Dana Scott

Data types as lattices

The meaning of many kinds of expressions in programming languages can be taken as elements of certain spaces of “partial” objects. In this report these spaces are modeled in one universal domain

Domains for Denotational Semantics

{\bf P} \omega

Advice on Modal Logic

, the set of all subsets of the integers. This domain renders the connection of this semantic theory with the ordinary theory of number theoretic (especially general recursive) functions clear and straightforward.

A type-theoretical alternative to ISWIM, CUCH, OWHY

The purpose of the theory of domains is to give models for spaces on which to define computable functions. The kinds of spaces needed for denotational sematics involve not only spaces of higher type (e.g. function spaces) but also spaces defined recursively (e.g. reflexive domains). Also required are many special domain constructs (or functors) in order to create the desired structures. There are several choices of a suitable category of domains, but the basic one which has the simplest properties is the one sometimes called consistently complete algebraic cpos. This category of domains is studied in this paper from a new, and it is to be hoped, simpler point of view incorporating the approaches of many authors into a unified presentation. Briefly, the domains of elements are represented set theoretically with the aid of structures called information systems. These systems are very familiar from mathematical logic, and their use seems to accord well with intuition. Many things that were done previously axiomatically can now be proved in a straightfoward way as theorems. The present paper discusses many examples in an informal way that should serve as an introduction to the subject.

Lectures on a Mathematical Theory of Computation

Everyone knows how much more pleasant it is to give advice than to take it. Everyone knows how little heed is taken of all the good advice he has to offer. Nevertheless, this knowledge seldom restrains anyone, least of all the present author. He has been noting the confusions, misdirections of emphasis, and duplications of effort current in studies of modal logic and is, by now, anxious to disseminate all kinds of valuable advice on the subject. Thus he is very happy that the Irving meeting has provided such a suitable and timely forum and hopes that all this advice can provoke some useful discussion — at least in self-defense. The time really seems to be ripe for a fruitful development of modal logic, if only we take care to purify and simplify the foundations. A quite flexible framework is indeed possible: the old puzzles can be brushed aside, and one can begin to provide meaningful applications.

Lambda Calculus: Some Models, Some Philosophy

Abstract The paper (first written in 1969 and circulated privately) concerns the definition, axiomatization, and applications of the hereditarily monotone and continuous functionals generated from the integers and the Booleans (plus “undefined” elements). The system is formulated as a typed system of combinators (or as a typed λ-calculus) with a recursion operator (the least fixed-point operator), and its proof rules are contrasted to a certain extent with those of the untyped λ-calculus. For publication (1993), a new preface has been added, and many bibliographical references and comments in footnotes have been appended.

Equilogical spaces

These notes were originally written for lectures on the semantics of programming languages delivered at Oxford during Michaelmas Term 1980. The purpose of the course was to provide the foundations needed for the method of denotational semantics; in particular I wanted to make the connections with recursive function theory more definite and to show how to obtain explicit, effectively given solutions to domain equations. Roughly, these chapters cover the first half of the book by Stoy, and he was able to continue the lectures the next term discussing semantical concepts following his text.

First Steps Towards Inferential Programming

Publisher Summary The chapter presents an exposition of why the λ-calculus has models. The A-calculus was one of the first areas of research of Professor Kleene, in which the experience gained by him was surely beneficial in his later development of the recursive function theory. The chapter discusses a very short historical summary, and it is found that there is considerable overlap with CURRY. There is a review of the theory of functions and relations as sets leading up to the important notion of a continuous set mapping. The problem of the self-application of a function to itself as an argument is discussed in the chapter from a new angle. The model (essentially due to PLOTKIN) of the basic laws of λ -calculus thus results. The chapter describes self-application to recursion by the proof of David Parks theorem to the effect that the least fixed-point operator and the paradoxical combinator are the same in a wide class of well-behaved models. The connection thus engendered to recursion theory (r. e. sets) is outlined, and some remarks on recent results about ill-behaved models and on induction principles are discussed. The theme of type theory and a construction of an (η)-model with fewer -type distinctions is presented. There is a brief discussion of how to introduce more type distinctions into models via equivalence relations. The chapter also presents various points of philosophical disagreement with Professor Curry.

Some Ordered Sets in Computer Science

It is well known that one can build models of full higher-order dependent-type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra. But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily define the category of ERs and equivalence-preserving continuous mappings over the standard category Top0 of topological T0-spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category--in contradistinction to Top0--is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein), a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of finite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models.

Combinators and classes

Our basic premise is that the ability to construct and modify programs will not improve without a new and comprehensive look at the entire programming process. Past theoretical research, say, in the logics of programs, has tended to focus on methods for reasoning about individual programs; little has been done, it seems to us, to develop a sound understanding of the process of programming — the process by which programs evolve in concept and in practice. At present, we lack the means to describe the techniques of program construction and improvement in ways that properly link verification, documentation and adaptability.