Domain theory

  title={Domain theory},
  author={Samson Abramsky and Achim Jung},
  booktitle={Logic in Computer Science},
bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as… 

A foundation for computation

This thesis is the study and application of domains with measurements and seeks to develop a mathematical setting in which the question “Does this program work, and if so, how does it compare to others which solve the same problem?” may be sensibly formalized and answered.

Directed complete poset congruences

On Some Constructions in Quantitative Domain Theory (Extended Abstract)

Domains introduced by Dana Scott [11] and independently by Yuri L. Ershov [3] are a structure modelling the notion of approximation and of computation. A computation performed using an algorithm


The present work can be seen as an attempt to develop a constructive theory of formal neighborhoods for continuous functionals, in a direct and intuitive style, to replace abstract domain theory by a more concrete, finitary theory of representations.

On Domain Theory over Girard Quantales

This paper demonstrates that the domain-theoretic construction of the Hoare, Smyth and Plotkin powerdomains of a continuous dcpo can be straightforwardly adapted to yield corresponding constructions for continuous Q-domains.

Topology in Computer Science Problems

We pose the problem of whether every FS-domain is a retract of a bifinite domain purely in terms of quasi-uniform spaces. 6.1 The problem and its history Ever since domains were introduced by Dana

Coalgebraic Theories of Sequences in PVS

This paper explains the setting of an extensive formalisation of the theory of sequences in the Prototype Veriication System based on the characterisation of sequences as a nal coalgebra, which is used as an axiom.

From Objects to Diagrams for Ranges of Functors

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for ''many'' objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general

Absolute retractness of automata on directed complete posets

The notion of retractness, which is about having left inverses (reflection) for monomorphisms, is crucial in most branches of mathematics. One very important notion related to it is injectivity,

A Cook's Tour of the Finitary Non-Well-Founded Sets

A topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets, is given, which contains non-well founded sets and a universal set and is closed under positive versions of the usual axioms of set theory.



The category-theoretic solution of recursive domain equations

  • M. SmythG. Plotkin
  • Mathematics
    18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
  • 1977
The purpose of the present paper is to set up a categorical framework in which the known techniques for solving equations find a natural place, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories.

The Largest First-Order-Axiomatizable Cartesian Closed Category of Domains

The purpose of this paper is to state and prove an analog to Smyth's theorem which says that the bounded complete domains form the largest cartesian closed category of domains.

A motivation and generalization of scott's notion of a continuous lattice

This work develops an indication of how to approach continuous lattices from a purely order-theoretic perspective and offers some arguments in support of the thesis that "continuous posets" (chain-complete posets with a basis) are the proper setting for an abstract theory of computation.

Universal domains in the theory of denotational semantics of programming languages

  • M. DrosteR. Göbel
  • Mathematics
    [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science
  • 1990
The authors present a categorical generalization of a well-known result in model theory, the Fraisse-Jonsson theorem, by which they characterize large classes of reasonable categories if they contain

Domain Theory in Logical Form

Concrete Domains

Completion of a construction of Johnstone

A complete lattice is constructed which is not sober in the Scott topology. Peter Johnstone has constructed [3] a (countable) partially ordered set X+ which admits a sober topology (i.e. the relation

DI-Domains as a Model of Polymorphism

This paper intends to demonstrate that Girard's generalized construction can be used to do denotational semantics in the ordinary way, but with the added feature of type polymorphism with a “types as domains” interpretation.

A Powerdomain Construction

  • G. Plotkin
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1976
A powerdomain construction is developed, which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains, and a restricted class of algebraic inductive partial orders is found which is closed under this construction.