Mathematical Theory of Domains

Abstract

Domain theory is an established part of theoretical computer science, used in giving semantics to programming languages and logics. In mathematics and logic it has also proved to be useful in the study of algorithms. This book is devoted to providing a unified and self-contained treatment of the subject. The theory is presented in a mathematically precise manner which nevertheless is accessible to mathematicians and computer scientists alike. The authors begin with the basic theory including domain equations, various domain representations and universal domains. They then proceed to more specialized topics such as effective and power domains, models of lambda-calculus and so on. In particular, the connections with ultrametric spaces and the Kleene–Kreisel continuous functionals are made precise. Consequently the text will be useful as an introductory textbook (earlier versions have been class-tested in Uppsala, Gothenburg, Passau, Munich and Swansea), or as a general reference for professionals in computer science and logic.

Construção de Domínios Bi-Scott via Completação por Ideais

“…Para cada x ∈ P , o conjunto ↓ x = {y ∈ P / x ≤ y}é um ideal, chamado de ideal principal gerado por x [17].…”

“…A noção dada aquié diferente da abordagem usual de domínios para este assunto (domínios efetivamente dados [16,18,17]). Isto, em princípio, proporcionaria uma nova abordagem para computabilidade no contínuo.…”

“…Em [7], foi provado que a completação por ideais de uma pré-ordem resulta num cpo algébrico e que todo cpo algébricoé ordem-isomorfaà completação por ideais de seus elementos compactos. Em [17,2] foi provado que a completação por ideais de um poset finitamente consistentemente completo, ou seja um poset em que todo conjunto finito majorado tem supremo, resulta em um domínio de Scott e que todo domínio de Scotté ordem isomorfaà completação por ideais de seus elementos compactos, que por sua vez são um poset finitamente consistentemente completo.…”

On the ubiquity of certain total type structures

“…This characterization suggests the possibility of an effective enumeration of the compact elements of P. We will fix on an enumeration which is similar in essentials to those given in e.g. (Plotkin 1977;Stoltenberg-Hansen et al 1994), but for our purposes it will be convenient to build in a few additional hygiene conditions on our representations of compact elements. , let ζ σ×τ : N P comp σ×τ be the smallest partial function such that…”

MATHS VS (META)MODELLING - Are we Reinventing the Wheel?

“…Denotational semantics [13,14] allows a detailed analysis of algorithms to be made and conclusions to be drawn regarding their behaviour and efficiency. However, the very mathematical nature of Denotational Semantics makes it highly inaccessible to traditional programmers and therefore its practical uptake has been limited.…”