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Martin Lück. Team Logic: Axioms, Expressiveness, Complexity

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Martin Lück. Team Logic: Axioms, Expressiveness, Complexity Bereitgestellt durch Technische Informationsbibliothek (TIB) Hannover. Technische Informationsbibliothek (TIB) Welfengarten 1 B, 30167 Hannover Postfach 6080, 30060 Hannover Web: https://www.tib.eu/de/ Lizenziert unter Creative Commons Namensnennung 3.0 Deutschland. Team Logic Axioms, Expressiveness, Complexity Von der Fakultät für Elektrotechnik und Informatik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Herrn M. Sc. Martin Lück geboren am 15.12.1988 in Güstrow Hannover 2020 Referent: Prof. Heribert Vollmer, Leibniz Universität Hannover Korreferenten: Prof. Lauri Hella, Universität Tampere (FI) Prof. Juha Kontinen, Universität Helsinki (FI) Tag der Promotion: 09.01.2020 Acknowledgments First of all, I wish to sincerely thank my supervisor, Professor Heribert Vollmer. I am indebted to him for granting me both the trust and liberty to pursue my own ideas in my area of research, as well as the time to develop these over the years and finally write this thesis. Also, I thank him for his deep, motivating and inspiring lectures that drew me to the field of theoretical computer science from the beginning of my studies. Next, I am thankful to my Finnish colleagues from the logic groups of Helsinki and Tampere for the enjoyable work together, for their hospitality and for many fruitful discussions. In particular, I want to thank Juha Kontinen, Miika Hannula, Jonni Virtema, Lauri Hella, and Miikka Vilander. I am especially grateful to Professor Juha Kontinen and Professor Lauri Hella for their advice and for agreeing to review this thesis on our hike at the Dagstuhl seminar on Logics for Dependence and Independence. Also, I thank my fellow students and now-colleagues from Hannover, Anselm Haak, Fabian Müller and Maurice Chandoo for many nice discussions, as well as Arne Meier, Irina Schindler, Yasir Mahmood, Timon Barlag and Rahel Becker from the Institute for Theoretical Computer Science. Finally, I want to thank my sister Maren and my girlfriend Lisa for their unbreakable support and their patience in often challenging times. vi Zusammenfassung Im letzten Jahrzehnt wurde die Teamsemantik kontinuierlich zu einem vielseitigen und mächtigen Rahmen entwickelt, um Begriffe wie Abhängigkeit und Unabhängigkeit mit den Mitteln formaler Logik beschreiben zu können. Bisher lag der Fokus dabei auf der Vielzahl nichtklassischer Atome, die Beziehungen zwischen einzelnen semantischen Einheiten beschreiben, wozu neben dem bekannten Abhängigkeitsatom auch Atome der Unabhängigkeit, des Ein- oder Ausschlusses, der Nichtleerheit sowie verschiedene Zählatome gehören. Das Ziel dieser Arbeit ist hingegen, die Natur der Teamsemantik auch und gerade ohne solche Atome zu untersuchen. Im ersten Teil beschäftigen wir uns dafür mit grundlegenden Fragen, wie etwa: Wann genau ist eine Logik eine teamsemantische Erweiterung einer klassischen Logik? Welche Freiheitsgrade gibt es beim Definieren einer solchen Erweiterung? Hierfür analysieren wir existierende Teamlogiken und zeigen mögliche Ansätze zur formalen Beantwortung dieser Fragen auf. Im Rest der Arbeit betrachten wir mit der Aussagenteamlogik PL(∼), der Modalteam- logik ML(∼) und der Prädikatenteamlogik FO(∼) drei konkrete Teamlogiken ohne nicht- klassische Atome, die aber unter den Boole’schen Verknüpfungen abgeschlossen sind. Wir untersuchen sie hinsichtlich dreier zentraler Fragestellungen aus dem Bereich der mathematischen Logik, nämlich Ausdrucksstärke, Berechnungskomplexität und Axio- matisierbarkeit. Dabei verwenden wir auch abstrakte Resultate aus dem ersten Teil. Ein wichtiger Teil der Arbeit ist das Ergebnis, dass ML(∼) nichtelementare Komplexität besitzt. Für den Beweis übertragen wir Begriffe aus der Modelltheorie auf die Teamlogik und konstruieren einen Schwerebeweis in mehreren Schritten, wobei wir als Zwischen- resultate festhalten, dass Eigenschaften wie Bisimilarität und Kanonizität in ML(∼) auf gewisse Weise effizient definierbar sind. Durch Übertragung der Filtrationsmethode auf Teamsemantik finden wir anschließend jedoch auch elementar entscheidbare Fragmente. Danach wird FO(∼) bezüglich Komplexitätsfragen betrachtet, wobei sich das Zwei- Variablen-Fragment wie im klassischen Fall als entscheidbar herausstellt, ebenso wie das Guarded Fragment GF(∼), das wir analog zum klassischen Fragment GF einführen. Des Weiteren wird im Bereich der Modelltheorie eine Variante des Ultraproduktsatzes von Łoś bewiesen, aus dem beispielsweise auch der Kompaktheitssatz für FO(∼) folgt. Zuletzt entwickeln wir ein modulares Beweissystem für die genannten Logiken PL(∼), ML(∼) und FO(∼). Wir zeigen insbesondere, dass sich die besondere Disjunktion der