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) N="left">IN additioN suitable coNteNts:
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