ABRAMSKY, Dept. of Computer Science, University of Oxford Rapporteurs: Mamuka JIBLADZE, Razmadze Mathematical Inst., Tbilisi State University Achim JUNG, School of Computer Science, University of Birmingham Directrice de thése: Mai GEHRKE, Lab. J. A. Dieudonné, CNRS et Université Côte d’Azur Co-encadrants de thése: Samuel J. VAN GOOL, ILLC, University of Amsterdam Daniela PETRI ¸SAN, IRIF, Université Paris Diderot Membres invités: Clemens BERGER, Lab. J. A. Dieudonné, Université Côte d’Azur Paul-André MELLIÈS, IRIF, CNRS et Université Paris Diderot ii Titre: Quantificateurs et dualité Résumé: Le thème central de la présente thèse est le contenu sémantique des quantificateurs logiques. Dans leur forme la plus simple, les quantificateurs permettent d’établir l’existence, ou la non-existence, d’individus répondant à une propriété. En tant que tels, ils incarnent la richesse et la complexité de la logique du premier ordre, par delà la logique propositionnelle. Nous contribuons à l’analyse sémantique des quantificateurs, du point de vue de la théorie de la dualité, dans trois domaines différents des mathématiques et de l’informatique théorique. D’une part, dans la théorie des langages formels à travers la logique sur les mots. D’autre part, dans la logique intuitionniste propositionnelle et dans l’étude de l’interpolation uniforme. Enfin, dans la topologie catégorique et dans la sémantique catégorique de la logique du premier ordre. Mots clefs: dualité de Stone, quantificateurs, théorie des langages, mesure, demi-anneau, algèbre profinie, monade de codensité, interpolation uniforme, logique intuitionniste propositionnelle, théorème de l’application ouverte, espaces compacts, pretopos. Title: Quantifiers and duality Abstract: The unifying theme of the thesis is the semantic meaning of logical quantifiers. In their basic form quantifiers allow to state the existence, or non-existence, of individuals satisfying a property. As such, they encode the richness and the complexity of predicate logic, as opposed to propositional logic. We contribute to the semantic un- derstanding of quantifiers, from the viewpoint of duality theory, in three different areas of mathematics and theoretical computer sci- ence. First, in formal language theory through the syntactic approach provided by logic on words. Second, in intuitionistic propositional logic and in the study of uniform interpolation. Third, in categorical topology and categorical semantics for predicate logic. Keywords: Stone duality, quantifiers, language theory, measure, semiring, profinite algebra, codensity monad, uniform inter- polation, intuitionistic propositional calculus, open mapping theorem, compact Hausdorff spaces, pretopos. iii “Je ne vois pas à quoi ca sert de rêver en arrière et à son âge elle ne pouvait plus rêver en avant.” Émile Ajar v Acknowledgements This thesis is the result of my Ph.D. research project, carried out between 2015 and 2018 in Paris and Nice. Many people have contributed, directly or indirectly, to the document that you are holding in your hands. I will now try to describe these contributions, hopefully forgetting as few as possible of them. First, I would like to thank Mai for her guidance during these three years. Her enthusiasm and energy, despite sometimes difficult circumstances, have been a great source of inspiration and motivation during my research project. Her intuitions and insights, which would often take me months to understand, have taught me a lot about doing research. One thing I have learnt is that very often it is more important, and difficult, to ask the right question, rather than giving the answer. My thanks also go to my other co-authors, Daniela, Sam, and Enzo. From each of them I have learnt a lot, both about doing and writing research. Several chapters of this thesis are based on joint works with them, which I enjoyed very much. Moreover, I want to thank Daniela for dedicating much of her time to supervising me during my first year of Ph.D., Sam for being always available to discuss with me and for giving me useful advices, and Enzo for several inspiring discussions and for his guidance since my Master’s studies. I would also like to thank the rapporteurs for my thesis, Mamuka Jibladze and Achim Jung, for kindly accepting the task of reading the manuscript and for their valuable