21st International Conference, FOSSACS 2018 Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018 Thessaloniki, Greece, April 14–20, 2018, Proceedings Foundations of Software Science and Computation Structures LNCS 10803 ARCoSS Christel Baier · Ugo Dal Lago (Eds.) Lecture Notes in Computer Science 10803 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison, UK Josef Kittler, UK Friedemann Mattern, Switzerland Moni Naor, Israel Bernhard Steffen, Germany Doug Tygar, USA Takeo Kanade, USA Jon M. Kleinberg, USA John C. Mitchell, USA C. Pandu Rangan, India Demetri Terzopoulos, USA Gerhard Weikum, Germany Advanced Research in Computing and Software Science Subline of Lecture Notes in Computer Science Subline Series Editors Giorgio Ausiello, University of Rome ‘ La Sapienza ’ , Italy Vladimiro Sassone, University of Southampton, UK Subline Advisory Board Susanne Albers, TU Munich, Germany Benjamin C. Pierce, University of Pennsylvania, USA Bernhard Steffen, University of Dortmund, Germany Deng Xiaotie, City University of Hong Kong Jeannette M. Wing, Microsoft Research, Redmond, WA, USA More information about this series at http://www.springer.com/series/7407 Christel Baier • Ugo Dal Lago (Eds.) Foundations of Software Science and Computation Structures 21st International Conference, FOSSACS 2018 Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018 Thessaloniki, Greece, April 14 – 20, 2018 Proceedings Editors Christel Baier TU Dresden Dresden Germany Ugo Dal Lago Universit à di Bologna Bologna Italy ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notes in Computer Science ISBN 978-3-319-89365-5 ISBN 978-3-319-89366-2 (eBook) https://doi.org/10.1007/978-3-319-89366-2 Library of Congress Control Number: 2018937398 LNCS Sublibrary: SL1 – Theoretical Computer Science and General Issues © The Editor(s) (if applicable) and The Author(s) 2018. This book is an open access publication. 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Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland ETAPS Foreword Welcome to the proceedings of ETAPS 2018! After a somewhat coldish ETAPS 2017 in Uppsala in the north, ETAPS this year took place in Thessaloniki, Greece. I am happy to announce that this is the fi rst ETAPS with gold open access proceedings. This means that all papers are accessible by anyone for free. ETAPS 2018 was the 21st instance of the European Joint Conferences on Theory and Practice of Software. ETAPS is an annual federated conference established in 1998, and consists of fi ve conferences: ESOP, FASE, FoSSaCS, TACAS, and POST. Each conference has its own Program Committee (PC) and its own Steering Com- mittee. The conferences cover various aspects of software systems, ranging from theoretical computer science to foundations to programming language developments, analysis tools, formal approaches to software engineering, and security. Organizing these conferences in a coherent, highly synchronized conference program facilitates participation in an exciting event, offering attendees the possibility to meet many researchers working in different directions in the fi eld, and to easily attend talks of different conferences. Before and after the main conference, numerous satellite work- shops take place and attract many researchers from all over the globe. ETAPS 2018 received 479 submissions in total, 144 of which were accepted, yielding an overall acceptance rate of 30%. I thank all the authors for their interest in ETAPS, all the reviewers for their peer reviewing efforts, the PC members for their contributions, and in particular the PC (co-)chairs for their hard work in running this entire intensive process. Last but not least, my congratulations to all authors of the accepted papers! ETAPS 2018 was enriched by the unifying invited speaker Martin Abadi (Google Brain, USA) and the conference-speci fi c invited speakers (FASE) Pamela Zave (AT & T Labs, USA), (POST) Benjamin C. Pierce (University of Pennsylvania, USA), and (ESOP) Derek Dreyer (Max Planck Institute for Software Systems, Germany). Invited tutorials were provided by Armin Biere (Johannes Kepler University, Linz, Austria) on modern SAT solving and Fabio Somenzi (University