Lecture Notes in Computer Science 7860Commenced Publication in 1973Founding and Former Series Editors:Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board

David HutchisonLancaster University, UK

Takeo KanadeCarnegie Mellon University, Pittsburgh, PA, USA

Josef KittlerUniversity of Surrey, Guildford, UK

Jon M. KleinbergCornell University, Ithaca, NY, USA

Alfred KobsaUniversity of California, Irvine, CA, USA

Friedemann MatternETH Zurich, Switzerland

John C. MitchellStanford University, CA, USA

Moni NaorWeizmann Institute of Science, Rehovot, Israel

Oscar NierstraszUniversity of Bern, Switzerland

C. Pandu RanganIndian Institute of Technology, Madras, India

Bernhard SteffenTU Dortmund University, Germany

Madhu SudanMicrosoft Research, Cambridge, MA, USA

Demetri TerzopoulosUniversity of California, Los Angeles, CA, USA

Doug TygarUniversity of California, Berkeley, CA, USA

Gerhard WeikumMax Planck Institute for Informatics, Saarbruecken, Germany

Bob Coecke Luke OngPrakash Panangaden (Eds.)

Computation, Logic,Games, andQuantum Foundations

The Many Facets of Samson Abramsky

Essays Dedicated to Samson Abramskyon the Occasion of His 60th Birthday

13

Volume Editors

Bob CoeckeLuke OngUniversity of Oxford, Department of Computer ScienceParks Road, Wolfson Building, Oxford OX1 3QD, UKE-mail: {bob.coecke, luke.ong}@cs.ox.ac.uk

Prakash PanangadenMcGill University, Department of Computer Science3480 Rue University, Montréal, QC, H3A 0E9, CanadaE-mail: prakash@cs.mcgill.ca

ISSN 0302-9743 e-ISSN 1611-3349ISBN 978-3-642-38163-8 e-ISBN 978-3-642-38164-5DOI 10.1007/978-3-642-38164-5Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2013936943

CR Subject Classification (1998): F.3, F.1, F.4, D.2.4, D.2-3, I.2.2-4, D.1, E.1

LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues

© Springer-Verlag Berlin Heidelberg 2013

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,in its current version, and permission for use must always be obtained from Springer. Violations are liableto prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.

Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Samson Abramsky

Preface

When one encounters Samson Abramsky it is hard to believe that he is 60, butwhen one attempts to survey his work it is equally hard to believe that he is only60! The three editors of this volume have known Samson for a long time, thelast two since the mid-1980s and the first since the early years of this century.In that period Samson’s interests have undergone a number of striking changes.The papers in this volume reflect this variety and give some indication of thedepth of his contributions in these areas.

His first publication in 1982 was on programming distributed systems; a sur-prise perhaps for all those who know him as an outstanding theoretician buta good indication of why his theoretical work has been grounded in computa-tional practice. The third editor was struggling to understand the semantics ofnondeterministic dataflow networks in 1983 when he encountered a paper called“Semantic Foundations of Applicative Multiprogramming.” His reaction at thetime was, “this guy thinks he has solved all the hard problems in this area,” buta closer look led him to exclaim, “he has solved all the hard problems!” Thispaper was an inspiration to him and led to his subsequent work on concurrency.They met shortly thereafter and have enjoyed a fruitful scientific relationship aswell as a warm friendship ever since.

The second editor was Samson Abramsky’s first PhD student and wrote athesis on the semantics of the lazy λ-calculus. This led to an interest in the fullabstraction problem, which they solved in spectacular fashion in two independentbut closely related approaches based on games. Subsequently Samson and hisstudents studied many different language features and showed how the gamesparadigm allows a unified view of many programming language features.

The first editor, while still in the process of making some suicidal careermoves within the then hibernating field of quantum foundations, came acrossSamson’s work with Vickers on quantales as algebras of processes in the late1990s. He was saved by Samson from the academic gutter in 2001 (on the adviceof the third editor!) and began his collaboration with Samson on categoricalquantum mechanics. This began with a famous paper in the Proceedings of IEEESymposium on Logic in Computer Science 2004, the first on quantum computingever accepted to LiCS, and this activity has grown into a large group at Oxfordwith close to 40 members today. It is fair to say that the categorical approachand the still growing group have been instrumental both in establishing quantumcomputing as a thriving field within the computer science community, as wellas in the revival of the field of quantum foundations as a whole, an area inwhich Samson has been primarily involved in the last few years. It has even hadsuccesses in database analysis, computational linguistics, and relativistic aspectsof quantum mechanics.

VIII Preface

Samson’s work in the 1980s was largely in the area of programming languagesemantics. He made key contributions to concurrency theory, domain theory,and abstract interpretation. Perhaps his most famous paper from this era was“Domain Theory in Logical Form,” which connected many different threads insemantics: modal logic, domain theory, Stone-type duality.

