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Proquest Dissertations u Ottawa L'Universitd canadienne Canada's university FACULTE DES ETUDES SUPERIEURES l==I FACULTY OF GRADUATE AND ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES L'Universit6 canadienne Canada's university Guy Beaulieu "MEWDE"UATH£S17XUTHORWTHE"STS" Ph.D. (Mathematics) GRADE/DEGREE Department of Mathematics and Statistics 7ACulTirfc6L17bT^ Probabilistic Completion of Nondeterministic Models TITRE DE LA THESE / TITLE OF THESIS Philip Scott DIREWEURlbTRECTRicirDE'iSTHESE"/ THESIS SUPERVISOR EXAMINATEURS (EXAMINATRICES) DE LA THESE /THESIS EXAMINERS Richard Blute _ _ Paul-Eugene Parent Michael Mislove Benjamin Steinberg G ^y.W:.Slater.. Le Doyen de la Faculte des etudes superieures et postdoctorales / Dean of the Faculty of Graduate and Postdoctoral Studies PROBABILISTIC COMPLETION OF NONDETERMINISTIC MODELS By Guy Beaulieu, B.Sc, M.S. Thesis submitted to the Faculty of Graduate and Postdoctoral Studies University of Ottawa in partial fulfillment of the requirements for the PhD degree in the Ottawa-Carleton Institute for Graduate Studies and Research in Mathematics and Statistics ©2008 Guy Beaulieu, B.Sc, M.S. 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: 978-0-494-48387-9 Our file Notre reference ISBN: 978-0-494-48387-9 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 Abstract Motivated by Moggi's work [34] on how monads can be used to capture computational behavior, there has been a growing interest in finding monads which capture the precise computational effects generated when combining the theory of probabilistic choice and the theory of nondeterministic choice. The resulting theory, called here the theory of mixed choice [23, 31, 33], captures the interaction between the two choice operators by requiring appropriate distributivity axioms (probabilistic choice over nondeterministic choice), in addition to the operations and axioms from both individual choice theories. Classically, the required monad has been defined by composing the adjunctions on the left hand side of the following diagram. Mod(P/in, Set) ug' QL Mod(ND}in, Set) This requires the composition of the distributions functor Vfin, which constructs free models of probabilistic choice over an arbitrary set X, and the finite convex set functor Cvxfin, which constructs free models of mixed choice over an arbitrary convex set (C, +A ), as seen in [31, 33, 49]. This construction allows us to build a free mixed choice model from an arbitrary model for probabilistic (P) choice, in such fashion as to preserve the P choice model's probabilistic structure. u We consider the dual approach to the above algorithm. Through general categor­ ical results from Barr and Wells [2], we observed that free models for mixed choice could also be constructed over arbitrary models of nondeterministic (ND) choice. This allows for an alternative algorithm for constructing models for mixed choice, which corresponds to going up the right hand side of the above commuting diagram. This requires the composition of the finite nonempty powerset functor Vjin, which constructs free models of nondeterministic choice over an arbitrary set X, and the V-stable functor Stblfin, which constructs free models of mixed choice over an arbi­ trary semilattice (S, V). This alternative approach allows for a wider range of mixed choice models to be studied: those that would arise from arbitrary semilattices, hence arbitrary nondeterministic models with a possible non-standard definition of nonde- terminism. By abstract categorical notions, it is known that the V-stable functor Stblfin exists (the functor in the N-E corner of the above diagram). However, up until now there have been no concrete presentations of this functor, nor any notion of how to construct free mixed choice models over arbitrary semilattices. These concepts represent the main focus of this thesis. We motivate and develop a new property for convex sets generated over a set equipped with a semilattice structure. This notion is called V-stability. In essence, a V-stable convex set preserves the ND structure on its underlying set (i.e., if it contains an element s such that x\/ y = s, then it must also contain the convex subset generated by x and y). Finally, we generalize many of our results in two directions. First, we construct the corresponding V-stable functors for infinitary mixed choice theories over Set. We then extend the entire development of the thesis to express posetal model categories (i.e over Poset) of our theories. This would include constructing models for mixed choice over models obtained from the convex (Plotkin), lower (Hoare), and upper (Smyth) variants of nondeterminism. Furthermore, we prove that the results in the literature obtained by following the left-hand side of the given diagram are equivalent to our results obtained by traversing its right-hand side. in Acknowledgements It is a difficult thing, looking back over the past years, to pinpoint every single person to whom I owe a debt of gratitude. This thesis has evolved a great deal from its inception, and was influenced by many individuals. There have been many people who have helped me significantly academically and also supported me morally. First and foremost, I must attest that my thesis supervisor, Philip J. Scott, has supported me both academically, financially and morally throughout this whole pro­ cess. Without his generosity I would not have been able to travel to far-away seminars and meet with so many well-known researchers in my field. Without his knowledge and guidance many of the results we've found and cultivated would not be as grand nor as significant as they are today. Without his patience and kindness I might not have had the opportunity to reach my goals at my own pace. Phil has been my su­ pervisor for many years and I hope to have him as a friend and colleague for many more. I have had the chance to discuss my ideas and approaches with many fine re­ searchers and I would like to thank as many as I can for their assistance, their time and their suggestions. I'd like to thank Michael Mislove and James Worrell who took an interest in my work and invited me to visit them in Tulane in order to discuss the huge possibilities and impact my work could have. It's through their comments and suggestions that I first came to realize that what I was working on was actually signif­ icant and interesting to others. I would also like to take special care to thank Daniele Varacca who took me aside after my recovery from a nasty bug at MFPS 2007 and showed me how important motivating my definitions and constructions were. Thus my focus on dealing with the "why's" as much as the "how's". There have been many IV more people who have sat down with me and took interest in my work and whose suggestions and comments have been greatly appreciated: Samson Abramsky, Glynn Winskel, Marcelo Fiore and Joel Ouaknine during my visit of Oxford and Cambridge, John Power and Catuscia Palamidessi during MFPS 2005, Vincent Danos, Alex Simp­ son, Peter Selinger, Paul-Eugene Parent, Jeff Egger, Robin Cockett, and many others. I apologize to all the people who I might have missed. Finally, I have had a lot of support from family and friends pushing and motivat­ ing me to continue when everything seemed their bleakest; when a certain possible solution wouldn't pan out; and, especially when finding mistakes would waste weeks and weeks of work. Among them, I owe my wife Daniela the most. Without her constant support, encouragement and considerable shouldering of the parental tasks (taking care of our two children while their father was working late nights and week­ ends) my work would never have seen the light of day. I'd also like to recognize my parents, numerous as they may be, for believing in me and nurturing my love for mathematics. To my friends, I'd like to give thanks as well, not only for supporting their jobless compatriot with minimal mockery but also for lending their support to my family repeatedly and selflessly. v Dedication I dedicate this work to my wonderful children Emma and Matteo.
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