of Colorado, Boulder, USA) on hardware veri fi cation. My sincere thanks to all these speakers for their inspiring and interesting talks! ETAPS 2018 took place in Thessaloniki, Greece, and was organised by the Department of Informatics of the Aristotle University of Thessaloniki. The university was founded in 1925 and currently has around 75,000 students; it is the largest uni- versity in Greece. ETAPS 2018 was further supported by the following associations and societies: ETAPS e.V., EATCS (European Association for Theoretical Computer Science), EAPLS (European Association for Programming Languages and Systems), and EASST (European Association of Software Science and Technology). The local organization team consisted of Panagiotis Katsaros (general chair), Ioannis Stamelos, Lefteris Angelis, George Rahonis, Nick Bassiliades, Alexander Chatzigeorgiou, Ezio Bartocci, Simon Bliudze, Emmanouela Stachtiari, Kyriakos Georgiadis, and Petros Stratis (EasyConferences). The overall planning for ETAPS is the main responsibility of the Steering Com- mittee, and in particular of its Executive Board. The ETAPS Steering Committee consists of an Executive Board and representatives of the individual ETAPS confer- ences, as well as representatives of EATCS, EAPLS, and EASST. The Executive Board consists of Gilles Barthe (Madrid), Holger Hermanns (Saarbr ü cken), Joost-Pieter Katoen (chair, Aachen and Twente), Gerald L ü ttgen (Bamberg), Vladimiro Sassone (Southampton), Tarmo Uustalu (Tallinn), and Lenore Zuck (Chicago). Other members of the Steering Committee are: Wil van der Aalst (Aachen), Parosh Abdulla (Uppsala), Amal Ahmed (Boston), Christel Baier (Dresden), Lujo Bauer (Pittsburgh), Dirk Beyer (Munich), Mikolaj Bojanczyk (Warsaw), Luis Caires (Lisbon), Jurriaan Hage (Utrecht), Rainer H ä hnle (Darmstadt), Reiko Heckel (Leicester), Marieke Huisman (Twente), Panagiotis Katsaros (Thessaloniki), Ralf K ü sters (Stuttgart), Ugo Dal Lago (Bologna), Kim G. Larsen (Aalborg), Matteo Maffei (Vienna), Tiziana Margaria (Limerick), Flemming Nielson (Copenhagen), Catuscia Palamidessi (Palaiseau), Andrew M. Pitts (Cambridge), Alessandra Russo (London), Dave Sands (G ö teborg), Don Sannella (Edinburgh), Andy Sch ü rr (Darmstadt), Alex Simpson (Ljubljana), Gabriele Taentzer (Marburg), Peter Thiemann (Freiburg), Jan Vitek (Prague), Tomas Vojnar (Brno), and Lijun Zhang (Beijing). I would like to take this opportunity to thank all speakers, attendees, organizers of the satellite workshops, and Springer for their support. I hope you all enjoy the proceedings of ETAPS 2018. Finally, a big thanks to Panagiotis and his local orga- nization team for all their enormous efforts that led to a fantastic ETAPS in Thessaloniki! February 2018 Joost-Pieter Katoen VI ETAPS Foreword Preface This volume contains the papers presented at the 21st International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2018), which was held April 16 – 19, 2018, in Thessaloniki, Greece. The conference is dedicated to foundational research with a clear signi fi cance for software science and brings together research on theories and methods to support the analysis, integration, synthesis, transformation, and veri fi cation of programs and software systems. The program consisted of 31 contributed papers, selected from among 103 sub- missions. Each submission was reviewed by at least three Program Committee mem- bers, with the help of external experts. After a three-day rebuttal phase, the selection was made based on discussions via the EasyChair conference management system, which was also used to assist with the compilation of the proceedings. We wish to thank all authors who submitted to FoSSaCS 2018, all the Program Committee members for their excellent work, and the external reviewers for their thorough evaluation of the submissions. In addition, we would like to thank the ETAPS organization for providing an excellent environment for FoSSaCS and other confer- ences and workshops. March 2018 Christel Baier Ugo Dal Lago Organization Program Committee Andreas Abel Gothenburg University, Sweden Christel Baier TU Dresden, Germany Nathalie Bertrand Inria, France Mikolaj Bojanczyk Warsaw