Mixins are modules that may contain deferred components, that is, components not defined in the module itself; moreover, in contrast to parameterised modules (like ML functors), they can be mutually dependent and allow their definitions to be overridden. In a preceding paper we defined a syntax and denotational semantics of a kernel language of mixin modules. Here, we take instead an axiomatic approach, giving a set of algebraic laws expressing the expected properties of a small set of primitive operators on mixins. Interpreting axioms as rewriting rules, we get a reduction semantics for the language and prove the existence of normal forms. Moreover, we show that the model defined in the earlier paper satisfies the given axiomatisation.

There are two main approaches to obtaining ‘topological’ cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed – for example, the category of sequential spaces. Under the other, one generalises the notion of space – for example, to Scott's notion of equilogical space. In this paper, we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.

We extend the proof-irrelevant model defined in Smith (1988) to the whole of Martin-Löf's logical framework. The main difference here is the existence of a type whose objects themselves represent types rather than proof-objects. This means that the model must now be able to distinguish between objects with different degree of relevance: those that denote proofs are irrelevant whereas those that denote types are not. In fact a whole hierarchy of relevance exists.

Another difference is the higher level of detail in the formulation of the formal theory, such as the explicit manipulation of contexts and substitutions. This demands an equally detailed definition of the model, including interpreting contexts and substitutions.

We are thus led to a whole reformulation of the proof-irrelevant model. We present a model that is built up from an arbitrary model of the untyped lambda calculus. We also show how to extend it when the logical framework itself is enlarged with inductive definitions. In doing so, a variant of Church numerals is introduced.

As in Smith (1988), the model can only be defined in the absence of universes, and it is useful to obtain an elementary proof of consistency and to prove the independence of Peano's fourth axiom.

This paper presents a simple implementation of the λ-calculus in the interaction net paradigm. It is based on a two-fold translation. λ-terms are coded (for duplication) or decoded (for execution), and reduction is achieved by switching between these two states: decoding corresponds to head reduction and encoding to left reduction.

We show that Friedman's proof of the existence of non-trivial βη-complete models of λ→ can be extended to system F. We isolate a set of conditions that are sufficient to ensure βη-completeness for a model of F (and α-completeness at the level of types), and we discuss which class of models we get. In particular, the model introduced in Barbanera and Berardi (1997), having as polymorphic maps exactly all possible Scott continuous maps, is βη-complete, and is hence the first known complete non-syntactic model of F. In order to have a suitable framework in which to express the conditions and develop the proof, we also introduce the very natural notion of ‘polymax models’ of System F.

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are defined for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This construction gives many examples, for coalgebras of polynomial functors on sets. More generally, it will be shown how ‘fuzzy’ predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner.