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39
Proofs as Polynomials
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Cited by 6 (1 self)
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
The Demonic Product of Probabilistic Relations
, 2001
"... The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the nondeterministic fringe of the probabilistic relations behaves properly: the ..."
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Cited by 5 (2 self)
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The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the nondeterministic fringe of the probabilistic relations behaves properly: the fringe of the product equals the demonic product of the fringes.
Pictures of complete positivity in arbitrary dimension
 In Quantum Programming Languages, Electronic Proceedings in Theoretical Computer Science
, 2011
"... Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this constructio ..."
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Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction. 1
Dagger categories and formal distributions
"... Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a cat ..."
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Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a category of tame formal distributions with coefficients in a commutative associative algebra and show that it is a dagger category. This gives access to a broad new class of models, with the abstract scalars in the sense of Abramsky being the elements of the algebra. We will also consider a subcategory of local formal distributions, based on the ideas of Kac. Locality has been of fundamental significance in various formulations of quantum field theory. Thus our work may provide the possibility of extending the abstract framework to QFT. We also show that these categories of formal distributions are monoidal and contain a nuclear ideal, a weak form of adjunction appropriate for analyzing categories
Deep Inference and Probabilistic Coherence Spaces
, 2009
"... This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor pr ..."
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This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We show that coherence spaces, with Retoré’s noncommutative tensor, is a model.We then consider Girard’s category of probabilistic coherence spaces and show that it contains a selfdual monoidal structure in addition to the ∗autonomous structure exhibited by Girard. This
Causal Theories: A Categorical Perspective on Bayesian Networks
"... It’s been an amazing year, and I’ve had a good time learning and thinking about the contents of this essay. A number of people have had significant causal influence on this. Foremost among these is my dissertation supervisor Jamie Vicary, who has been an excellent guide throughout, patient as I’ve j ..."
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It’s been an amazing year, and I’ve had a good time learning and thinking about the contents of this essay. A number of people have had significant causal influence on this. Foremost among these is my dissertation supervisor Jamie Vicary, who has been an excellent guide throughout, patient as I’ve jumped from idea to idea and with my vague questions, and yet careful to ensure I’ve stayed on track. We’ve had some great discussions too, and I thank him for them. John Baez got me started on this general topic, has responded enthusiastically and generously to probably too many questions, and, with the support of the Centre for Quantum Technologies, Singapore, let me come visit him to pester him with more. Bob Coecke has been a wonderful and generous general supervisor, always willing to talk and advise, and has provided many of the ideas that lurk in the background of those here. I thank both of them too. I also thank Rob Spekkens, Dusko Pavlovic, Prakash Panangaden, and Samson Abramsky for some interesting discussions
COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION
"... Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large n ..."
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Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choiceofduals functor on the compact
A Categorical Foundation for Bayesian Probability
 Applied Categorical Structures
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HigherOrder and Reflexive Action Calculi: Their Type Theory and Models
, 1998
"... Action calculi have been introduced by Milner as a framework for representing models of interactive behaviour. Two natural extensions of action calculi have been proposed: the higherorder action calculi, which add higherorder features to the basic setting, and the reflexive action calculi, which a ..."
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Action calculi have been introduced by Milner as a framework for representing models of interactive behaviour. Two natural extensions of action calculi have been proposed: the higherorder action calculi, which add higherorder features to the basic setting, and the reflexive action calculi, which allow circular bindings of processes and gives recursion in the presense of higherorder features. In this paper, we present simple type theories for action calculi, higherorder action calculi and reflexive action calculi. We also give the categorical models of the extensions, by enriching Power's models of action calculi. As applications, we give a semantic proof of the conservativity of higherorder action calculi over action calculi, and a precise connection with Moggi's computational lambda calculus and notions of computation. We also relate the models of higherorder reflexive action calculi to models of recursive computation, by following the observation that the trace operator of Joya...