University, Poland Udi Boker Interdisciplinary Center (IDC) Herzliya, Israel Luis Caires Universidade NOVA de Lisboa, Portugal Ugo Dal Lago University of Bologna, Italy Yuxin Deng East China Normal University, China Mariangiola Dezani-Ciancaglini Universit à di Torino, Italy Ichiro Hasuo National Institute of Informatics, Japan Radha Jagadeesan DePaul University, UK Stefan Kiefer University of Oxford, UK Barbara K ö nig Universit ä t Duisburg-Essen, Germany David Monniaux CNRS, VERIMAG, France Andrzej Murawski The University of Warwick, UK Joel Ouaknine Max Planck Institute for Software Systems, Germany Catuscia Palamidessi Inria, France Kirstin Peters TU Berlin, Germany Damien Pous CNRS, ENS Lyon, France Jean-Francois Raskin Universit é Libre de Bruxelles, Belgium Helmut Seidl Technical University of Munich, Germany Alexandra Silva University College London, UK Alex Simpson University of Ljubljana, Slovenia Jiri Srba Aalborg University, Denmark Jean-Marc Talbot Aix-Marseille Universit é , France Christine Tasson Universit é Denis Diderot, France Kazushige Terui Kyoto University, Japan Additional Reviewers Aler Tubella, Andrea Almagor, Shaull Asada, Kazuyuki Atkey, Robert Bacci, Giorgio Bacci, Giovanni Bagnol, Marc Baldan, Paolo Basold, Henning Bavera, Francisco Beffara, Emmanuel Benveniste, Albert Beohar, Harsh Berardi, Stefano Bertolissi, Clara Berwanger, Dietmar Blondin, Michael Bocchi, Laura Boreale, Michele Boulm é , Sylvain Bouyer, Patricia Brazdil, Tomas Brotherston, James Brunet, Paul Bruni, Roberto Bucchiarone, Antonio Busatto-Gaston, Damien B ø nneland, Frederik M. Cabrera, Benjamin Cadilhac, Micha ë l Carayol, Arnaud Castellan, Simon Chen, Tzu-Chun Clouston, Ranald Cockx, Jesper Coppo, Mario Corbineau, Pierre Cristescu, Ioana Doumane, Amina Dubut, J é r é my Eberhart, Clovis Emmi, Michael Enea, Constantin Enevoldsen, S ø ren Enqvist, Sebastian Exibard, L é o Falcone, Ylies Feng, Yuan Figueira, Diego Fijalkow, Nathana ë l Fournier, Paulin Fujii, Soichiro Galmiche, Didier Geeraerts, Gilles Genest, Blaise Gorogiannis, Nikos Graham-Lengrand, St é phane Grellois, Charles Haar, Stefan Haase, Christoph Halfon, Simon Hartmann, Nico Hautem, Quentin Hirschkoff, Daniel Hirschowitz, Tom Hsu, Justin Huang, Mingzhang Jacobs, Bart Jacquemard, Florent Jansen, Nils Jaskelioff, Mauro Jecker, Isma ë l Junges, Sebastian Kakutani, Yoshihiko Kanovich, Max Kaufmann, Isabella Kerjean, Marie King, Andy Klein, Felix Klin, Bartek Ko ł odziejczyk, Leszek Kretinsky, Jan Krivine, Jean Kupke, Clemens Kutsia, Temur K ü pper, Sebastian Laarman, Alfons Laird, Jim Lanese, Ivan Lang, Frederic Lazic, Ranko Lefaucheux, Engel Leifer, Matthew Lepigre, Rodolphe Letouzey, Pierre Levy, Paul Blain Li, Xin Liang, Hongjin Licata, Daniel R. Litak, Tadeusz Lohrey, Markus Lombardy, Sylvain Long, Huan Luttik, Bas L ó pez, Hugo A. Mackie, Ian Madnani, Khushraj Maggi, Fabrizio Maria Mallet, Frederic Maranget, Luc Markey, Nicolas Martens, Wim Mayr, Richard Mazowiecki, Filip Miku č ionis, Marius Milius, Stefan Mio, Matteo Moggi, Eugenio Monmege, Benjamin Muniz, Marco Nestmann, Uwe New, Max Nielsen, Mogens Nolte, Dennis Nordvall Forsberg, Fredrik Nyman, Ulrik Okudono, Takamasa Orchard, Dominic Oualhadj, Youssouf Padovani, Luca Panangaden, Prakash Pang, Jun Pavlovic, Dusko Perez, Guillermo Pitts, Andrew Plump, Detlef Pouly, Amaury Power, John Pruekprasert, Sasinee Ramsay, Steven Regnier, Laurent Rehak, Vojtech Roggenbach, Markus Rot, Jurriaan Sacerdoti Coen, Claudio Sammartino, Matteo Sankur, Ocan Saurin, Alexis Schalk, Andrea Scherer, Gabriel Schmidt-Schau ß , Manfred Selinger, Peter Shirmohammadi, Mahsa Sickert, Salomon X Organization Sighireanu, Mihaela Sistla, A. Prasad Sojakova, Kristina Soloviev, Sergei Sozeau, Matthieu Sprunger, David Strassburger, Lutz Tang, Qiyi Torres Vieira, Hugo Tsuiki, Hideki Tsukada, Takeshi Turrini, Andrea Tzevelekos, Nikos Valencia, Frank Valiron, Beno î t van Ditmarsch, Hans Varacca, Daniele Vial, Pierre Vicary, Jamie Vijayaraghavan, Muralidaran Villevalois, Didier Waga, Masaki Wagner, Christoph Wojtczak, Dominik Wolff, Sebastian Worrell, James Yamada, Akihisa Yang, Pengfei Yoshimizu, Akira Yu, Tingting Zimmermann, Martin Organization XI Contents Semantics Non-angelic Concurrent Game Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simon Castellan, Pierre Clairambault, Jonathan Hayman, and Glynn Winskel A Trace Semantics for System F Parametric Polymorphism . . . . . . . . . . . . . 