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

Coalgebras are categorical presentations of state-based systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final coalgebra as an object in it. This phenomenon is what Baez and Dolan have called the microcosm principle, a prototypical example of which is “a monoid in a monoidal category.” In this paper we obtain a formalization of the microcosm principle in which such a nested model is expressed categorically as a suitable lax natural transformation. An application of this account is a general compositionality result which supports modular verification of complex systems.

Abstract. We present an algebraic treatment of exception handlers and, more generally, introduce handlers for other computational effects representable by an algebraic theory. These include nondeterminism, interactive input/output, concurrency, state, time, and their combinations; in all cases the computation monad is the free-model monad of the theory. Each such handler corresponds to a model of the theory for the effects at hand. The handling construct, which applies a handler to a computation, is based on the one introduced by Benton and Kennedy, and is interpreted using the homomorphism induced by the universal property of the free model. This general construct can be used to describe previously unrelated concepts from both theory and practice. 1

Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.

A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational λ-calculus, with computational effects. We develop the theory of Freyd-categories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freyd-category. We then give canonical, universal embeddings of Freyd-categories into closed Freyd-categories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λc-calculus. Our leading examples of signatures arise from side-effects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism. Key words: Freyd-category, enriched Yoneda embedding, conical colimit completion, canonical model

Starting with Moggi’s work on monads as refined to Lawvere theories, we give a general construct that extends denotational semantics for a global computational effect canonically to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state. Given any Lawvere theory L, possibly countable and possibly enriched, we first give a universal construction that extends L, hence the global operations and equations of a given effect, to incorporate worlds of arbitrary finite size. Then, making delicate use of the final comodel of the ordinary Lawvere theory L, we give a construct that uniformly allows us to model block, the universality of the final comodel yielding a universal property of the construct. We illustrate both the universal extension of L and the canonical construction of block by seeing how they work in the case of state.

Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multi-sorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with name-binding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a second-order extension of the Maude system [37], a first-order semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)

In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effect-free coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributing program syntax over effects, again inducing a coalgebraic operational semantics, but this time in the Kleisli category for the monad derived from the algebraic effects. The final coalgebra in this Kleisli category then serves as the denotational model. For it to exist, we ensure that the the Kleisli category is enriched over CPOs by considering the monad of possibly infinite terms, extended with a bottom element. Unlike the effectless setting, not all operational specifications give rise to adequate and compositional semantics. We give a proof of adequacy and compositionality provided the specifications can be described by evaluation-in-context. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.