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 seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi’s side-effects monad transformer. Finally we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.

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...

To my parents Understanding procedure calls is crucial in computer science and everyday pro-gramming. Among the most common strategies for passing procedure argu-ments (‘evaluation strategies’) are ‘call-by-name’, ‘call-by-need’, and ‘call-by-value’, where the latter is the most commonly used. While reasoning about procedure calls is simple for call-by-name, problems arise for call-by-need and call-by-value, because it matters how often and in which order the arguments of a procedure are evaluated. We shall classify these problems and see that all of them occur for call-by-value, some occur for call-by-need, and none occur for call-by-name. In that sense, call-by-value is the ‘greatest common denominator ’ of the three evaluation strategies. Reasoning about call-by-value programs has been tackled by Eugenio Moggi’s ‘computational lambda-calculus’, which is based on a distinction between ‘values’