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The Weakest Completion Approach to the Probabilistic Semantics
, 2000
"... A standard program starts its execution in an initial state, and terminates (if it ever does) in one of a set of final states. Its behaviour can be modelled by a binary relation between the initial and final states. The difference between the standard and the probabilistic semantics is that the form ..."
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A standard program starts its execution in an initial state, and terminates (if it ever does) in one of a set of final states. Its behaviour can be modelled by a binary relation between the initial and final states. The difference between the standard and the probabilistic semantics is that the former tells us which final states are or are not possible, whereas the latter tells us the probability with which they may occur. This paper presents a link between the probabilistic and the imperative programming using the weakest completion. We demonstrate how the probabilistic semantics can be derived directly from the standard relational one using the type embedding and healthiness condition of real programs. Carroll Morgan is an adjunct professor in the department of computer science at New South Wales University in Australia. He conducts research in the area of refinement theories and formal methods applied to software engineering and applications to parallel and distributed computing, ...
A Programming Language for Probabilistic Computation
, 2005
"... As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages to facilitate their modeling. Most of the existing probabilistic languages, however, focus only on discrete distributions, and there has been little effort to develop ..."
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As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages to facilitate their modeling. Most of the existing probabilistic languages, however, focus only on discrete distributions, and there has been little effort to develop probabilistic languages whose expressive power is beyond discrete distributions. This dissertation presents a probabilistic language, called PTP (ProbabilisTic Programming), which supports all kinds of probability distributions.
Tracebased Semantics for Probabilistic Timed I/O Automata Submitted for review. Full version http://theory.lcs.mit.edu/ ∼mitras/ research/PTIOA06full.pdf
"... Abstract. We propose the Probabilistic Timed I/O Automaton (PTIOA) framework for modelling and analyzing discretely communicating probabilistic hybrid systems. State transition of a PTIOA can be nondeterministic or probabilistic. Probabilistic choices can be based on continuous distributions. Contin ..."
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Abstract. We propose the Probabilistic Timed I/O Automaton (PTIOA) framework for modelling and analyzing discretely communicating probabilistic hybrid systems. State transition of a PTIOA can be nondeterministic or probabilistic. Probabilistic choices can be based on continuous distributions. Continuous evolution of a PTIOA is purely nondeterministic. PTIOAs can communicate through shared actions. By supporting external nondeterminism, the framework allows us to model arbitrary interleaving of concurrently executing automata. The framework generalizes several previously studied automata models of its class. We develop the tracebased semantics for PTIOAs which involves measure theoretic constructions on the space of executions of the automata. We introduce a new notion of external behavior for PTIOAs and show that PTIOAs have simple compositionality properties with respect this external behavior. 1
Categories for Imperative Semantics PLDG Seminar
"... The aim of these notes is to provide an introduction to category theory, and a motivation for its use in denotational semantics. I will do this by showing how to apply it to give an abstract semantics to a simple imperative language. These notes are loosely based ..."
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The aim of these notes is to provide an introduction to category theory, and a motivation for its use in denotational semantics. I will do this by showing how to apply it to give an abstract semantics to a simple imperative language. These notes are loosely based
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.
The Expectation Monad in Quantum Foundations
"... The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two pr ..."
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The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to socalled Banach effect algebras. These structures capture states and effects in quantum foundations, and the duality between them. Moreover, the approach leads to a new reformulation of Gleason’s theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.
Logical Relations for Monadic Types †
, 2004
"... Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi’s computational lambdacalculus. Th ..."
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Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi’s computational lambdacalculus. The treatment is categorical, and is based on notions of subsconing, mono factorization systems, and monad morphisms. Our approach has a number of interesting applications, including cases for lambdacalculi with nondeterminism (where being in logical relation means being bisimilar), dynamic name
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
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