DOCSLIB.ORG

Proquest Dissertations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Proquest Dissertations u Ottawa L'Universite canadienne Canada's university FACULTE DES ETUDES SUPERIEURES nm FACULTY OF GRADUATE AND ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES L'Universitt eanodienne Canada's university Benoit Valiron AUTEUR DE LA THESE / AUTHOR OF THESIS Ph.D. (Mathematics) GRADE/DEGREE Department of Mathematics and Statistics FACULTE, ECOLE, DEPARTEMENT / FACULTY, SCHOOL, DEPARTMENT Semantics for a Higher-Order Functional Programming Language for Quantum Computation TITRE DE LA THESE / TITLE OF THESIS Peter Selinger DIRECTEUR (DIRECTRICE) DE LA THESE / THESIS SUPERVISOR CO-DIRECTEUR (CO-DIRECTRICE) DE LA THESE / THESIS CO-SUPERVISOR EXAMINATEURS (EXAMINATRICES) DE LA THESE/THESIS EXAMINERS Richard Blute Douglas Howe Pieter Hofstra Prakash Panangaden Gary W. Slater Le Doyen de la Faculte des etudes superieures et postdoctorales / Dean of the Faculty of Graduate and Postdoctoral Studies Semantics for a Higher-Order Functional Programming Language for Quantum Computation. Benoit Valiron Thesis Submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics1 Department of Mathematics and Statistics Faculty of Science University of Ottawa © Benoit Valiron, Ottawa, Canada, 2008 1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-46525-7 Our file Notre reference ISBN: 978-0-494-46525-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada Pour mon pere. Abstract The objective of this thesis is to develop a semantics for higher-order quantum information. Following the work done in the author's M.Sc. thesis, we study a lambda calculus for quantum computation with classical control. The language features two important properties. The first one, arising from the so-called no-cloning theorem of quantum computation, is the need for a distinction between duplicable and non-duplicable elements. For keeping track of duplicability at higher order, we use a type system inspired by the resource-sensitive linear logic. The second important aspect is the probability inherent to measurement, the only operation for retrieving classical data from quan­ tum data. This forces us into choosing a reduction strategy for being able to define an operational semantics. We address the question of a denotational semantics in two respects. First, we restrict the study to the strictly linear aspect of the language. Doing so, we suppress the need for distinguishing between duplicable and non-duplicable elements and we can focus on the description of quantum features at higher order. Using the category of completely positive maps as a framework, we build a fully abstract denotational model of the strictly linear fragment of the language. The study of the full language is more demanding. For dealing with the probabilistic aspect of the language, we use a method inspired by Moggi and build a computational model with a distinction between values and computations. For the distinction between duplicability and non-duplicability in the calculus, we adapt Bierman's linear category, where the duplication is considered as a comonad with specific properties. The resulting model is what we call a linear category for duplication. Finally, we only focus on the fragment of the language that contains the aforementioned elements, and remove the classical Boolean and quantum Boolean features to get a generic computational linear lambda-calculus. In this idealized setting, we show that such a language have a full and complete interpretation in a linear category for duplication. m Acknowledgements Numerous people allowed me to write this thesis. In particular, I would like to thank: • Peter, who supported me for yet another degree; • Caroline, who came all the way to Halifax; • my parents, for being supportive all that time; • Rob, for feeding me with coffee; • the administrative staff of both Dalhousie and Ottawa University, for being so helpful; • and finally Gilles, for printing more that 1000 pages of this opus of content. IV Contents Abstract iii Acknowledgements iv List of Figures ix List of Tables x 1 Introduction 1 2 Notions of Category Theory 6 2.1 Categories and Functors 6 2.2 Natural Transformations 7 2.3 Adjoint Functors 8 2.4 Products and Coproducts 9 2.5 Monads 10 2.6 Comonads 11 2.7 Symmetric Monoidal Categories and Comonoids 13 2.7.1 Monoidal Categories 13 2.7.2 