In the early 1990s he began to think deeply about linear logic, which had beenintroduced by Girard in 1987. He produced his influential unpublished work onlinear realizability algebra, which allowed one to think of linear logic as a program-ming language. In 1991 Radha Jagadeesan completed his PhD under the supervi-sion of the third editor and went on to be Samson’s post-doc at Imperial Collegeand joined in the linear logic enterprise. They produced another classic influen-tial paper, “Games and Full Completeness for Multiplicative Linear Logic,” whichbuilt on an earlier paper “New Foundations for the Geometry of Interaction” in theProceedings of IEEE Symposium on Logic in Computer Science 1992.

The game approach led to the resolution of the long-standing PCF full ab-straction problem by Abramsky, Jagadeesan, and Malacaria, and by Hylandand Ong, and by Nickau. These papers laid the foundations of game semanticsand have led to several PhD theses and a revolution in the understanding ofprogramming language semantics.

In 2004, Samson was elected a Fellow of the Royal Society. In 2007 he receivedthe LiCS “Test ofTime” award for his paper on “Domain Theory in Logical Form.”

The papers in this volume represent his manifold contributions to semantics,logic, games, and quantum mechanics. The papers of Hoare, Plotkin, Mislove,Jung, Honsell and Lenisa, Martin, and Vickers represent the programming lan-guages and domain theory phase of Samson’s interests. His interest in categorytheory is represented by the papers of Fiore and Devesas Campos, Melies, andPavlovic. His interest in games both for logic and for computation is well rep-resented by the papers of Hankin and Malacaria, van Benthem, Clairambault,Gutierrez and Winskel, Murawski and Tzevelekos, Vaananen, and Ghica whoeven applies these ideas to hardware design.

The remaining papers are all from the “quantum phase” of Samson’s interestsand include papers from physicists (Hardy), former physicists (Panangaden),economists (Brandenburger), category theorists (Malherbe, Scott and Selinger,and Yanofsky), computer scientists (Nagarajan and Gay) and logicians (Keisler),as well as current and former members of the Oxford group (Coecke, Kissingerand Heunen, and Hines). We hope that Samson enjoys reading these papers asmuch as we enjoyed putting this collection together.

March 2013 Bob CoeckeLuke Ong

Prakash Panangaden

Organization

Program Committee

Bob Coecke University of Oxford, UKRoss Duncan Universite Libre Bruxelles, BelgiumDan Ghica University of Birmingham, UKChris Heunen University of Oxford, UKPeter Hines University of York, UKRadha Jagadeesan DePaul University, USAAchim Jung University of Birmingham, UKAleks Kissinger University of Oxford, UKLuke Ong University of Oxford, UKPrakash Panangaden McGill University, CanadaNoson Yanofsky Brooklyn College, USA

Samson Abramsky’s Doctoral Students

Former Students and Their Theses

– Luke Ong: The Lazy Lambda Calculus: an Investigation into the Foundationsof Functional Programming, Imperial College London, 1988.

– David Fuller: Partial Evaluation and Logic Programming, Imperial CollegeLondon, 1989.

– Bent Thomsen: Calculi for Higher Order Communicating Systems, ImperialCollege London, 1991.

– Ian Mackie: The Geometry of Implementation: Applications of the Geometryof Interaction to Language Implementation, Imperial College London, 1994.

– Simon Gay: Linear Types for Communicating Processes, Imperial CollegeLondon, 1995.

– Guy McCusker: Games and Full Abstraction for a Functional Metalanguagewith Recursive Types, Imperial College London, 1996.

– Rajagopal Nagarajan: Typed Concurrent Programs: Specification & Verifi-cation, Imperial College London, 1998.

– Jim Laird: A Semantic Analysis of Control, University of Edinburgh, 1999.– Jose Espırito Santo: Conservative Extensions of the Lambda-Calculus for the

Computational Interpretation of Sequent Calculus, University of Edinburgh,2002.

– Juliusz Chroboczek: Game Semantics and Subtyping, University of Edin-burgh, 2003.

– Jan Jurjens: Principles for Secure Systems Design, University of Edinburgh,2004.

– Ross Duncan: Types for Quantum Computing, University of Oxford, 2007.– Nikos Tzevelekos: Nominal Game Semantics, University of Oxford, 2008.– William Edwards: Non-locality in Categorical Quantum Mechanics, Univer-

sity of Oxford, 2009.– Jacob Biamonte: Categorical Models of Quantum Information in Many-Body

Systems, University of Oxford, 2010.– Alexander Kissinger: Pictures of Processes: Automated Graph Rewriting for

Monoidal Categories and Applications to Quantum Computing, Universityof Oxford, 2011.

– Andrei Akhvlediani: Relating Types in Categorical Universal Algebra, Uni-versity of Oxford, 2012.

XII Samson Abramsky’s Doctoral Students

Current Students at the University of Oxford

– Philip Atzemoglou: Higher-Order Semantics for Quantum Programming Lan-guages with Classical Control.

– Miriam Backens: Classical vs. Quantum in Graphical Models.– Carmen Constantin: Aspects of the Topos Approach to Quantum Theory.– Abhishek Dasgupta: Frameworks for Parallelising Machine Learning Using

Generic Inference.– Nadish de Silva: Geometric Aspects of Quantum Phase Space.– Raymond Lal: Causal Structure in Categorical Quantum Mechanics.– Shane Mansfield: Approaches to Non-locality and Contextuality in Possibilis-

tic Theories, and Extensions to Probabilistic Theories.– Daniel Marsden: Parameterized Logics and Applications to Quantum Sys-

tems.– Yoshihiro Maruyama: Duality, Categorical Logic, and Quantum Symmetry .– Alexander Merry: Foundations for an Interactive Theorem Prover for Graph-

ical Calculi.– Hugo Nava Kopp: Abstract Approach to Entropy.– Johan Paulsson: Between Probabilities and Categories; A Diagrammatic Ap-

proach to Foundations of Quantum Theory.– Roman Priebe: The Regular Histories Formulation of Quantum Theory.– Rui Soares Barbosa: Contextuality in Quantum Mechanics.– Colin Stephen: Categories for Tropical Quantum Computing.– Norihiro Yamada: Constructive Mathematics and Proofs as Programs.– Vladimir Zamdzhiev: Graph Grammars and Their Applications to Quantum

Computing.

Tabula Gratulatori

Tony Hoare: Best wishes for your birthday and for long continuation of yourinspiring research career: from your precursor at Oxford, Tony.

Alexandru BaltagJon BarrettJohan van BenthemRichard BluteRichard BornatAdam BrandenburgerStephen BrookesPeter BunemanBob CoeckeRobert ConstableRoss DuncanMartin EscardoMarcelo FioreSimon GayDan GhicaJeremy GibbonsGeorg GottlobChris HankinLucien HardyChris HeunenJane HillstonPeter HinesFurio Honsell

Ian HorrocksMartin HylandRadha JagadeesanAchim JungElham KashefiJuliette KennedyAleks KissingerDexter KozenDaniel KroeningMarta KwiatkowskaMarina LenisaPaul LevyIan MackiePasquale MalacariaKeye MartinPaul-Andre MelliesAlbert MeyerMichael MisloveJohn MitchellAndrzej MurawskiRajagopal

NagarajanPeter O’Hearn

Luke OngJoel OuakninePrakash PanangadenDusko PavlovicGordon PlotkinJohn ReynoldsBill RoscoeMehrnoosh SadrzadehDana ScottPhilip ScottRobert SeelyPeter SelingerAlex SimpsonSonja SmetsNikos TzevelekosJouko VannanenJamie VicaryGlynn WinskelBen WorrellNoson YanofskyMingsheng Ying

Samson and His Research Group

XVI Samson and His Research Group

Samson and His Friends

XVIII Samson and His Friends

Table of Contents

Use of a Canonical Hidden-Variable Space in Quantum Mechanics . . . . . . 1Adam Brandenburger and H. Jerome Keisler

Imperfect Information in Logic and Concurrent Games . . . . . . . . . . . . . . . 7Pierre Clairambault, Julian Gutierrez, and Glynn Winskel

Compositional Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Bob Coecke, Chris Heunen, and Aleks Kissinger

The Algebra of Directed Acyclic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Marcelo Fiore and Marco Devesas Campos

Diagrammatic Reasoning for Delay-Insensitive AsynchronousCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Dan R. Ghica

Payoffs, Intensionality and Abstraction in Games . . . . . . . . . . . . . . . . . . . . 69Chris Hankin and Pasquale Malacaria

On the Theory of Composition in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Lucien Hardy

On the Functor �2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Chris Heunen

Quantum Speedup and Categorical Distributivity . . . . . . . . . . . . . . . . . . . . 122Peter Hines

Unifying Semantics for Concurrent Programming . . . . . . . . . . . . . . . . . . . . 139Tony Hoare

Unfixing the Fixpoint: The Theories of the λY -Calculus . . . . . . . . . . . . . . 150Furio Honsell and Marina Lenisa

Continuous Domain Theory in Logical Form . . . . . . . . . . . . . . . . . . . . . . . . . 166Achim Jung

Presheaf Models of Quantum Computation: An Outline . . . . . . . . . . . . . . . 178Octavio Malherbe, Philip Scott, and Peter Selinger

Nothing Can Be Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Keye Martin

XX Table of Contents

Dialogue Categories and Frobenius Monoids . . . . . . . . . . . . . . . . . . . . . . . . . 197Paul-Andre Mellies

Anatomy of a Domain of Continuous Random Variables II . . . . . . . . . . . . 225Michael Mislove

Towards Nominal Abramsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Andrzej S. Murawski and Nikos Tzevelekos

Techniques for Formal Modelling and Analysis of Quantum Systems . . . 264Simon J. Gay and Rajagopal Nagarajan

Quantum Field Theory for Legspinners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Prakash Panangaden

Bicompletions of Distance Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Dusko Pavlovic

Partial Recursive Functions and Finality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Gordon Plotkin

Breaking the Atom with Samson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Jouko Vaananen

Reasoning about Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336Johan van Benthem

Domain Theory in Topical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348Steve Vickers

Kolmogorov Complexity of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350Noson S. Yanofsky

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

On the Functor �2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chris Heunen

E1

Erratum