20 Guilhem Jaber and Nikos Tzevelekos Categorical Combinatorics for Non Deterministic Strategies on Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Cl é ment Jacq and Paul-Andr é Melli è s A Syntactic View of Computational Adequacy . . . . . . . . . . . . . . . . . . . . . . 71 Marco Devesas Campos and Paul Blain Levy Linearity A New Linear Logic for Deadlock-Free Session-Typed Processes . . . . . . . . . 91 Ornela Dardha and Simon J. Gay A Double Category Theoretic Analysis of Graded Linear Exponential Comonads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Shin-ya Katsumata Depending on Session-Typed Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Bernardo Toninho and Nobuko Yoshida Interoperability for ML and a Linear Language . . . . . . . . . . . . . . 146 Gabriel Scherer, Max New, Nick Rioux, and Amal Ahmed Concurrency Automata for True Concurrency Properties . . . . . . . . . . . . . . . . . . . . . . . . . 165 Paolo Baldan and Tommaso Padoan A Theory of Encodings and Expressiveness (Extended Abstract). . . . . . . . . . 183 Rob van Glabbeek A Framework for Parameterized Monitorability. . . . . . . . . . . . . . . . . . . . . . 203 Luca Aceto, Antonis Achilleos, Adrian Francalanza, and Anna Ing ó lfsd ó ttir Logics for Bisimulation and Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Xinxin Liu, Tingting Yu, and Wenhui Zhang Lambda-Calculi and Types Call-by-Need, Neededness and All That . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Delia Kesner, Alejandro R í os, and Andr é s Viso Fitch-Style Modal Lambda Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Ranald Clouston Realizability Interpretation and Normalization of Typed Call-by-Need k -calculus with Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 É tienne Miquey and Hugo Herbelin Quotient Inductive-Inductive Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, Nicolai Kraus, and Fredrik Nordvall Forsberg Category Theory and Quantum Control Guarded Traced Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Sergey Goncharov and Lutz Schr ö der Proper Semirings and Proper Convex Functors . . . . . . . . . . . . . . . . . . . . . . 331 Ana Sokolova and Harald Woracek From Symmetric Pattern-Matching to Quantum Control . . . . . . . . . . . . . . . . 348 Amr Sabry, Beno î t Valiron, and Juliana Kaizer Vizzotto Quantitative Models The Complexity of Graph-Based Reductions for Reachability in Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 St é phane Le Roux and Guillermo A. P é rez A Hierarchy of Scheduler Classes for Stochastic Automata. . . . . . . . . . . . . . 384 Pedro R. D ’ Argenio, Marcus Gerhold, Arnd Hartmanns, and Sean Sedwards Symbolically Quantifying Response Time in Stochastic Models Using Moments and Semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Hugo Bazille, Eric Fabre, and Blaise Genest Comparator Automata in Quantitative Verification. . . . . . . . . . . . . . . . . . . . 420 Suguman Bansal, Swarat Chaudhuri, and Moshe Y. Vardi XIV Contents Logics and Equational Theories Modular Tableaux Calculi for Separation Theories . . . . . . . . . . . . . . . . . . . 441 Simon Docherty and David Pym Differential Calculus with Imprecise Input and Its Logical Framework. . . . . . 459 Abbas Edalat and Mehrdad Maleki The Effects of Adding Reachability Predicates in Propositional Separation Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 St é phane Demri, É tienne Lozes, and Alessio Mansutti The Equational Theory of the Natural Join and Inner Union is Decidable. . . . 494 Luigi Santocanale Graphs and Automata Minimization of Graph Weighted Models over Circular Strings . . . . . . . . . . 513 Guillaume Rabusseau Games on Graphs with a Public Signal Monitoring . . . . . . . . . . . . . . . . . . . 530 Patricia Bouyer WQO Dichotomy for 3-Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 S ł awomir Lasota and Rados ł aw Pi ó rkowski Verifying Higher-Order Functions with Tree Automata . . . . . . . . . . . . . . . . 