domain theory

1 definition found

domain theory - Free On-line Dictionary of Computing (26 May 2007) :

  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 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)