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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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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.
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
Random Measurable Selections
"... Abstract. We make the first steps towards showing a general “randomness for free ” theorem for stochastic automata. The goal of such theorems is to replace randomized schedulers by averages of pure schedulers. Here, we explore the case of measurable multifunctions and their measurable selections. T ..."
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Abstract. We make the first steps towards showing a general “randomness for free ” theorem for stochastic automata. The goal of such theorems is to replace randomized schedulers by averages of pure schedulers. Here, we explore the case of measurable multifunctions and their measurable selections. This involves constructing probability measures on the measurable space of measurable selections of a given measurable multifunction, which seems to be a fairly novel problem. We then extend this to the case of IT automata, namely, nondeterministic (infinite) automata with a historydependent transition relation. Throughout, we strive to make our assumptions minimal. 1
Characterizing the EilenbergMoore Algebras for a Monad of Stochastic Relations
, 2004
"... We investigate the category of EilenbergMoore algebras for the Giry monad associated with stochastic relations over Polish spaces with continuous maps as morphisms. The algebras are characterized through convex partitions of the space of all probability measures. Examples are investigated, and it ..."
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We investigate the category of EilenbergMoore algebras for the Giry monad associated with stochastic relations over Polish spaces with continuous maps as morphisms. The algebras are characterized through convex partitions of the space of all probability measures. Examples are investigated, and it is shown that finite spaces usually do not have algebras at all.
Pipes and Filters: Modelling a Software Architecture Through Relations
, 2002
"... A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream proces ..."
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A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream processing has been made popular through UNIX pipes that serially connect independent components for performing a sequence of tasks. We show in this paper how to formalize this architecture in terms of monads, hereby including relational specifications as special cases. The system is given through a directed acyclic graph the nodes of which carry the computational structure by being labelled with morphisms from the monad, and the edges provide the data for these operations. It is shown how fundamental compositional operations like combining pipes and filers, and refining a system by replacing simple parts through more elaborate ones, are supported through this construction.
in a probabilistic logic
, 2002
"... Reasoning about probabilistic sequential programs ..."
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Labelled Markov Processes as Generalised Stochastic Relations
"... Labelled Markov processes (LMPs) are labelled transition systems in which each transition has an associated probability. In this paper we present a universal LMP as the spectrum of a commutative C ∗algebra consisting of formal linear combinations of labelled trees. This yields a simple tracetree s ..."
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Labelled Markov processes (LMPs) are labelled transition systems in which each transition has an associated probability. In this paper we present a universal LMP as the spectrum of a commutative C ∗algebra consisting of formal linear combinations of labelled trees. This yields a simple tracetree semantics for LMPs that is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with distinguished entry and exit points as stateful stochastic relations. This allows us to define a category LMP, with measurable spaces as objects and LMPs as morphisms. Our main result in this context is to provide a predicatetransformer duality for