Commutative Comonoids 13 2.8 Monoidal Comonads and Coalgebras 14 3 Quantum Computation 17 3.1 Mathematical Formalism 17 3.1.1 Generalities on Finite Dimensional Hilbert Spaces 17 3.1.2 Tensor Products of Hilbert Spaces 19 3.1.3 Completely Positive Maps 19 3.1.4 Superoperators 20 3.1.5 The 2-Dimensional Hilbert Space 21 3.2 Quantum Foundations 23 3.2.1 Quantum Bits 23 3.2.2 Operations 23 3.2.3 Mixed States 25 3.3 Quantum Effects 26 3.3.1 No Cloning 26 3.3.2 Entanglement 26 3.3.3 Bell's Inequalities 27 3.4 Some Algorithms and Uses of Quantum Effects 28 3.4.1 Teleportation 28 v CONTENTS vi 3.4.2 Dense Coding 30 3.4.3 The Deutsch Algorithm 31 4 A Tour in Existing Models of Quantum Computation 32 4.1 The Various Paradigms 32 4.1.1 Unitary Gates as Computation 32 4.1.2 Concurrent Quantum Computation 34 4.1.3 Measurement-Based Quantum Computation 34 4.1.4 The QRAM Model 35 4.2 Formalism of Hilbert Spaces 35 4.2.1 Dagger Compact-Closed Categories 36 4.2.2 Classical Objects 37 4.3 A Flow-Chart Language 37 4.3.1 The language 37 4.3.2 The Category of Superoperators 37 4.3.3 Interpretation of the Flow-Chart Language 39 4.4 Extension to Higher Order 40 5 Lambda Calculus and Semantics of Higher-Order Computation 41 5.1 Lambda Calculus 41 5.1.1 The Language 41 5.1.2 Free and Bound Variables 42 5.1.3 Alpha-Equivalence 42 5.1.4 Operational Meaning of Lambda Calculus 42 5.1.5 Typed Lambda Calculus 43 5.2 Proofs as Computations 45 5.2.1 Intuitionistic Logic 46 5.2.2 Curry-Howard Isomorphism 46 5.3 Categorical Logic 47 5.4 Lambda Calculus and Side Effects 48 5.4.1 Pure Versus Impure Calculus 48 5.4.2 Reduction Strategies 49 5.4.3 Towards a Semantics 49 5.4.4 Computational Model for Call-By-Value 50 5.5 Intuitionistic Linear Logic 51 5.6 Linear Calculi and their Interpretations 51 5.6.1 Earlier Models 52 5.6.2 Bierman's Linear Category 52 6 A Lambda Calculus for Quantum Computation 54 6.1 The Language 54 6.2 Operational Semantics 56 6.2.1 Abstract Machine 56 6.2.2 Evaluation Strategy 56 6.2.3 Probabilistic Reduction Systems 58 6.2.4 Reduction System 59 6.3 Type System 60 6.3.1 Types 60 6.3.2 Typing Rules 61 6.4 Properties of the Type System 62 CONTENTS vii 6.4.1 Safety Properties 62 6.4.2 Type Inference Algorithm 63 6.5 Examples of Algorithms 64 6.5.1 The Deutsch Algorithm 64 6.5.2 The Teleportation Procedure 64 6.5.3 Type Derivation of the Teleportation Protocol 65 6.5.4 Reduction of the Teleportation Term 67 6.5.5 Reduction of the Superdense Coding Term 68 6.6 Towards a Denotational Semantics 69 7 The Linear Fragment 70 7.1 A Linear Lambda Calculus for Quantum Computation 70 7.2 Operational Semantics 75 7.2.1 Small Step Semantics 75 7.2.2 Safety Properties 77 7.2.3 Normalization 79 7.2.4 Quantum Context and Reduction 81 7.2.5 Reduction to Values 82 7.3 Denotational Semantics 83 7.3.1 Modeling the Linear Quantum Lambda Calculus 83 7.3.2 Fullness of the First-Order Fragment 89 7.3.3 Fullness up to Scalar Multiple 91 7.4 Equivalence Classes of Terms 94 7.4.1 Axiomatic Equivalence 94 7.4.2 Operational Context 95 7.4.3 Operational Equivalence 95 7.4.4 Denotational Equivalence 96 7.5 Soundness and Full Abstraction 96 7.5.1 Proof of the Soundness Theorem 97 7.5.2 Full Abstraction: Preliminary Lemmas 97 7.5.3 Proof of the Full Abstraction Theorem 98
Load more

Recommended publications
  • Brief Curriculum Vitae
    Brief Curriculum Vitae Name Vasilakopoulou Christina E-mail christie DOT vasi AT gmail EMPLOYMENT Jun 2019 – now University of Patras, Greece Researcher, Department of Mathematics PI of research grant “Duality and enrichment in operadic theory; interactions with Hopf structures and abstract machines modeling”, H.F.R.I. (ELIDEK) Oct 2017 – June 2019 University of Calfornia at Riverside (USA) Visiting Assistant Professor, Department of Mathematics Oct 2016 – Sep 2017 Université Libre de Bruxelles (Belgium) Postdoctoral Fellow, Department of Mathematics Oct 2015 – Jun 2016 Massachusetts Institute of Technology (USA) Postdoctoral Associate, Department of Mathematics May – Sep 2015 Athens University of Economics and Business (Greece) Postdoctoral Researcher, Department of Informatics Sep 2014 – Feb 2015 University of Hawaii at Manoa (USA) Postdoctoral Research Fellow, Department of Information and Computer Sciences EDUCATION 2010 – 2014 University of Cambridge (UK) – PhD in Category Theory Supervisors: Martin Hyland, Ignacio Lopez Franco Thesis title: “Generalization of