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.