Game Semantics in String Diagrams
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Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation
1 Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation Patrick Cousota aD´epartement d’Informatique, Ecole´ Normale Sup´erieure, 45 rue d’Ulm, 75230 Paris cedex 05, France, [email protected], http://www.di.ens.fr/~cousot We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step seman- tics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational se- mantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conser- vative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these seman- tics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski fixpoint approximation transfer theorems. Contents 1 Introduction 2 2 Abstraction of Fixpoint Semantics 3 2.1 Fixpoint Semantics ............................... 3 2.2 Fixpoint Semantics Approximation ...................... 4 2.3 Fixpoint Semantics Transfer .......................... 5 2.4 Semantics Abstraction ............................. 7 2.5 Fixpoint Semantics Fusion ........................... 8 2.6 Fixpoint Iterates Reordering .......................... 8 3 Transition/Small-Step Operational Semantics 9 4 Finite and Infinite Sequences 9 4.1 Sequences .................................... 9 4.2 Concatenation of Sequences .......................... 10 4.3 Junction of Sequences ............................. 10 5 Maximal Trace Semantics 10 2 5.1 Fixpoint Finite Trace Semantics ....................... -
Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
Duality and Traces for Indexed Monoidal Categories
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES KATE PONTO AND MICHAEL SHULMAN Abstract. By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a re- finement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal cate- gory, while the Reidemeister trace is a trace in a bicategory. In this paper, we show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious “fiberwise" traces by incorporating the action of the fundamental group of the base space. Our abstract framework lays the foundation for generalizations of these ideas to other contexts. Contents 1. Introduction2 2. Indexed monoidal categories5 3. Indexed coproducts8 4. Symmetric monoidal traces 12 5. Shadows from indexed monoidal categories 15 6. Fiberwise duality 22 7. Base change objects 27 8. Total duality 29 9. String diagrams for objects 39 10. String diagrams for morphisms 42 11. Proofs for fiberwise duality and trace 49 12. Proofs for total duality and trace 63 References 65 Date: Version of October 31, 2011. Both authors were supported by National Science Foundation postdoctoral fellowships during the writing of this paper. This version of this paper contains diagrams that use dots and dashes to distinguish different features, and is intended for printing on a black and white printer. -
A Game Theoretical Semantics for a Logic of Formal Inconsistency
A game theoretical semantics for a logic of formal inconsistency CAN BA¸SKENT∗, Department of Computer Science, University of Bath, BA2 7AY, England. Downloaded from https://academic.oup.com/jigpal/article/28/5/936/5213088 by guest on 25 September 2020 PEDRO HENRIQUE CARRASQUEIRA∗∗, Center for Logic, Epistemology and the History of Science (CLE), Institute of Philosophy and the Humanities (IFCH), State University of Campinas (UNICAMP), Campinas 13083-896, Brazil. Abstract This paper introduces a game theoretical semantics for a particular logic of formal inconsistency called mbC. Keywords: Game theoretical semantics, logic of formal inconsistency, mbC, non-deterministic runs, oracles 1 Motivation Paraconsistent logics are the logics of inconsistencies. They are designed to reason about and with inconsistencies and generate formal systems that do not get trivialized under the presence of inconsistencies. Formally, a logic is paraconsistent if the explosion principle does not hold—in paraconsistent systems, there exist some formulas ϕ, ψ such that ϕ, ¬ϕ ψ. Within the broad class of paraconsistent logics, Logics of Formal Inconsistency (LFIs, for short) present an original take on inconsistencies. Similar to the da Costa systems, LFIs form a large class of paraconsistent logics and make use of a special operator to control inconsistencies [8, 9]. Another interesting property of LFIs is that they internalize consistency and inconsistency at the object level. To date, the semantics for LFIs has been suggested using truth tables and algebraic structures. What we aim for in this paper is to present a game theoretical semantics for a particular logic within the class of LFIs. By achieving this, we attempt at both presenting a wider perspective for the semantics of paraconsistency and a broader, more nuanced understanding of semantic games. -
Game Semantics of Homotopy Type Theory
