We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the a-calculus, 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 multi-sorted logic with higher-order value and computation types, as in Levy’s call-by-push-value, a principle of induction over computations, a free algebra principle, and predicate fixed points. This logic embraces Moggi’s computational λ-calculus, and also, via definable modalities, Hennessy-Milner logic, and evaluation logic, though Hoare logic presents difficulties. 1

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.

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.

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 side-effects monad transformer (an application is the combination of side-effects with nondeterminism). And we give a theory...

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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.

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.

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...

The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by

This paper provides a case-study in the eld of metric semantics for probabilistic programming. Both an operational and a denotational semantics are presented for an abstract process language L pr , which features action refinement and probabilistic choice. The two models are constructed in the setting of complete ultrametric spaces, here based on probability measures of compact support over sequences of actions. It is shown that the standard toolkit for metric semantics works well in the probabilistic context of L pr , e.g. in establishing the correctness of the denotational semantics with respect to the operational one. In addition, it is shown how the method of proving full abstraction -- as proposed recently by the authors for a nondeterministic language with action refinement -- can be adapted to deal with the probabilistic language L pr as well.