This special issue of Mathematical Structures in Computer Science contains the final Proceedings of the Tutorial Workshop on Realizability Semantics and Applications, which was held between June 30 and July 1, 1999 in Trento, Italy, as one of the satellite workshops associated to the Federated Logic Conference.

The purpose of this short paper is to sketch the development of a few basic topics in the history of realizability. The number of topics is quite limited and reflects very much my own personal taste, prejudices and area of competence.

The notion of tripos (Hyland et al. 1980; Pitts 1981) was motivated by the desire to explain in what sense Higg's description of sheaf toposes as H-valued sets and Hyland's realizability toposes are instances of the same construction. The construction itself can be seen as the universal solution to the problem of realizing the predicates of a first order hyperdoctrine as subobjects in a logos with effective equivalence relations. In this note it is shown that the resulting logos is actually a topos if and only if the original hyperdoctrine satisfies a certain comprehension property. Triposes satisfy this property, but there are examples of non-triposes satisfying this form of comprehension.

We survey the main results on computability and totality in Scott–Eršov-domains as well as their applications to the theory of functionals of higher types and the semantics of functional programming languages. A new density theorem is proved and applied to show the equivalence of the hereditarily computable total continuous functionals with the hereditarily effective operations over a large class of base types.

Realizability and related functional interpretations provide models for constructive mathematics. Generally, these models do not validate the axiom of choice for propositions taken over hierarchies of extensional functionals. We describe simple classes of models where the axiom is validated.

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability.

In standard realizability one works with respect to an untyped universe of realizers called a partial combinatory algebra (pca). It is well known that a pca [Ascr ] gives rise to a categorical model of impredicative type theory via the category Asm([Ascr ]) of assemblies over [Ascr ] or the realizability topos over [Ascr ].

Recently, John Longley introduced a typed version of pca's (Longley 1999b). The above mentioned construction of categorical models extends to the typed case. However, in general these are no longer impredicative. We show that for a typed pca [Tscr ] the ensuing models are impredicative if and only if [Tscr ] has a universal type U. Such a type U can be endowed with the structure of an untyped pca such that U and [Tscr ] induce equivalent realizability models: in other words, a typed pca [Tscr ] with a universal type is essentially untyped. Thus, a posteriori it turns out that nothing is lost by restricting to (untyped) pca's as far as realizability models of impredicative type theories are concerned.

For instance, we show that for a typed pca [Tscr ] the fibred category of discrete families in Asm([Tscr ]) is small if and only if [Tscr ] has a universal type. As the category of ¬¬-separated objects of the modified realizability topos is equivalent to Asm([Tscr ]) for an appropriate typed pca [Tscr ] without a universal type, it follows that the discrete families in the subcategory of ¬¬-separated objects of the modified realizability topos do not provide a model of polymorphic λ-calculus.