algebraic operations via enrichment” 2009 – 2010 University of Cambridge (UK) – MASt in Mathematics 2005 – 2009 University of Patras (Greece) – Ptychion in Mathematics (B.Sc.) Direction: Pure Mathematics, Final mark: 9,26/10 “Excellent” PAPERS To appear Mitchell Buckley, Timmy Fieremans, Christina Vasilakopoulou and Joost Vercruysse, Oplax Hopf Algebras, Higher Structures 2021 Mitchell Buckley, Timmy Fieremans, Christina Vasilakopoulou and Joost Vercruysse, A Larson- Sweedler Theorem for Hopf V-categories, Advances in Mathematics 376 2021 Georgios Bakirtzis, Cody H. Fleming and Christina Vasilakopoulou, Categorical Semantics of Cyber-Physical Systems Theory, ACM Transactions on Cyber-Physical Systems (3) 5, 1–32 2020 Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck Construction, Theory and Applications of Categories 35, no.
    [Show full text]
  • Some Reasons for Generalizing Domain Theory
    Under consideration for publication in Math. Struct. in Comp. Science Some Reasons for Generalizing Domain Theory Martin Hyland Received 9 October 2009 One natural way to generalize Domain Theory is to replace partially ordered sets by categories. This kind of generalization has recently found application in the study of concurrency. An outline is given of the elegant mathematical foundations which have been developed. This is specialized to give a construction of cartesian closed categories of domains, which throws light on standard presentations of Domain Theory. Introduction This paper is one of what I hope will be a series written in conscious homage to Kreisel’s paper (Kreisel 1971). That paper was prepared for the European Meeting of the ASL in Manchester 1969, was widely aired at the time and was finally published in the pro- ceedings in 1971. It is a survey of the state of Recursion Theory and its generalizations around 1970. At one level it is a remarkable attempt to influence the development of logic, providing concrete proposals and problems as well as more general reflections on future directions. At times, for example in connection with the theory of admissible sets, the discussion foreshadows developments whose significance was only later fully appreciated. At other points, for example in connection with axiomatics, an approach is promoted which has been explored extensively but has in my view been less successful. (It is of course a positive feature that suggestions can even now be regarded as contentious.) However one can read the paper on another level as providing a case study of the value of generalization in a specific area of mathematics.
    [Show full text]
  • From Brouwer's Thesis to the Fan Functional
    From Brouwer's Thesis to the Fan Functional Ulrich Berger Swansea University University of Birmingham Theoretical Computer Science Seminar February 21, 2020 1 / 36 Overview 1. Introduction 2. Brouwer's thesis 3. Abstract bar induction 4. Vacuous truth 5. Proving uniform continuity 6. Extracting the fan functional 2 / 36 Introduction What are the logical roots of intriguing algorithms/computing principles? I Primitive recursion comes from induction on N. I General recursion comes from wellfounded induction. I The extended Euklidean algorithm comes from a classical proof that Z is a principal ideal ring. I Normalization by evaluation (for the typed lambda-calculus) comes from the Tait/Girard proof of strong normalization, respectively a completeness proof for intuitionistic logic. I ... Where does Tait's fan functional come from? 3 / 36 The fan functional The fan functional computes for every continuous function on Cantor space with values in N its least modulus of uniform continuity: FAN :(f0; 1gN ! N) ! N FAN(F ) = µn 8α; β (α =n β ! F α = F β) Def where α =n β = 8k 2 N (k < n ! α k = β k). So, clearly, this must come from: Fan theorem: Every continuous function on Cantor space with values in N is uniformly continuous. The real question is what are the right logical and mathematical principles and what is the right formal system for a proof of this theorem in order to extract the fan functional, more precisely, a purely functional program that computes it? 4 / 36 Brief history of the fan functional Tait introduced the Fan functional in 1963 and showed that it is recursively continuous but not computable by Kleene's schemata S1-S9, thus shattering Kleene's hope that S1-S9 is a universal notion of computation in higher types.
