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Definitions for domain theory:

The Free On-line Dictionary of Computing (27 SEP 03) :
domain theory A branch of mathematics introduced by Dana Scott in 1970 as a mathematical theory of programming languages, and for nearly a quarter of a century developed almost exclusively in connection with denotational semantics in computer science. In denotational semantics of programming languages, the meaning of a program is taken to be an element of a domain. A domain is a mathematical structure consisting of a set of values (or "points") and an ordering relation, <= on those values. Domain theory is the study of such structures. ("<=" is written in LaTeX as \subseteq) Different domains correspond to the different types of object with which a program deals. In a language containing functions, we might have a domain X -> Y which is the set of functions from domain X to domain Y with the ordering f <= g iff for all x in X, f x <= g x. In the {pure lambda-calculus} all objects are functions or applications of functions to other functions. To represent the meaning of such programs, we must solve the recursive equation over domains, D = D -> D which states that domain D is (isomorphic to) some {function space} from D to itself. I.e. it is a fixed point D = F(D) for some operator F that takes a domain D to D -> D. The equivalent equation has no non-trivial solution in {set theory}. There are many definitions of domains, with different properties and suitable for different purposes. One commonly used definition is that of Scott domains, often simply called domains, which are {omega-algebraic}, consistently complete CPOs. There are domain-theoretic computational models in other branches of mathematics including dynamical systems, fractals, measure theory, integration theory, probability theory, and stochastic processes. See also abstract interpretation, bottom, {pointed domain}. (1999-12-09)

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