domain theory

domain theory

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domain theory

Source: 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)