565 Thomas Genet, Timoth é e Haudebourg, and Thomas Jensen Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Contents XV Semantics Non-angelic Concurrent Game Semantics Simon Castellan 1( B ) , Pierre Clairambault 2 , Jonathan Hayman 3 , and Glynn Winskel 3 1 Imperial College London, London, UK simon@phis.me 2 Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France 3 Computer Laboratory, University of Cambridge, Cambridge, UK Abstract. The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten. We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation . Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism , and still get a category – this partial hiding remains sound up to weak bisim- ulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences. 1 Introduction Game semantics represents programs as strategies for two player games deter- mined by the types. Traditionally, a strategy is simply a collection of execution traces, each presented as a play (a structured sequence of events) on the corre- sponding game. Beyond giving a compositional framework for the formal seman- tics of programming languages, game semantics proved exceptionally versatile, providing very precise (often fully abstract) models of a variety of languages and programming features. One of its rightly celebrated achievements is the reali- sation that combinations of certain effects, such as various notions of state or control, could be characterised via corresponding conditions on strategies (inno- cence, well bracketing, . . . ) in a single unifying framework. This led Abramsky to propose the semantic cube programme [1], aiming to extend this success to fur- ther programming features: concurrency, non-determinism, probabilities, etc. . . However, this elegant picture soon showed some limitations. While indeed the basic category of games was successfully extended to deal with concurrency [10,13], non-determinism [11], and probabilities [9] among others, these exten- sions (although fully abstract) are often incompatible with each other, and really, incompatible as well with the central condition of innocence. Hence a semantic c © The Author(s) 2018 C. Baier and U. Dal Lago (Eds.): FOSSACS 2018, LNCS 10803, pp. 3–19, 2018. https://doi.org/10.1007/978-3-319-89366-2 _ 1 4 S. Castellan et al. hypercube encompassing all these effects remained out of reach. It is only recently that some new progress has been made with the discovery that some of these effects could be reconciled in a more refined, more intensional games framework. For instance, in [6,16] innocence is reconciled with non-determinism, and in [15] with probabilities. In [7], innocence is reconciled with concurrency. But something is still missing: the works above dealing with non-deterministic innocence consider only may-convergence ; they ignore execution branches lead- ing to divergence. To some extent this seems to be a fundamental limitation of the game semantics methodology: at the heart of the composition of strategies lies the hiding operation that removes unobservable events. Diverging paths, by nature non-observable, are forgotten by hiding. Some models of must-testing do exist for particular languages, notably McCusker and Harmer’s model for non-deterministic Idealized Algol [11]; the model works by annotating strategies with stopping traces , recording where the program may diverge. But this app- roach again mixes poorly with other constructions (notably innocence), and more importantly, is tied to may and must equivalences. It is not clear how it could be extended to support richer notions of convergence, such as fair-testing [2]. Our aim is to present a basis for non-deterministic game semantics which, besides being compatible with innocence, concurrency, etc ., is not biased towards may-testing; it is non-angelic . It should not be biased towards must-testing either; it should in fact be agnostic with respect to the testing equivalence, and support them all. Clearly, for this purpose it is paramount to remember the non-deterministic branching information; indeed in the absence of that infor- mation, notions such as fair-testing are lost. In fact, there has been a lot of