Teamlogik wie ein modaler Operator axiomatisieren lässt. Für die Vollständigkeit des Systems spielt auch die Widerlegungsvollständigkeit auf der Ebene der Literale eine wichtige Rolle, die für Teamlogiken im Gegensatz zu klassischer Logik nicht trivial gegeben ist. Es werden zwei Möglichkeiten vorgestellt, wie sie für obige Logiken dennoch erreicht werden kann. Schlagworte: Team-Semantik; Axiome; Ausdrucksstärke; Komplexität 2010 MSC: 03B45; 03B60; 03B70; 03C20; 68Q15; 68Q17 vii Abstract In the last decade, team semantics has been continuously developed into a flexible and powerful framework to describe concepts of dependence and independence by means of formal logic. Until now, research mostly focused on the manifold non-classical atoms that express relationships between the semantical units. Besides the prominent dependence atom, among these there also are atoms of independence, inclusion, exclusion, non- emptiness as well as various counting atoms. The aim of this thesis is instead to study the nature of team semantics on its own and especially without such atoms. In the first part, we consider basic questions, such as: What makes a logic a team-semantical extension of a classical logic? Which degrees of freedom exist when defining such an extension? For this, we analyze existing team logics and point out possible approaches to answer these questions formally. In the remainder of the thesis, we study propositional team logic PL(∼), modal team logic ML(∼) and first-order team logic FO(∼), which are three concrete team logics without non-classical atoms, yet which are closed under the Boolean operations. We investigate three central questions from the area of mathematical logic, that is, expressive power, computational complexity, and axiomatizability. For this, we also utilize abstract results from the first part. An important part of the thesis is the result that ML(∼) has non-elementary complexity. For the proof, we generalize concepts of model theory to team logic and in several steps construct a hardness proof, with key steps such as showing that bisimilarity and canonic- ity are, in a certain sense, efficiently definable in ML(∼). That being said, by transferring the well-known filtration method to team semantics, we also find elementarily decidable fragments. Afterwards, the complexity aspects of FO(∼) are studied. Its two-variable fragment turns out to be decidable as in the classical case, and likewise for the Guarded Fragment GF(∼) that we introduce analogously to the classical fragment GF. Furthermore, as a model-theoretic result, we prove a variant of Łoś’s ultraproduct theorem, which entails, for example, the compactness theorem for FO(∼). Finally, we develop a modular proof system for the mentioned logics PL(∼), ML(∼) and FO(∼). In particular, we show that the special disjunction of team logic can be axiomatized like a modal operator. In order to prove that this system is complete, the refutation completeness on the level of literals plays a crucial role, which, in contrast to classical logics, does not trivially hold for team logics. We present two methods to achieve this property for the above logics nonetheless. Keywords: Team semantics; axioms; expressiveness; complexity 2010 MSC: 03B45; 03B60; 03B70; 03C20; 68Q15; 68Q17 viii Contents 1 Introduction 1 1.1 Team logic . 2 1.2 Contributions . 6 1.3 Further notes . 11 2 Preliminaries 12 2.1 Complexity theory . 12 2.2 Team logic . 15 3 Abstract team logic 29 3.1 Basic definitions . 29 3.2 Teamification . 37 3.3 Operators . 43 3.4 Transversals . 51 3.5 Relaxations . 53 3.6 Strict and lax standard transversals . 57 3.7 Quasi-flatness . 59 3.8 Outlook: Linear Temporal Logic . 73 3.9 Summary and outlook . 76 4 The complexity of modal team logic 78 4.1 Types and canonical models . 78 4.2 Scopes and subteam quantifiers . 82 4.3 Implementing bisimulation . 85 4.4 Enforcing a canonical model . 87 4.5 Defining an order on types . 91 4.6 Encoding non-elementary computations . 97 4.7 Hardness under strict semantics . 109 4.8 Hardness on restricted frame classes . 110 4.9 Filtration in team semantics . 113 4.10 Summary and outlook . 119 5 First-order team logic 122 5.1 Upper bounds for satisfiability and validity . 122 5.2 A standard translation for team semantics . 133 5.3 Łoś’s theorem for team semantics . 139 5.4 Summary and outlook . 145 Contents 6 An axiomatization of team logic 148 6.1 Introduction . 148 6.2 Axioms of the Boolean connectives . 151 6.3 Operator elimination . 159 6.4 A remark on the empty team . 171 6.5 Summary and outlook . 173 7 Conclusion 177 Bibliography 179 Appendix 190 Index 200 Curriculum Vitae 203 List of publications 204 x List of Tables and Figures 2.1 Complexity of satisfiability and validity of propositional and modal team logics . 26 2.2 Complexity of model
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