This paper investigates the transformation of λν-terms into continuation-passing style (CPS). We show that by appropriate η-expansion of Fisher and Plotkin's two-pass equational specification of the CPS transform, we can obtain a static and context-free separation of the result terms into “essential” and “administrative” constructs. Interpreting the former as syntax builders and the latter as directly executable code, We obtain a simple and efficient one-pass transformation algorithm, easily extended to conditional expressions, recursive definitions, and similar constructs. This new transformation algorithm leads to a simpler proof of Plotkin's simulation and indifference results.

We go on to show how CPS-based control operators similar to, more general then, Scheme's call/cc can be naturally accommodated by the new transformation algorithm. To demonstrate the expressive power of these operators, we use them to present an equivalent but even more concise formulation of the efficient CPS transformation algorithm. Finally, we relate the fundamental ideas underlying this derivation to similar concepts from other work on program manipulation; we derive a one-pass CPS transformation of λn-terms; and we outline some promising areas for future research.

We focus attention on a language ℒ generated by a set of elementary actions under operations of sequential composition, external binary choice, iteration, and non-nested disjoint parallel composition: that is, on an especially simple parallel language without explicit internal, or ‘local’, nondeterminism. ℒ is nondeterministic, despite the restriction to external choice, since a given action may occur in both subterms F and G of the parallel composition F ║ G. We exhibit a single set of equations that axiomatizes both step bisimulation and pomset equivalence on ℒ Given that step bisimulation and pomset equivalence are incomparable on the language ℒ+ obtained from ℒ by relaxing just the constraint on choice, the coincidence of these equivalences on ℒ suggests that the elimination of explicit internal choice can result in simplifications at the semantic level. We reinforce this impression by showing for ℒ that step bisimulation (i) coincides with pomset bisimulation equivalence, (ii) is real-time consistent, and (iii) has step trees as concrete representatives of its equivalence classes. Moreover, we show that none of these results holds for the language ℒ+ Finally, step trace equivalence is proved not to coincide with step bisimulation equivalence on ℒ.

This paper introduces the following new constructions on stable domains and event structures: the tensor product; the linear function space; and the exponential. These give rise to a monoidal closed category of dI-domains and to stable event structures, which can be used to interpret intuitionistic linear logic. Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models are discussed.

The categorical data types in models of second order lambda calculus are studied. We prove that Reynolds parametricity is a sufficient and necessary condition for the categorical data types to fulfill the universal properties.

By establishing an appropriate equivalence, we observe that the theory of semi-functors can be fully embedded in the theory of (ordinary) functors. As a result, standard properties and constructions on functors extend automatically to semi-functors.

This paper considers two main aspects of the lower power locale PL(X): first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of X with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from o(X) to Sub(1), and a new proof of a theorem of Jibladze relating them to Ω-nuclei.

Having started life as a way of reconstructing a topological space from a pair of complementary subspaces, the glueing construction has found employment in a wide range of different roles, from the construction of free distributive lattices to a supporting part in the 2-categorical analysis of types theories. In this latter role the construction appears to be a fundamental factor in the behaviour of higher order proof theory. What is going on here? Before that can be answered we need at least a less ad hoc description of the construction. In this paper I set down what is, I believe, the beginnings of a coherent account of the algebraic version of glueing. As well as the abstract theory, I give a good selection of different examples to illustrate the diverse nature of the uses of the construction.

This paper surveys several different variants of order sorted algebra (abbreviated OSA), comparing some of the main approaches (overloaded OSA, universe OSA, unified algebra, term declaration algebra, etc.), emphasising motivation and intuitions, and pointing out features that distinguish the original ‘overloaded’ OSA approach from some later developments. These features include sort constraints and retracts; the latter is particularly useful for handling multiple data representations (including automatic coercions among them). Many examples are given, for most of which, runs are shown on the OBJ3 system.

This paper also significantly generalises overloaded OSA by dropping the regularity and monotonicity assumptions, and by adding signatures of non-monotonicities, which support simple semantics for some aspects of object oriented programming. A number of new results for this generalisation are proved, including initiality, variety, and quasi-variety theorems. Axiomatisability results à la Birkhoff are also proved for unified algebras.

We propose a formalization of standard database management systems using topos theory. In this treatment, all constructions take place within an ambient topos, which thereby serves as the ‘universe of discourse’. A database schema is defined using objects and morphisms in the ambient topos. A database state for a given schema involves not only the ambient topos but also an internal category within the topos. This approach neatly separates the schema from the state data by placing them in distinct category structures. It is shown that database states can either be regarded syntactically as objects in an external topos or semantically as morphisms in an internal slice category. A number of operations are introduced that correspond to operations used in standard database systems. Extraction selects some of the tables, attributes and domains of a database state. The squeeze operation performs an ‘elimination of duplicates’, which can be combined with extraction to obtain an operation called ‘projection’ in standard relational database systems. A join operation is defined, which generalizes the relational join operation and can be used for the cartesian product and selection operations. Finally, ‘boolean’ operations of intersection, union and difference are introduced and related to the other operations.

Various Theories of Types are introduced, by stressing the analogy ‘propositions-as-types’: from propositional to higher order types (and Logic). In accordance with this, proofs are described as terms of various calculi, in particular of polymorphic (second order) λ-calculus. A semantic explanation is then given by interpreting individual types and the collection of all types in two simple categories built out of the natural numbers (the modest sets and the universe of ω-sets). The first part of this paper (syntax) may be viewed as a short tutorial with a constructive understanding of the deduction theorem and some work on the expressive power of first and second order quantification. Also in the second part (semantics, §§6–7) the presentation is meant to be elementary, even though we introduce some new facts on types as quotient sets in order to interpret ‘explicit polymorphism’. (The experienced reader in Type Theory may directly go, at first reading, to §§6–8).

We establish functional completeness of Locally Cartesian Closed Categories by axiomatising the theory of LCCC's in a constructive style. As a consequence we show that there is essentially only one interpretation of Martin-Löf's theory of types with extensional equality in the theory of LCCC's.