[1] : 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 = fd
        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
     
     
see also:
denotational semantics latex \subseteq pure lambda-calculus application recursive 
isomorphic function space fixed point set theory omega-algebraic 
consistently complete cpo dynamical systems fractals measure theory 
integration theory probability theory stochastic processes abstract interpretation bottom 
pointed domain 

Domain Theory >> D Words
Domain Theory, definition of term: Domain Theory
domain+theory_pag1.html

Domain >> D Words
Domain, definition of term: Domain
domain_pag1.html

Theory >> T Words
Theory, definition of term: Theory
theory_pag1.html

Domain Name >> D Words
Domain Name, definition of term: Domain Name
domain+name_pag1.html

Group Theory >> G Words
Group Theory, definition of term: Group Theory
group+theory_pag1.html

Kinetic Theory >> K Words
Kinetic Theory, definition of term: Kinetic Theory
kinetic+theory_pag1.html

Public Domain >> P Words
Public Domain, definition of term: Public Domain
public+domain_pag1.html

Set Theory >> S Words
Set Theory, definition of term: Set Theory
set+theory_pag1.html

Wave Theory >> W Words
Wave Theory, definition of term: Wave Theory
wave+theory_pag1.html

Glacier Theory >> G Words
Glacier Theory, definition of term: Glacier Theory
glacier+theory_pag1.html

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