Game Semantics of Homotopy Type Theory Norihiro Yamada [email protected] University of Minnesota Homotopy Type Theory Electronic Seminar Talks (HoTTEST) Department of Mathematics, Western University February 11, 2021 N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 1 / 19 Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation). Proofs/objects as computations (e.g., succ : : max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies. Introduction Background: MLTT vs. HoTT N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 Proofs/objects as computations (e.g., succ : : max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies. Introduction Background: MLTT vs. HoTT Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation). N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 MLTT as a foundation of constructive maths. -
The Joy of String Diagrams Pierre-Louis Curien
The joy of string diagrams Pierre-Louis Curien To cite this version: Pierre-Louis Curien. The joy of string diagrams. Computer Science Logic, Sep 2008, Bertinoro, Italy. pp.15-22. hal-00697115 HAL Id: hal-00697115 https://hal.archives-ouvertes.fr/hal-00697115 Submitted on 14 May 2012 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. The Joy of String Diagrams Pierre-Louis Curien Preuves, Programmes et Syst`emes,CNRS and University Paris 7 May 14, 2012 Abstract In the past recent years, I have been using string diagrams to teach basic category theory (adjunctions, Kan extensions, but also limits and Yoneda embedding). Us- ing graphical notations is undoubtedly joyful, and brings us close to other graphical syntaxes of circuits, interaction nets, etc... It saves us from laborious verifications of naturality, which is built-in in string diagrams. On the other hand, the language of string diagrams is more demanding in terms of typing: one may need to introduce explicit coercions for equalities of functors, or for distinguishing a morphism from a point in the corresponding internal homset. So that in some sense, string diagrams look more like a language ”`ala Church", while the usual mathematics of, say, Mac Lane's "Categories for the working mathematician" are more ”`ala Curry". -
The Game Semantics of Game Theory
The game semantics of game theory Jules Hedges We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by n ≥ 0 noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a com- pact closed category of ‘computable open games’ by replacing the underly- ing dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the traced cartesian category of directed-complete partial orders. 1 Introduction Although the mathematics of games shares a common ancestor in Zermelo’s work on backward induction, it split early on into two subjects that are essentially disjoint: game theory and game semantics. Game theory is the applied study of modelling real-world interacting agents, for example in economics or artificial intelligence. Game semantics, by contrast, uses agents to model – this time in the sense of semantics rather than mathematical modelling – situations in which a system interacts with an environment, but neither would usually be thought of as agents in a philosophical sense. On a technical level, game semantics not only restricts to the two-player zero-sum case, but moreover promotes one of the players to be the Player, and demotes the other to mere Opponent. -
Category Theory Using String Diagrams
Category Theory Using String Diagrams Dan Marsden November 11, 2014 Abstract In [Fokkinga, 1992a,b] and [Fokkinga and Meertens, 1994] a calcula- tional approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting re- tain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphi- cal representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation. Our approach is to proceed primarily by example, systematically ap- plying graphical techniques to many aspects of category theory. We de- velop string diagrammatic formulations of many common notions, includ- ing adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. These graphical tools are then used to explicitly prove many standard results in our proposed diagrammatic style. 1 Introduction arXiv:1401.7220v2 [math.CT] 9 Nov 2014 This work develops in some detail many aspects of basic category theory, with the aim of demonstrating the combined effectiveness of two key concepts: 1. The calculational reasoning approach to mathematics. 2. The use of string diagrams in category theory. -
Semantics for a Lambda Calculus for String Diagrams
Semantics for a Lambda Calculus for String Diagrams Bert Lindenhovius Michael Mislove Department of Computer Science Department of Computer Science Tulane University Tulane University Vladimir Zamdzhiev Université de Lorraine, CNRS, Inria, LORIA Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical en- richment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragment of the language which corresponds to a modified version of Benton and Wadler’s adjoint calculus with recursion. In keeping with the purpose of the special issue, we also describe the influence of Samson Abram- sky’s research on these results, and on the overall project of which this is a part. 1 Dedication We are pleased to contribute to this volume honoring SAMSON ABRAMSKY’s many contributions to logic and its applications in theoretical computer science. This contribution is an extended version of the paper [28], which concerns the semantics of high-level functional quantum programming languages. This research is part of a project on the same general theme. The project itself would not exist if it weren’t for Samson’s support, guidance and participation. And, as we document in the final section of this contribution, Samson’s research has had a direct impact on this work, and also points the way for further results along the general line we are pursuing. -
