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domain theory
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 functionspace 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 settheory.