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) Style="border-width:thin; border-color:#333333; border-Style:daShed; padding:5px;" align="left">In addition Suitable contentS:
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