Constructive Game Logic ⋆
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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 ....................... -
Game-Based Verification and Synthesis
Downloaded from orbit.dtu.dk on: Sep 30, 2021 Game-based verification and synthesis Vester, Steen Publication date: 2016 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Vester, S. (2016). Game-based verification and synthesis. Technical University of Denmark. DTU Compute PHD-2016 No. 414 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Ph.D. Thesis Doctor of Philosophy Game-based verification and synthesis Steen Vester Kongens Lyngby, Denmark 2016 PHD-2016-414 ISSN: 0909-3192 DTU Compute Department of Applied Mathematics and Computer Science Technical University of Denmark Richard Petersens Plads Building 324 2800 Kongens Lyngby, Denmark Phone +45 4525 3031 [email protected] www.compute.dtu.dk Summary Infinite-duration games provide a convenient way to model distributed, reactive and open systems in which several entities and an uncontrollable environment interact. -
The Unintended Interpretations of Intuitionistic Logic
THE UNINTENDED INTERPRETATIONS OF INTUITIONISTIC LOGIC Wim Ruitenburg Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI 53233 Abstract. We present an overview of the unintended interpretations of intuitionis- tic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and higher order logic without choice principles or not satisfying the right choice sequence properties. We conclude with remarks on the quest for a correct interpretation of intuitionistic logic. §1. The Origins of Intuitionistic Logic Intuitionism was more than twenty years old before A. Heyting produced the first complete axiomatizations for intuitionistic propositional and predicate logic: according to L. E. J. Brouwer, the founder of intuitionism, logic is secondary to mathematics. Some of Brouwer’s papers even suggest that formalization cannot be useful to intuitionism. One may wonder, then, whether intuitionistic logic should itself be regarded as an unintended interpretation of intuitionistic mathematics. I will not discuss Brouwer’s ideas in detail (on this, see [Brouwer 1975], [Hey- ting 1934, 1956]), but some aspects of his philosophy need to be highlighted here. According to Brouwer mathematics is an activity of the human mind, a product of languageless thought. One cannot be certain that language is a perfect reflection of this mental activity. This makes language an uncertain medium (see [van Stigt 1982] for more details on Brouwer’s ideas about language). In “De onbetrouwbaarheid der logische principes” ([Brouwer 1981], pp. -
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. -
Survey of the Computational Complexity of Finding a Nash Equilibrium
Survey of the Computational Complexity of Finding a Nash Equilibrium Valerie Ishida 22 April 2009 1 Abstract This survey reviews the complexity classes of interest for describing the problem of finding a Nash Equilibrium, the reductions that led to the conclusion that finding a Nash Equilibrium of a game in the normal form is PPAD-complete, and the reduction from a succinct game with a nice expected utility function to finding a Nash Equilibrium in a normal form game with two players. 2 Introduction 2.1 Motivation For many years it was unknown whether a Nash Equilibrium of a normal form game could be found in polynomial time. The motivation for being able to due so is to find game solutions that people are satisfied with. Consider an entertaining game that some friends are playing. If there is a set of strategies that constitutes and equilibrium each person will probably be satisfied. Likewise if financial games such as auctions could be solved in polynomial time in a way that all of the participates were satisfied that they could not do any better given the situation, then that would be great. There has been much progress made in the last fifty years toward finding the complexity of this problem. It turns out to be a hard problem unless PPAD = FP. This survey reviews the complexity classes of interest for describing the prob- lem of finding a Nash Equilibrium, the reductions that led to the conclusion that finding a Nash Equilibrium of a game in the normal form is PPAD-complete, and the reduction from a succinct game with a nice expected utility function to finding a Nash Equilibrium in a normal form game with two players. -
Equilibrium Computation in Normal Form Games
Tutorial Overview Game Theory Refresher Solution Concepts Computational Formulations Equilibrium Computation in Normal Form Games Costis Daskalakis & Kevin Leyton-Brown Part 1(a) Equilibrium Computation in Normal Form Games Costis Daskalakis & Kevin Leyton-Brown, Slide 1 Tutorial Overview Game Theory Refresher Solution Concepts Computational Formulations Overview 1 Plan of this Tutorial 2 Getting Our Bearings: A Quick Game Theory Refresher 3 Solution Concepts 4 Computational Formulations Equilibrium Computation in Normal Form Games Costis Daskalakis & Kevin Leyton-Brown, Slide 2 Tutorial Overview Game Theory Refresher Solution Concepts Computational Formulations Plan of this Tutorial This tutorial provides a broad introduction to the recent literature on the computation of equilibria of simultaneous-move games, weaving together both theoretical and applied viewpoints. It aims to explain recent results on: the complexity of equilibrium computation; representation and reasoning methods for compactly represented games. It also aims to be accessible to those having little experience with game theory. Our focus: the computational problem of identifying a Nash equilibrium in different game models. We will also more briefly consider -equilibria, correlated equilibria, pure-strategy Nash equilibria, and equilibria of two-player zero-sum games. Equilibrium Computation in Normal Form Games Costis Daskalakis & Kevin Leyton-Brown, Slide 3 Tutorial Overview Game Theory Refresher Solution Concepts Computational Formulations Part 1: Normal-Form Games -
Constructivity in Homotopy Type Theory