comments, and the other members of my Ph.D. com- mittee, Samson Abramsky, Clemens Berger, and Paul-André Melliès, for the interest they showed in my Ph.D. research. During the last three years I have profited from stimulating discussions with several researchers, from which I learnt a lot. I would like to thank all of them, and in particular: Samson Abramsky, Matthias Baaz, Clemens Berger, Olivia Caramello, Olivier Carton, Thomas Colcombet, Paul-André Melliès, Frédéric Patras, Jean-Éric Pin, and Marek Zawadowski. I spent the first two years of my Ph.D. studies in Paris, and the third one in Nice. I am very grateful to both the Institut de Recherche en Informa- tique Fondamentale (IRIF) of Paris-Diderot, and to the Laboratoire J. A. Dieudonné in Nice, for hosting me and providing me with the best envi- ronment to focus on my Ph.D. project. I am also grateful to the organisers and participants of the local seminars for the lively and stimulating atmo- sphere they provided. vi I had the opportunity to carry out my Ph.D. project thanks to Univer- sité Sorbonne Paris Cité, which funded my research through a Volant In- ternational Fellowship, agreement IDEX-REGGI15RDXMTSPC1GEHRKE. Also, I had the chance to travel, present my own research at conferences, meet researchers in different fields, and discover new places, thanks to the project DuaLL, funded by the European Research Council under the Eu- ropean Union’s Horizon 2020 research and innovation programme (grant agreement No.670624). In addition to the people I have already mentioned, I would like to thank Célia Borlido, Leo Cabrer, Tomáš Jakl, and Mehdi Zaïdi, for many discus- sions about mathematics, life, movies, cultures, languages, football, climb- ing, coffee, pizza, and francesinha (which I am still waiting to taste), which made these years even more pleasant. Outside academia, in the ‘real world’, I was lucky to have many peo- ple around me to make my days better. In particular, I would to thank Christina, Giacomo e Alessandra, Luca e Giulia, Lucia e Claudio, and Mima, for hanging out, visiting me, and hosting me. Grazie alla mia famiglia per tutto l’affetto, il caffè e il grana forniti in questi anni, e per il supporto costante e incondizionato, scandito a ritmo di traslochi. Merci à Chloé qui m’a soutenu et supporté même quand je ne faisais qu’écrire cette thèse, manger et râler. Tu es mon bonheur ! Merci aussi à sa famille qui m’a aidé à me distraire avec de bonnes conversations, de bons livres et du bon whisky. vii Contents Overview of the thesis 1 I Formal languages and duality 5 1 Introduction: duality and recognition 7 1.1 Stone duality for Boolean algebras . 9 1.2 The algebraic approach to automata theory . 17 1.3 Logic on words . 23 1.4 Boolean spaces with internal monoids . 25 1.5 An overview of the following chapters . 32 2 On existentially quantified languages 33 2.1 The unary Schützenberger product of a BiM . 35 2.2 The Schützenberger product of two BiMs . 40 2.3 Ultrafilter equations . 45 3 Profinite algebras, and semiring-valued measures 49 3.1 Codensity and profinite monads . 52 3.1.1 Codensity monads: a brief introduction . 52 3.1.2 Profinite monads and their algebras . 56 3.2 Measures with values in profinite semirings . 62 3.3 The case of finite semirings: the main result . 70 3.4 The case of profinite idempotent semirings: algebras of con- tinuous functions . 73 4 Semiring quantifiers and measures 79 4.1 Extending Set-monads to BiMs . 80 4.2 Extending the free semimodule monad to BiMs . 86 4.3 Recognisers for operations given by S-valued transductions 88 4.3.1 Recognising operations modelled by a monad . 88 4.3.2 Recognising quantified languages via S-transductions 91 4.4 Duality-theoretic account of the construction . 95 4.4.1 The BiM ♦X by duality . 96 4.4.2 A Reutenauer theorem for ♦X . 102 viii II Logic, spaces and coherent categories 105 5 Introduction: coherent categories and their neighbourhoods 107 6 Uniform interpolation for IPC, topologically 117 6.1 Esakia duality for Heyting algebras . 118 6.2 Open maps and uniform interpolation . 122 6.3 Clopen up-sets step-by-step . 124 6.4 Reduction to finite Kripke models . 126 6.5 Proof of Lemma 6.15 . 128 7 An abstract characterisation of compact Hausdorff spaces 133 7.1 The topological representation .
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