activity in the past five years or so around games model that do observe the branching information. It is a feature of Hirschowitz’s work presenting strategies as presheaves or sheaves on certain categories of cospans [12]; of Tsukada and Ong’s work on nondeterministic innocence via sheaves [16]; and of our own line of work presenting strategies as certain event structures [5,7,14]. But observing branching information is not sufficient. Of the works mentioned above, those of Tsukada and Ong and our own previous work are still angelic, because they rely on hiding for composition. On the other hand, Hirschowitz’s work gets close to achieving our goals; by refraining from hiding altogether, his model constructs an agnostic and precise representation of the operational behaviour of programs, on which he then considers fair-testing. But by not con- sidering hiding he departs from the previous work and methods of game seman- tics, and from the methodology of denotational semantics. In contrast, we would like an agnostic games model that still has the categorical structure of traditional semantics. A games model with partial hiding was also recently introduced by Yamada [18], albeit for a different purpose: he uses partial hiding to represent normalization steps, whereas we use it to represent fine-grained nondeterminism. Contributions. In this paper, we present the first category of games and strate- gies equipped to handle non-determinism, but agnostic with respect to the notion of convergence (including fair convergence). We showcase our model by interpreting APCF + , an affine variant of non-deterministic PCF: it is the Non-angelic Concurrent Game Semantics 5 simplest language featuring the phenomena of interest. We show adequacy with respect to may, must and fair convergences. The reader will find in the first author’s PhD thesis [3] corresponding results for full non-deterministic PCF (with detailed proofs), and an interpretation of a higher-order language with shared memory concurrency. In [3], the model is proved compatible with our earlier notions of innocence, by establishing a result of full abstraction for may equivalence, for nondeterministic PCF. We have yet to prove full abstraction in the fair and must cases; finite definability does not suffice anymore. Outline. We begin Sect. 2 by introducing APCF + . To set the stage, we describe an angelic interpretation of APCF + in the category CG built in [14] with strategies up to isomorphism, and hint at our two new interpretations. In Sect. 3, starting from the observation that the cause of “angelism” is hiding, we omit it altogether, constructing an uncovered variant of our concurrent games, similar to that of Hirschowitz. Despite not hiding, when restricting the location of non- deterministic choices to internal events, we can still obtain a category up to weak bisimulation . But weak bisimulation is not perfect: it does not preserve must- testing, and is not easily computed. So in Sect. 4, we reinstate some hiding: we show that by hiding all synchronised events except some dubbed essential , we arrive at the best of both worlds. We get an agnostic category of games and strategies up to isomorphism , and we prove our adequacy results. 2 Three Interpretations of Affine Nondeterministic PCF 2.1 Syntax of APCF + The language APCF + extends affine PCF with a nondeterministic boolean choice, choice . Its types are A, B ::= B | A B , where A B represents affine functions from A to B . The following grammar describes terms of APCF + : M, N ::= x | M N | λx. M | tt | ff | if M N 1 N 2 | choice | ⊥ Typing rules are standard, we show application and conditionals. As usual, a conditional eliminating to arbitrary types can be defined as syntactic sugar. Γ M : A B Δ N : A Γ, Δ M N : B Γ M : B Δ N 1 : B Δ N 2 : B Γ, Δ if M N 1 N 2 : B The first rule is multiplicative : Γ and Δ are disjoint. The operational semantics is that of PCF extended with the (only) two nondeterministic rules choice → tt and choice → ff 2.2 Game Semantics and Event Structures Game semantics interprets an open program by a strategy, recording the behaviour of the program (Player) against the context (Opponent) in a 2- player