Abstract In this paper, we show that the Day monoidal product generalises in a straightforward way to other algebraic constructions and partial algebraic constructions on categories. This generalisation was motivated by its applications in logic, for example, in hybrid and separation logic. We use the description of the Day monoidal product using profunctors to show that the definition generalises to an extension of an arbitrary algebraic structure on a category to a pseudo-algebraic structure on a functor category. We provide two further extensions. First, we consider the case where some of the operations on the category are partial, and second, we show that the resulting operations on the functor category have adjoints (they are residuated).

Abstract We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation R between two transition systems on state spaces S and T is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from R. We show that this approach works for the original Park–Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.

The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory.

We describe a polymorphic, typed lambda calculus with substructural features. This calculus extends the first-order substructural lambda calculus αλ associated with bunched logic. A particular novelty of our new calculus is the substructural treatment of second-order variables. This is accomplished through the use of bunches of type variables in typing contexts. Both additive and multiplicative forms of polymorphic abstraction are then supported. The calculus has sensible proof-theoretic properties and a straightforward categorical semantics using indexed categories. We produce a model for additive polymorphism with first-order bunching based on partial equivalence relations. We consider additive and multiplicative existential quantifiers separately from the universal quantifiers.

This paper introduces a notion of proof net for classical logic, provides a static correctness condition for these nets, and analyses theconnection between nets and conventional sequent calculus. The main surprise of the paper is that there are no surprises at the static level. Subsequent work reveals that there are few at the dynamic either.

Abstract. This is a largely tutorial paper about the categorical notion of monad and the ways in which monads on different categories correspond to variations on the standard notion of algebraic theory.

We d e v elop a notion of equivalence between interpretations of the simply typed -calculus together with an equationally de ned abstract data-type, and we s h o w that two i n terpretations are equivalent if and only if they are linked by a logical relation.We s h o w that our construction generalises from the simply typed -calculus to include the linear -calculus and calculi with additional type and term constructors, such a s those given by s u m t ypes or by a strong monad for modelling phenomena such as partiality or nondeterminism.This is all done in terms of category theoretic structure, usingbrations to model logical relations following Hermida, and adapting Jung and Tiuryn's logical relations of varying arity to provide the completeness results, which form the heart of the work.

We introduce the notions of premonoidal category and premonoidal functor, and show how these can be used in the denotational semantics of programming languages. We characterize the semantic definitions of Eugenio Moggi's monads as notions of computation, exhibit a representation theorem for our premonoidal setting in terms of monads, and give a fibrational setting for the structure.

The pioneering work on relational parametricity for the second order lambda calculus was done by Reynolds (1983) under the assumption of the existence of set-based models, and subsequently reformulated by him, in conjunction with his student Ma, using the technology of PL-categories. The aim of this paper is to use the different technology of internal category theory to re-examine Ma and Reynolds' definitions. Apart from clarifying some of their constructions, this view enables us to prove that if we start with a non-parametric model which is left exact and which satisfies a completeness condition corresponding to Ma and Reynolds "suitability for polymorphism", then we can recover a parametric model with the same category of closed types. This implies, for example, that any suitably complete model (such as the PER model) has a parametric counterpart.>

The authors consider the idea of treating a parametrized type as an arbitrary functor from some parametrizing category to a category of types, and giving elements semantics as natural transformations. They show that under reasonable hypotheses this is only possible when the parametrizing category is a groupoid. This suggests a semantics for a semiparametric form of polymorphism. They discuss the interpretation of this form of parametricity in a PER model, and show that it coincides with the ostensibly stronger form derived from dinaturality.>

Set-theoretic (or rather the more general topos-theoretic) models of polymorphic lambda-calculi are discussed under the assumption that the datatypes of the language are to be interpreted as sets and the operations as partial functions. It is shown that it is not possible to obtain a model in which function spaces are interpreted by the full partial function space, but that it is nevertheless possible to have models which incorporate a usefully large class of partial functions. The main result is that set-theoretic models do not exist, even constructively. This is a much stronger result than holds for the classical sound-order lambda calculus.>

The family of readability toposes, of which the effective topos is the best known, was discovered by Martin Hyland in the late 1970's. Since then these toposes have been used for several purposes. The effective topos itself was originally intended as a category in which various recursion-theoretic or effective constructions would live as natural parts of the higher-order type structure. For example the hereditary effective operators become the higher types over N (Hyland [1982]), and effective domains become the countably-based domains in the topos (McCarty [1984], Rosolini [1986]). However, following the discovery by Moggi and Hyland that it contained nontrivial small complete categories, the effective topos has also been used to provide natural models of polymorphic type theories, up to and including the theory of constructions (Hyland [1987], Hyland, Robinson and Rosolini [1987], Scedrov [1987], Bainbridge et al. [1987]). Over the years there have also been several different constructions of the topos. The original approach, as in Hyland [1982], was to construct the topos by first giving a notion of Pω-valued set. A Pω-valued set is a set X together with a function =x: X × X → Pω. The elements of X are to be thought of as codes, or as expressions denoting elements of some “real underlying” set in the topos. Given a pair (x,x′) of elements of X, the set =x (x,x′) (generally written ) is the set of codes of proofs that the element denoted by x is equal to the element denoted by x′.