Least fixpoints as meanings of recursive definitions.

this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of higher-type functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as well---and that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the so-called untyped -calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary way---avoiding heavy prerequisites (such as category theory) but still maintaining some level of abstraction---with the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...

We consider type universes as examples of regular algebras in the area of denotational semantics. The paper concentrates on our method which was used implicitly to prove that the interesting domain equations have solutions in the domain universes underlying MetaSoft, cf. [BBP90], and BSI/VDM, cf. [TW90]. Technically speaking the method allows to prove regularity of a universe. It is demonstrated by means of an example that the method applies even to universes which are essentially regular, i.e., which are neither cpo's, nor the images of the initial regular algebra. 1 Introduction 1.1 The Problem It is a usual practice in the area of programming languages to assign types to the manipulated objects. The typing procedure yields the first, naive, explanation of the notion of type: each type stands for the set of objects that have the type assigned to them. Consequently, one demands that the type forming operators should also be interpreted as operations on sets. It was discovere...

The category of L-domains was discovered by A. Jung while solving the problem of finding maximal cartesian closed categories of algebraic CPO's and continuous functions. In this note we analyse properties of the lossless powerdomain construction, that is closed on the algebraic L-domains. The powerdomain is shown to be isomorphic to a collection of subsets of the domain on which the construction was done. The proof motivates a certain finiteness condition on the inconsistency relations of elements. It is shown that all algebraic CPO's D whose basis B(D) has property M satisfy the condition. In particular, the coherent L- domains satisfy the condition. 1 Introduction Recent work by A. Jung and C. Gunter shows that the L-domains discovered independently by A. Jung[6] and T. Coquand[1] form an interesting cartesian closed category. P. Buneman proposed the lossless powerdomain construction that is closed on L-domains. Buneman's construction was based on intuitions from databases. We were...