game. Usually, the executions recorded are represented as plays , i.e. linear 6 S. Castellan et al. sequences of computational events called moves ; a strategy being then a set of such plays. For instance, the nondeterministic boolean would be represented as the (even-prefix closure of the) set of plays { q − · tt + , q − · ff + } on the game for booleans. In the play q − · tt + , the context starts the computation by asking the value of the program (q − ) and the program replies ( tt + ). Polarity indicates the origin (Program (+) or Opponent/Environment ( − )) of the event. Being based on sequences of moves, traditional game semantics handles con- currency via interleavings [10]. In contrast, in concurrent games [14], plays are generalised to partial orders which can express concurrency as a primitive. For instance, the execution of a parallel implementation of and against the context ( tt , tt ) gives the following partial order: B ⇒ B ⇒ B q ( − ) q q (+) tt tt ( − ) tt (+) In this picture, the usual chronological linear order is replaced by an explicit partial order representing causality . Moves are concurrent when they are incom- parable (as the two Player questions here). Following the longstanding conven- tion in game semantics, we show which component of the type a computational event corresponds to by displaying it under the corresponding occurrence of a ground type. For instance in this diagram, Opponent first triggers the com- putation by asking the output value, and then and concurrently evaluates his two arguments. The arguments having evaluated to tt , and can finally answer Opponent’s initial question and provide the output value. In [7], we have shown how deterministic pure functional parallel programs can be interpreted (in a fully abstract way) using such representations. Partial-Orders and Non-determinism. To represent nondeterminism in this par- tial order setting, one possibility is to use sets of partial orders [4]. This rep- resentation suffers however from two drawbacks: firstly it forgets the point of non-deterministic branching; secondly, one cannot talk of an occurrence of a move independently of an execution. Those issues are solved by moving to event structures [17], where the nondeterministic boolean can be represented as: B q ( − ) tt ff (+) The wiggly line ( ) indicates conflict : the boolean values cannot coexist in an execution. Together this forms an event structure , defined formally later. Non-angelic Concurrent Game Semantics 7 2.3 Interpretations of APCF + with Event Structures Let us introduce informally our interpretations by showing which event struc- tures they associate to certain terms of APCF + Angelic Covered Interpretation. Traditional game semantics interpretations of nondeterminism are angelic (with exceptions, see e.g. [11]); they only describe what terms may do, and forget where they might get stuck. The interpretation of M = ( λb. if b tt ⊥ ) choice for instance, in usual game semantics is the same as that of tt . This is due to the nature of composition which tends to forget paths that do not lead to a value. Consider the strategy for the function λb. if b tt ⊥ : B ⇒ B q ( − ) q (+) ff tt ( − ) tt (+) The interpretation of M arises as the composition of this strategy with the nondeterministic boolean. Composition is defined in two steps: interaction (Fig. 1a) and then hiding (Fig. 1b). Hiding removes intermediate behaviour which does not correspond to visible actions in the output type of the composition. Hiding is crucial in order for composition to satisfy basic categorical proper- ties (without it, the identity candidate, copycat, is not even idempotent). Strate- gies on event structures are usually considered up to isomorphism , which is the strongest equivalence relation that makes sense. Without hiding, there is no hope to recover categorical laws up to isomorphism. However, it turns out that, treating events in the middle as τ -transitions ( ∗ in Fig. 1a), weak bisimulation equates enough strategies to get a category. Following these ideas, a category of uncovered strategies up to weak bisimilarity is built in Sect. 3. Fig. 1. Three interpretations of ( λb. if b tt ⊥ ) choice