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)