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25
Adequacy for algebraic effects
 In 4th FoSSaCS
, 2001
"... We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the acalculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to ..."
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Cited by 32 (16 self)
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We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the acalculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to obtain the logic, which is a classical firstorder multisorted logic with higherorder value and computation types, as in Levy’s callbypushvalue, a principle of induction over computations, a free algebra principle, and predicate fixed points. This logic embraces Moggi’s computational λcalculus, and also, via definable modalities, HennessyMilner logic, and evaluation logic, though Hoare logic presents difficulties. 1
Computational Effects and Operations: An Overview
, 2004
"... We overview a programme to provide a unified semantics for computational effects based upon the notion of a countable enriched Lawvere theory. We define the notion of countable enriched Lawvere theory, show how the various leading examples of computational effects, except for continuations, give ris ..."
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Cited by 26 (8 self)
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We overview a programme to provide a unified semantics for computational effects based upon the notion of a countable enriched Lawvere theory. We define the notion of countable enriched Lawvere theory, show how the various leading examples of computational effects, except for continuations, give rise to them, and we compare the definition with that of a strong monad. We outline how one may use the notion to model three natural ways in which to combine computational effects: by their sum, by their commutative combination, and by distributivity. We also outline a unified account of operational semantics. We present results we have already shown, some partial results, and our plans for further development of the programme.
Axioms for Probability and Nondeterminism
 ENTCS
, 2003
"... This paper presents a domain model for a process algebra featuring both probabilistic and nondeterministic choice. The former is modelled using the probabilistic powerdomain of Jones and Plotkin, while the latter is modelled by a geometrically convex variant of the Plotkin powerdomain. The main resu ..."
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Cited by 23 (1 self)
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This paper presents a domain model for a process algebra featuring both probabilistic and nondeterministic choice. The former is modelled using the probabilistic powerdomain of Jones and Plotkin, while the latter is modelled by a geometrically convex variant of the Plotkin powerdomain. The main result is to show that the expected laws for probability and nondeterminism are sound and complete with respect to the model. We also present an operational semantics for the process algebra, and we show that the domain model is fully abstract with respect to probabilistic bisimilarity.
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
"... ..."
Combining Computational Effects: Commutativity and Sum
, 2002
"... We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations ..."
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Cited by 19 (4 self)
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We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi's sideeffects monad transformer (an application is the combination of sideeffects with nondeterminism). And we give a theory...
Characterising testing preorders for finite probabilistic processes
 In LICS’07: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, Los Alamitos, CA
"... In 1992 Wang & Larsen extended the may and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to find complete axiomatisations and alternative cha ..."
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Cited by 14 (7 self)
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In 1992 Wang & Larsen extended the may and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to find complete axiomatisations and alternative characterisations for these preorders. This paper solves both problems for finite processes with silent moves. It characterises the may preorder in terms of simulation, and the must preorder in terms of failure simulation. It also gives a characterisation of both preorders using a modal logic. Finally it axiomatises both preorders over a probabilistic version of CSP. 1.
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 10 (2 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
A Logic for Probabilities in Semantics
, 2003
"... Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Pl ..."
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Cited by 9 (1 self)
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Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Plotkin represents a significant advance. Further work, especially by AlvarezManilla, has greatly improved our understanding of the probabilistic powerdomain, and has helped clarify its relation to classical measure and integration theory. On the practical side, many researchers such as Kozen, Segala, Desharnais, and Kwiatkowska, among others, study problems of verification for probabilistic computation by defining various suitable logics for the classes of processes under study. The work reported here begins to bridge the gap between the domain theoretic and verification (model checking) perspectives on probabilistic computation by exhibiting sound and complete logics for probabilistic powerdomains that arise directly from given logics for the underlying domains. The category in which the construction is carried out generalizes Scott’s Information Systems by taking account of full classical sequents. Via Stone duality, following Abramsky’s Domain Theory in Logical Form, all known interesting categories of domains are embedded as subcategories. So the results reported here properly generalize similar constructions on specific categories of domains. The category offers a promising universe of semantic domains characterized by a very rich structure and good preservation properties of standard constructions. Furthermore, because the logical constructions make use of full classical sequents, the morphisms have a natural nondeterministic interpretation. Thus the category is a natural one in which to investigate the relationship between probabilistic and nondeterministic computation. We discuss the problem of integrating probabilistic and nondeterministic computation after presenting the construction of logics for probabilistic powerdomains.
Continuous Previsions
"... We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they m ..."
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Cited by 6 (5 self)
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We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they model exactly the interaction between probabilistic and (demonic, angelic, chaotic) choice.
Probability, Nondeterminism and Concurrency: Two Denotational Models for Probabilistic Computation
 PHD THESIS, UNIV. AARHUS, 2003. BRICS DISSERTATION SERIES
, 2003
"... Nondeterminism is modelled in domain theory by the notion of a powerdomain, while probability is modelled by that of the probabilistic powerdomain. Some problems arise when we want to combine them in order to model computation in which both nondeterminism and probability are present. In particular t ..."
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Cited by 5 (1 self)
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Nondeterminism is modelled in domain theory by the notion of a powerdomain, while probability is modelled by that of the probabilistic powerdomain. Some problems arise when we want to combine them in order to model computation in which both nondeterminism and probability are present. In particular there is no categorical distributive law between them. We introduce the powerdomain of indexed valuations which modifies the usual probabilistic powerdomain to take more detailed account of where probabilistic choices are made. We show the existence of a distributive law between the powerdomain of indexed valuations and the nondeterministic powerdomain. By means of an equational theory we give an alternative characterisation of indexed valuations and the distributive law. We study the relation between valuations and indexed valuations. Finally we use indexed valuations to give a semantics to a programming language. This semantics reveals the computational intuition lying behind the mathematics. In the second part of the thesis we provide an operational reading of continuous valuations on certain domains (the distributive concrete domains of Kahn and Plotkin) through the model of probabilistic event structures. Event structures are a model for concurrent computation that account for causal relations between events. We propose a way of adding probabilities to confusion free event structures, defining the notion of probabilistic event structure. This leads to various ideas of a run for probabilistic event structures. We show a confluence theorem for such runs. Configurations of a confusion free event structure form a distributive concrete domain. We give a representation theorem which characterises completely the powerdomain of valuations of such concrete domains in terms of prob...