    [Show full text]
  • Wellfounded Trees and Dependent Polynomial Functors
    Wellfounded Trees and Dependent Polynomial Functors Nicola Gambino? and Martin Hyland Department of Pure Mathematics and Mathematical Statistics University of Cambridge {N.Gambino,M.Hyland}@dpmms.cam.ac.uk Abstract. We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by in- vestigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed cat- egories. 1 Introduction Types of wellfounded trees, or W-types, are one of the most important com- ponents of Martin-L¨of’s dependent type theories. First, they allow us to define a wide class of inductive types [5, 15]. Secondly, they play an essential role in the interpretation of constructive set theories in dependent type theories [3]. Finally, from the proof-theoretic point of view, they represent the paradigmatic example of a generalised inductive definition and contribute considerably to the proof-theoretic strength of dependent type theories [8]. In [16] a categorical counterpart of the notion of W-type was introduced. In a locally cartesian closed category, W-types are defined as the initial algebras for endofunctors of a special kind, to which we shall refer here as polynomial functors. The purpose of this paper is to study polynomial endofunctors and W-types more closely. In particular, we set out to explore some of the consequences of the assumption that a locally cartesian closed category has W-types, i.e. that every polynomial endofunctor has an initial algebra.
    [Show full text]
  • The Continuum Hypothesis in Algebraic Set Theory
    The Continuum Hypothesis in Algebraic Set Theory T. P. Kusalik Department of Mathematics and Statistics McGill University Montreal,Quebec December 31, 2008 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Science ©T. Kusalik. 2008 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-53761-9 Our file Notre reference ISBN: 978-0-494-53761-9 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation.
    [Show full text]
  • Curriculum Vitae
    10221 Phar Lap Dr Cupertino 95014 H 408 219 6912 T 408 725 0460 B [email protected] Í http://vcvpaiva.github.io/ Valeria de Paiva http://www.cs.bham.ac.uk/~vdp Profile: Mathematician with strong industrial and academic background, and over 100 publications. Project leader with a rare ability to get people from different technical backgrounds communicating and working together. Education 1990 PhD, Pure Mathematics, University of Cambridge, Cambridge, UK. Thesis: The Dialectica Categories, supervisor: Prof M. J. E. Hyland 1985 Part III Mathematical Tripos (MSc), University of Cambridge, Cambridge, UK. Lucy Cavendish College 1984 MSc, Algebra, Pontifical Catholic University (PUC), Rio de Janeiro, Brazil. Thesis: The Hurwitz-Radon Transformations, supervisor: Dr D. Randall 1982 BSc, Pure Mathematics, Pontifical Catholic University (PUC), Rio de Janeiro, Brazil. Work Experience Mar Principal Research Scientist, Samsung Research America, Mountain View, CA. 2019–Jan Research, design and deployment of conversational systems, related to smart devices in the 2020 connected home. I also serve a a connection to the academic community. Sep Senior Research Scientist, Nuance Communications, Sunnyvale, CA. 2012–Dec Research, design and deployment of conversational systems. My first project was on a "living 2018 room" assistant such as Amazon’s Alexa. The most recent project was on a car concierge. In the NAIL lab I am the bridge between the Natural Language processing (NLP) experts and the knowledge representation (KR) experts. April Senior Research Scientist, Rearden Commerce, Foster City, CA. 2011–2012 Design and deployment of sentiment analysis tools. { Sentiment analysis mechanisms, especially geared towards to reviews of hotels, using logical knowledge representation and ontologies, as well as statistical natural language processing modules.