Game Theoretical Semantics for Paraconsistent Logics
Game Theoretical Semantics for Paraconsistent Logics Can Ba¸skent Department of Computer Science, University of Bath, England [email protected], www.canbaskent.net/logic 1 Introduction Game theoretical semantics suggests a very intuitive approach to formal seman- tics. The semantic verification game for classical logic is played by two players, verifier and falsifier who we call Heloise and Abelard respectively. The goal of Heloise in the game is to verify the truth of a given formula in a given model whereas for Abelard it is to falsify it. The rules are specified syntactically based on the form of the formula. During the game, the given formula is broken into subformulas step by step by the players. The game terminates when it reaches the propositional literals and when there is no move to make. If the game ends up with a propositional literal which is true in the model in question, then Heloise wins the game. Otherwise, Abelard wins. Conjunction is ssociated with Abelard, disjunction with Heloise. That is, when the main connective is a conjunction, it is Abelard’s turn to choose and make a move, and similarly, disjunction yields a choice for Heloise. The negation operator switches the roles of the players: Heloise becomes the falsifier, Abelard becomes the verifier. The major result of this approach states that Heloise has a winning strategy if and only if the given formula is true in the given model. The semantic verification game and its rules are shaped by classical logic and consequently by its restrictions. In this work, we first observe how the verification games change in non-classical, especially propositional paraconsistent logics, and give Hintikka-style game theoretical se- mantics for them. -
2-Categorical Aspects of Quasi-Categories
2-Categorical Aspects of Quasi-Categories Zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften von der KIT-Fakultät für Mathematik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation von Tobias Columbus Tag der mündlichen Prüfung: 28. Juni 2017 1. Referent: PD Dr. Fabian Januszewski 2. Referent: Prof. Dr. Frank Herrlich 3. Referent: Dr. Richard Garner Acknowledgements First and foremost, I want to thank Fabian Januszewski, Frank Herrlich and Gabriela Weitze-Schmithüsen1 for allowing me latitude in pursuing my research. During my time in Karlsruhe, I enjoyed a lot of freedom, which is not something to be taken for granted. I am also indebted to Richard Garner who unhesitatingly accepted to serve as external reviewer for this thesis. I confess with embarrassement that those aspects of my work that are closest to his did not make it into this text. Each member of the number theory and algebraic geometry group in Karls- ruhe has had a stake in the completion of this work — be it by taking o some load of me during the last months, by discussing the very topic treated in this text or by simply having fun together during lunch or while having a coee. From this group of my colleagues, I have to single out my ocemate, Felix Wellen, with whom I had a lot of interesting discussions2 and without 1in alphabetical order 2More often than not, I was spluttering callow thoughts and it was only with the help of Felix that I was able to arrange my ideas. iii whom the last years would probably have been less fun. -
A Game Theoretical Semantics for Logics of Nonsense
A Game Theoretical Semantics for Logics of Nonsense Can Bas¸kent Department of Computer Science, Middlesex University, London, UK [email protected] Logics of non-sense allow a third truth value to express propositions that are nonsense. These logics are ideal formalisms to understand how errors are handled in programs and how they propagate throughout the programs once they appear. In this paper, we give a Hintikkan game semantics for logics of non-sense and prove its correctness. We also discuss how a known solution method in game theory, the iterated elimination of strictly dominated strategies, relates to semantic games for logics of nonsense. Finally, we extend the logics of nonsense only by means of semantic games, developing a new logic of nonsense, and propose a new game semantics for Priest’s Logic of Paradox. Keywords Game semantics, logics of nonsense, logic of paradox, iterated elimination of strictly dominated strategies. 1 Introduction Logics of nonsense allow a third truth value to express propositions that are nonsense. The initial mo- tivation behind introducing nonsensical propositions was to capture the logical behavior of semantic paradoxes that were thought to be nonsensical. As argued by Ferguson, nonsense is “infectious” [8]: “Meaninglessness [...] propagates through the language; that a subformula is meaningless entails that any complex formula in which it finds itself is likewise meaningless.” Logics of nonsense, for this reason, are intriguing. Nonsense appears in various contexts. In mathematics, the expression “1/x” is considered mean- ingless when x = 0. Certain approaches to truth disregard paradoxes of self-reference and exclude them as meaningless.