Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses, they are totally equivalent operationally iff they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott's model of continuous functions, Milner's fully abstract model of PCF and their effective submodels. We investigate how totality differs between these models. Some apparently rather difficult open problems arise that essentially concern whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivalence.

The theory of algebraic module specifications and modular systems was developed initially mainly on the basis of equational algebraic specifications. We show that it is in fact almost independent of what kind of underlying specification framework is chosen. More specifically, we present a formulation where this framework appears as an indexed category or, equivalently, specification frame. The ensuing theory is called the theory of abstract module specifications. We are able to prove main results concerning the correctness and compositionality of abstract module specifications in a purely categorical way, assuming the existence of pushouts of morphisms between abstract specifications that allow model amalgamation, functor extension and/or suitable free constructions. Then, by instantiating the theory of abstract module specifications to the behaviour specification frame in the sense of Nivela and Orejas, we obtain a theory of behaviour module specifications.

This paper presents a filter model for π-calculus and shows its full abstraction with respect to a ‘may’ operational semantics. The model is introduced in the form of a type assignment system. Types are related by a preorder that mimics the operational behaviour of terms. A subject expansion theorem holds. Terms are interpreted as filters of types: this interpretation is compositional. The proof of full abstraction relies on a notion of realizability of types and on the construction of terms, which test when an arbitrary term has a fixed type.

In constructive theories, an apartness relation is often taken as basic and its negation used as equality. An apartness relation should be continuous in its arguments, as in the case of computable reals. A similar approach can be taken to order relations. We shall here study the partial order on open intervals of computable reals. Since order on reals is undecidable, there is no simple uniformly applicable lattice meet operation that would always produce non-negative intervals as values. We show how to solve this problem by a suitable definition of apartness for intervals. We also prove the strong extensionality of the lattice operations, where by strong extensionality of an operation f on elements a, b we mean that apartness of values implies apartness in some of the arguments: f(a, b)≠f(c, d) ⊃a≠c∨b≠d.

Most approaches to computable reals start from a concrete definition. We shall instead represent them by an abstract axiomatically introduced order structure.

We give a formulation of simple type theory in which types are just sets. This theory is obtained by relativizing quantifiers and replacing typing predicates by membership of some sets. It can easily be extended to include predicate subtyping.