    [Show full text]
  • The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
    The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads Martin Hyland2 Dept of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge, ENGLAND John Power1 ,3 Laboratory for the Foundations of Computer Science University of Edinburgh Edinburgh, SCOTLAND Abstract Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future. Keywords: Universal algebra, Lawvere theory, monad, computational effect. 1 Introduction There have been two main category theoretic formulations of universal algebra. The earlier was by Bill Lawvere in his doctoral thesis in 1963 [23]. Nowadays, his central construct is usually called a Lawvere theory, more prosaically a single-sorted finite product theory [2,3]. It is a more flexible version of the universal algebraist’s notion 1 This work is supported by EPSRC grant GR/586372/01: A Theory of Effects for Programming Languages.
    [Show full text]
  • Valeria De Paiva
    10221 Phar Lap Dr Cupertino 95014 H 408 219 6912 T 408 725 0460 B [email protected] Í http://vcvpaiva.github.io/ Valeria de Paiva http://www.cs.bham.ac.uk/~vdp Profile: Mathematician with strong industrial and academic background, and over forty publications. Project leader with a rare ability to get people from different technical backgrounds communicating and working together. Education 1990 PhD, Pure Mathematics, University of Cambridge, Cambridge, UK. Thesis: The Dialectica Categories, supervisor: Prof M. J. E. Hyland 1985 Part III Mathematical Tripos (MSc), University of Cambridge, Cambridge, UK. Lucy Cavendish College 1984 MSc, Algebra, Pontifical Catholic University (PUC), Rio de Janeiro, Brazil. Thesis: The Hurwitz-Radon Transformations, supervisor: Dr D. Randall 1982 BSc, Pure Mathematics, Pontifical Catholic University (PUC), Rio de Janeiro, Brazil. Work Experience Sep Senior Research Scientist, Nuance Communications, Sunnyvale, CA. 2012–now Research, design and deployment of conversational systems. My first project here was on "living room" assistants such as Amazon’s Alexa. A new project is on a car concierge. April Senior Research Scientist, Rearden Commerce, Foster City, CA. 2011–2012 Design and deployment of sentiment analysis tools. { Sentiment analysis mechanisms, especially geared towards to reviews of hotels, using logical knowledge representation and ontologies, as well as statistical natural language processing modules. May Search Analyst, Cuil, Inc., Menlo Park, CA. 2008–2010 Design and deployment of mechanisms for testing quality of search results. { Effectively coordinated team of ranking, data mining and front-end engineers to achieve quality testing tasks. { Pre-screening of results when new data or algorithms are released.
    [Show full text]
  • Basic Category Theory
    Basic Category Theory TOMLEINSTER University of Edinburgh arXiv:1612.09375v1 [math.CT] 30 Dec 2016 First published as Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2014. ISBN 978-1-107-04424-1 (hardback). Information on this title: http://www.cambridge.org/9781107044241 c Tom Leinster 2014 This arXiv version is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 c Tom Leinster 2014, 2016 Preface to the arXiv version This book was first published by Cambridge University Press in 2014, and is now being published on the arXiv by mutual agreement. CUP has consistently supported the mathematical community by allowing authors to make free ver- sions of their books available online. Readers may, in turn, wish to support CUP by buying the printed version, available at http://www.cambridge.org/ 9781107044241. This electronic version is not only free; it is also freely editable. For in- stance, if you would like to teach a course using this book but some of the examples are unsuitable for your class, you can remove them or add your own. Similarly, if there is notation that you dislike, you can easily change it; or if you want to reformat the text for reading on a particular device, that is easy too. In legal terms, this text is released under the Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). The licence terms are available at the Creative Commons website, https:// creativecommons.org/licenses/by-nc-sa/4.0.
    [Show full text]
  • PREFACE 1 . Framework
    Theory and Applications of Categories comma Vol period .. 1 7 comma 2006 comma pp period .. 1 endash 9 period nnoindentPREFACETheory and Applications of Categories , Vol . nquad 17 , 2006 , pp . nquad 1 −− 9 . VALERIA DE PAIVA AND VAUGHAN PRATT n centerline fPREFACE g TheoryThis special and Applications volume of Theory of Categories and Applications , Vol . .. 1 7of , Categories 2006 , pp . .. open 1 { 9 parenthesis . TAC closing parenthesis .. addresses the theme .. quotedblleft Chu .. Spaces : .. Theory andPREFACE Applications period quotedblright .. The .. idea for the volume grew out .. of n centerlinethe .. workshopfVALERIA .. of that DE .. PAIVAVALERIA name .. AND organized DEVAUGHAN PAIVA by the PRATT .. AND volumeg VAUGHAN quoteright s PRATT.. editors .. in .. association .. with the conferenceThis special Logic involume Computer of Theory .... Science and .... Applications open parenthesis of LiCS Categories quoteright 2000 ( TAC closing ) parenthesis addresses the.... held theme at .... Santa Barbara in 2000 period ....This\ We Chu special Spaces volume : of Theory Theory and and Applications Applications . "nquad Theof idea Categories for the volumenquad grew( TAC out ) nquad of theaddresses the themeworkshopconceivednquad the s of` cope ` that Chu of thenquad name volumeSpaces as organized not : beingnquad l by imitedTheory the to the volume and material Applications ' s presented editors at the. in'' nquad associationThe nquad withidea for the volume grew out nquad o f thetheworkshopnquad itworkshop self but extendingnquad beyondo f that it ton embracequad name a widernquad rangeorganized of topics related by to the Chunquad volume ' s nquad e d i t o r s nquad in nquad association nquad with the conferencespaces and Dialectica Logic in constructions Computer period Science ( LiCS ' 2000 ) held at Santa Barbara in 2000 .
    [Show full text]
  • Wellfounded Trees and Dependent Polynomial Functors
    This is a repository copy of Wellfounded Trees and Dependent Polynomial Functors. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/113158/ Version: Accepted Version Proceedings Paper: Gambino, N orcid.org/0000-0002-4257-3590 and Hyland, M (2004) Wellfounded Trees and Dependent Polynomial Functors. In: Lecture Notes in Computer Science. TYPES 2003: International Workshop on Types for Proofs and Programs, 30 Apr - 04 May 2003, Torino, Italy. Springer Nature , pp. 210-225. ISBN 978-3-540-22164-7 https://doi.org/10.1007/978-3-540-24849-1_14 © Springer-Verlag Berlin Heidelberg 2004. This is an author produced version of a conference paper published in Lecture Notes in Computer Science. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Wellfounded Trees and Dependent Polynomial Functors Nicola Gambino⋆ and Martin Hyland Department of Pure Mathematics and Mathematical Statistics University of Cambridge {N.Gambino,M.Hyland}@dpmms.cam.ac.uk Abstract.
    [Show full text]
  • Théorie De L'argumentation
    UNIVERSITÉ D’ARTOIS ECOLE DOCTORALE SPI 072 SCIENCES POUR L’INGÉNIEUR THÈSE pour l’obtention du titre de Docteur en Sciences de l’Université d’Artois Mention : Informatique Présentée par Badran Raddaoui Contributions aux approches logiques de l’argumentation en intelligence artificielle soutenue publiquement le 21 novembre 2013 Composition du jury : Rapporteurs : Jérôme Lang (Directeur de Recherche) CNRS – LAMSADE Igor Stephan´ (Maître de Conférences, HdR) Université d’Angers Examinateurs : Philippe Besnard (Directeur de Recherche) CNRS – IRIT (Co-directeur) Éric Gregoire´ (Professeur des Universités) Université d’Artois (Co-directeur) Sébastien Konieczny (Directeur de Recherche) CNRS – CRIL Pierre Marquis (Professeur des Universités) Université d’Artois Yves Moinard (Chargé de Recherche) INRIA – IRISA Centre de Recherche en Informatique de Lens (CRIL CNRS UMR 8188) Université d’Artois, rue Jean Souvraz, S.P. 18 F-62307, Lens Cedex France Secrétariat : Tél. : +33 (0)3 21 79 17 23 – Fax : +33 (0)3 21 79 17 70 http://www.cril.univ-artois.fr A la mémoire de mon cher frère MAHER (1997-2013) Remerciements Un travail tel que celui-ci n’est pas seulement l’œuvre de son signataire. Je tiens à remer- cier tous ceux qui m’ont aidé et soutenu, sans qu’ils aient eu forcément pleinement conscience de l’importance que cela pouvait avoir pour moi. Je remercie – M. Pierre Marquis, Professeur des Universités à l’Université d’Artois, pour l’honneur qu’il m’a fait en acceptant de présider le jury de cette thèse. – M. Jérôme Lang, Directeur de Recherche CNRS au Laboratoire d’Analyse et Modéli- sation de Systèmes pour l’Aide à la DEcision (LAMSADE), Université Paris Dauphine et – M.
    [Show full text]