A. John Power

Computational Effects and Operations: An Overview

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.

From Comodels to Coalgebras: State and Arrays

We investigate the notion of a comodel of a (countable) Lawvere theory, an evident dual to the notion of model. By taking the forgetful functor from the category of comodels to Set, every (countable) Lawvere theory generates a comonad on Set. But while Lawvere theories are equivalent to finitary monads on Set, and that result extends to higher cardinality, no such result holds for comonads, and that is not only for size reasons: it is primarily because, while Set is cartesian closed, Set^o^p is not. So every monad with rank on Set generates a comonad on Set, but not conversely. Our leading example is given by the countable Lawvere theory for global state: its category of comodels is the category of arrays, yielding a precise relationship between global state and arrays. Restricting from arbitrary comonads to those comonads generated by Lawvere theories allows us to study new and interesting constructions, in particular that of tensor product.

An Axiomatic Approach to Binary Logical Relations with Applications to Data Refinement

We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion of “relation” between models of such a sketch in a semantic category. We then prove three results: (i) such models lift to the whole language together with the sketch; (ii) any such relation satisfies a soundness condition, and (iii) such relations compose. We do this for both equality of data representations and for an ordered version. Finally, we compare our formulation of data refinement with that of Hoare.

A Fibrational Semantics for Logic Programs

We introduce a new semantics for logic programming languages. It generalises the traditional Herbrand universe semantics, and specialises the semantics of logical relations, as used in analysing parametricity in functional and imperative programming languages. We outline a typed logic programming language, give it this semantics, and show how it supports structured development of logic programs as advocated by Sterling et al. In particular, it gives semantics for some dynamic aspects of logic programs.

Environments, Continuation Semantics and Indexed Categories

There have traditionally been two approaches to modelling environments, one by use of finite products in Cartesian closed categories, the other by use of the base categories of indexed categories with structure. Recently, there have been more general definitions along both of these lines: the first generalising from Cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κ-categories. The added generality is not of the purely mathematical kind; in fact it is necessary to extend semantics from the logical calculi studied in, say, Type Theory to more realistic programming language fragments. In this paper, we establish an equivalence between these two recent notions. We then use that equivalence to study semantics for continuations. We give three category theoretic semantics for modelling continuations and show the relationships between them. The first is given by a continuations monad. The second is based on a symmetric premonoidal category with a self-adjoint structure. The third is based on a κ-category with indexed self-adjoint structure. We extend our result about environments to show that the second and third semantics are essentially equivalent, and that they include the first.

Complete cuboidal sets in axiomatic domain theory

We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, via, that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domain-theoretic structures.

Modularity of Behaviours for Mathematical Operational Semantics

Some years ago, Turi and Plotkin gave a precise mathematical formulation of a notion of structural operational semantics: their formulation is equivalent to a distributive law of the free monad on a signature over the cofree copointed endofunctor on a behaviour endofunctor. From such a distributive law, one can readily induce a distributive law of the monad over the cofree comonad on the behaviour endofunctor, and much of their analysis can be carried out in the latter terms, adding a little more generality that proves to be vital here. Here, largely at the latter level of generality, we investigate the situation in which one has two sorts of behaviours, with operational semantics possibly interacting with each other. Our leading examples are given by combining action and timing, with a modular account of the operational semantics for the combination induced by that of each of the two components. Our study necessitates investigation and new results about products of comonads and liftings of monads to categories of coalgebras for the product of comonads, providing constructions with which one can readily calculate.

A representation result for free cocompletions

Abstract Given a class F of weights, one can consider the construction that takes a small category C to the free cocompletion of C under weighted colimits, for which the weight lies in F . Provided these free F -cocompletions are small, this construction generates a 2-monad on Cat , or more generally on V - Cat for monoidal biclosed complete and cocomplete V . We develop the notion of a dense 2-monad on V - Cat and characterise free F -cocompletions by dense KZ -monads on V - Cat . We prove various corollaries about the structure of such 2-monads and their Kleisli 2-categories, as needed for the use of open maps in giving an axiomatic study of bisimulation in concurrency. This requires the introduction of the concept of a pseudo-commutativity for a strong 2-monad on a symmetric monoidal 2-category, and a characterisation of it in terms of structure on the Kleisli 2-category.

Categories with Algebraic Structure

We give an exposition of a unified study of categories with extra structure that arise in computer science and mathematics. We consider several examples of structures that arise, showing that with a precise formulation of the notion of category with algebraic structure, they are categories with algebraic structure. We then outline the central results and issues in the study of categories with algebraic structure, with an account of why those issues are of computational interest. We illustrate general theorems that yield substantial results in examples given by familiar structures. In particular, we explain known mathematics that supports the idea of a programming language being freely generated by basic data and specified algebraic structure. We then show in detail how the concept of algebraic structure may be used in defining new category theoretic structures by showing how it affected the precise formulation of premonoidal category, as has recently been proposed to account for contexts.