Ludwig Maximilian University of Munich Munich Center for Mathematical Philosophy Constructivity in Homotopy Type Theory Author: Supervisors: Maximilian Doré Prof. Dr. Dr. Hannes Leitgeb Prof. Steve Awodey, PhD Munich, August 2019 Thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Logic and Philosophy of Science contents Contents 1 Introduction1 1.1 Outline................................ 3 1.2 Open Problems ........................... 4 2 Judgements and Propositions6 2.1 Judgements ............................. 7 2.2 Propositions............................. 9 2.2.1 Dependent types...................... 10 2.2.2 The logical constants in HoTT .............. 11 2.3 Natural Numbers.......................... 13 2.4 Propositional Equality....................... 14 2.5 Equality, Revisited ......................... 17 2.6 Mere Propositions and Propositional Truncation . 18 2.7 Universes and Univalence..................... 19 3 Constructive Logic 22 3.1 Brouwer and the Advent of Intuitionism ............ 22 3.2 Heyting and Kolmogorov, and the Formalization of Intuitionism 23 3.3 The Lambda Calculus and Propositions-as-types . 26 3.4 Bishop’s Constructive Mathematics................ 27 4 Computational Content 29 4.1 BHK in Homotopy Type Theory ................. 30 4.2 Martin-Löf’s Meaning Explanations ............... 31 4.2.1 The meaning of the judgments.............. 32 4.2.2 The theory of expressions................. 34 4.2.3 Canonical forms ...................... 35 4.2.4 The validity of the types.................. 37 4.3 Breaking Canonicity and Propositional Canonicity . 38 4.3.1 Breaking canonicity .................... 39 4.3.2 Propositional canonicity.................. 40 4.4 Proof-theoretic Semantics and the Meaning Explanations . 40 5 Constructive Identity 44 5.1 Identity in Martin-Löf’s Meaning Explanations......... 45 ii contents 5.1.1 Intensional type theory and the meaning explanations 46 5.1.2 Extensional type theory and the meaning explanations 47 5.2 Homotopical Interpretation of Identity ............ -
Local Constructive Set Theory and Inductive Definitions
Local Constructive Set Theory and Inductive Definitions Peter Aczel 1 Introduction Local Constructive Set Theory (LCST) is intended to be a local version of con- structive set theory (CST). Constructive Set Theory is an open-ended set theoretical setting for constructive mathematics that is not committed to any particular brand of constructive mathematics and, by avoiding any built-in choice principles, is also acceptable in topos mathematics, the mathematics that can be carried out in an arbi- trary topos with a natural numbers object. We refer the reader to [2] for any details, not explained in this paper, concerning CST and the specific CST axiom systems CZF and CZF+ ≡ CZF + REA. CST provides a global set theoretical setting in the sense that there is a single universe of all the mathematical objects that are in the range of the variables. By contrast a local set theory avoids the use of any global universe but instead is formu- lated in a many-sorted language that has various forms of sort including, for each sort α a power-sort Pα, the sort of all sets of elements of sort α. For each sort α there is a binary infix relation ∈α that takes two arguments, the first of sort α and the second of sort Pα. For each formula φ and each variable x of sort α, there is a comprehension term {x : α | φ} of sort Pα for which the following scheme holds. Comprehension: (∀y : α)[ y ∈α {x : α | φ} ↔ φ[y/x] ]. Here we use the notation φ[a/x] for the result of substituting a term a for free oc- curences of the variable x in the formula φ, relabelling bound variables in the stan- dard way to avoid variable clashes. -
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. -
Proof Theory of Constructive Systems: Inductive Types and Univalence
Proof Theory of Constructive Systems: Inductive Types and Univalence Michael Rathjen Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England [email protected] Abstract In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a cen- tral role. One objective of this article is to describe the connections with Martin-L¨of type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the rela- tionship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-L¨of type theory. Key words: Explicit mathematics, constructive Zermelo-Fraenkel set theory, Martin-L¨of type theory, univalence axiom, proof-theoretic strength MSC 03F30 03F50 03C62 1 Introduction Intuitionistic systems of inductive definitions have figured prominently in Solomon Feferman’s program of reducing classical subsystems of analysis and theories of iterated inductive definitions to constructive theories of various kinds. In the special case of classical theories of finitely as well as transfinitely iterated inductive definitions, where the iteration occurs along a computable well-ordering, the program was mainly completed by Buchholz, Pohlers, and Sieg more than 30 years ago (see [13, 19]). For stronger theories of inductive 1 i definitions such as those based on Feferman’s intutitionic Explicit Mathematics (T0) some answers have been provided in the last 10 years while some questions are still open. -
Well-Supported Vs. Approximate Nash Equilibria: Query Complexity of Large Games∗
Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games∗ Xi Chen†1, Yu Cheng‡2, and Bo Tang§3 1 Columbia University, New York, USA [email protected] 2 University of Southern California, Los Angeles, USA [email protected] 3 Oxford University, Oxford, United Kingdom [email protected] Abstract In this paper we present a generic reduction from the problem of finding an -well-supported Nash equilibrium (WSNE) to that of finding an Θ()-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player. Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE. This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions. As an application we prove a 2Ω(n/ log n) lower bound on the randomized query complexity of finding an -ANE in binary-action n-player games, for some constant > 0. This answers an open problem posed by Hart and Nisan [23] and Babichenko [2], and is very close to the trivial upper bound of 2n. Previously for WSNE, Babichenko [2] showed a 2Ω(n) lower bound on the randomized query complexity of finding an -WSNE for some constant > 0. Our result follows directly from combining [2] and our new reduction from WSNE to ANE. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity Keywords and phrases Equilibrium Computation, Query Complexity, Large Games Digital Object Identifier 10.4230/LIPIcs.ITCS.2017.57 1 Introduction The celebrated theorem of Nash [29] states that every finite game has an equilibrium point.