domain theory
eory> 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 ef="module.php?name=Lexikon&file=search&eid=1&query=denotational semantics">denotational semantics in computer science. In ef="module.php?name=Lexikon&file=search&eid=1&query=denotational semantics">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 ef="module.php?name=Lexikon&file=search&eid=1&query=LaTeX">LaTeX as ef="module.php?name=Lexikon&file=search&eid=1&query=subseteq">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 ef="module.php?name=Lexikon&file=search&eid=1&query=pure lambda-calculus">pure lambda-calculus all objects are functions or ef="module.php?name=Lexikon&file=search&eid=1&query=application">applications of functions to other functions. To represent the meaning of such programs, we must solve the ef="module.php?name=Lexikon&file=search&eid=1&query=recursive">recursive equation over domains, D = D -> D which states that domain D is (ef="module.php?name=Lexikon&file=search&eid=1&query=isomorphic">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 ef="module.php?name=Lexikon&file=search&eid=1&query=omega-algebraic">omega-algebraic, ef="module.php?name=Lexikon&file=search&eid=1&query=consistently complete">consistently complete ef="module.php?name=Lexikon&file=search&eid=1&query=CPO">CPOs. There are domain-theoretic computational models in other branches of mathematics including ef="module.php?name=Lexikon&file=search&eid=1&query=dynamical systems">dynamical systems, ef="module.php?name=Lexikon&file=search&eid=1&query=fractals">fractals, ef="module.php?name=Lexikon&file=search&eid=1&query=measure theory">measure theory, ef="module.php?name=Lexikon&file=search&eid=1&query=integration theory">integration theory, ef="module.php?name=Lexikon&file=search&eid=1&query=probability theory">probability theory, and ef="module.php?name=Lexikon&file=search&eid=1&query=stochastic processes">stochastic processes. See also ef="module.php?name=Lexikon&file=search&eid=1&query=abstract interpretation">abstract interpretation, ef="module.php?name=Lexikon&file=search&eid=1&query=bottom">bottom, {pointed domain}. (1999-12-09) e="border-width:thin; border-color:#333333; border-style:dashed; padding:5px;" align="left">In addition suitable contents:
[ ef="module.php?name=Lexikon&op=content&tid=31">2 ] [ ef="module.php?name=Lexikon&op=content&tid=134">= ] [ ef="module.php?name=Lexikon&op=content&tid=177">abstract interpretation ] [ ef="module.php?name=Lexikon&op=content&tid=396">ag ] [ ef="module.php?name=Lexikon&op=content&tid=411">ai ] [ ef="module.php?name=Lexikon&op=content&tid=433">al ] [ ef="module.php?name=Lexikon&op=content&tid=464">algebra ] [ ef="module.php?name=Lexikon&op=content&tid=466">algebraic ] [ ef="module.php?name=Lexikon&op=content&tid=544">am ] [ ef="module.php?name=Lexikon&op=content&tid=592">an ] [ ef="module.php?name=Lexikon&op=content&tid=683">app ] [ ef="module.php?name=Lexikon&op=content&tid=700">application ] [ ef="module.php?name=Lexikon&op=content&tid=740">ar ] [ ef="module.php?name=Lexikon&op=content&tid=743">arc ] [ ef="module.php?name=Lexikon&op=content&tid=800">as ] [ ef="module.php?name=Lexikon&op=content&tid=894">at ] [ ef="module.php?name=Lexikon&op=content&tid=996">av ] [ ef="module.php?name=Lexikon&op=content&tid=1026">b ] [ ef="module.php?name=Lexikon&op=content&tid=1034">ba ] [ ef="module.php?name=Lexikon&op=content&tid=1177">bd ] [ ef="module.php?name=Lexikon&op=content&tid=1181">be ] [ ef="module.php?name=Lexikon&op=content&tid=1269">bi ] [ ef="module.php?name=Lexikon&op=content&tid=1368">bj ] [ ef="module.php?name=Lexikon&op=content&tid=1444">bo ] [ ef="module.php?name=Lexikon&op=content&tid=1501">bot ] [ ef="module.php?name=Lexikon&op=content&tid=1503">bottom ] [ ef="module.php?name=Lexikon&op=content&tid=1535">br ] [ ef="module.php?name=Lexikon&op=content&tid=1550">branch ] [ ef="module.php?name=Lexikon&op=content&tid=1606">bs ] [ ef="module.php?name=Lexikon&op=content&tid=1695">by ] [ ef="module.php?name=Lexikon&op=content&tid=1708">C ] [ ef="module.php?name=Lexikon&op=content&tid=1724">ca ] [ ef="module.php?name=Lexikon&op=content&tid=1863">cat ] [ ef="module.php?name=Lexikon&op=content&tid=2001">ch ] [ ef="module.php?name=Lexikon&op=content&tid=2099">ci ] [ ef="module.php?name=Lexikon&op=content&tid=2138">cl ] [ ef="module.php?name=Lexikon&op=content&tid=2247">co ] [ ef="module.php?name=Lexikon&op=content&tid=2330">com ] [ ef="module.php?name=Lexikon&op=content&tid=2441">complete ] [ ef="module.php?name=Lexikon&op=content&tid=2491">computer ] [ ef="module.php?name=Lexikon&op=content&tid=2545">con ] [ ef="module.php?name=Lexikon&op=content&tid=2594">connect ] [ ef="module.php?name=Lexikon&op=content&tid=2606">cons ] [ ef="module.php?name=Lexikon&op=content&tid=2609">consistently complete ] [ ef="module.php?name=Lexikon&op=content&tid=2767">CP ] [ ef="module.php?name=Lexikon&op=content&tid=2900">cu ] [ ef="module.php?name=Lexikon&op=content&tid=2976">D ] [ ef="module.php?name=Lexikon&op=content&tid=3151">de ] [ ef="module.php?name=Lexikon&op=content&tid=3282">denotational semantics ] [ ef="module.php?name=Lexikon&op=content&tid=3371">diff ] [ ef="module.php?name=Lexikon&op=content&tid=3436">ding ] [ ef="module.php?name=Lexikon&op=content&tid=3565">do ] [ ef="module.php?name=Lexikon&op=content&tid=3595">domain ] [ ef="module.php?name=Lexikon&op=content&tid=3752">du ] [ ef="module.php?name=Lexikon&op=content&tid=3865">ec ] [ ef="module.php?name=Lexikon&op=content&tid=3896">ed ] [ ef="module.php?name=Lexikon&op=content&tid=3929">ee ] [ ef="module.php?name=Lexikon&op=content&tid=3946">eg ] [ ef="module.php?name=Lexikon&op=content&tid=4008">element ] [ ef="module.php?name=Lexikon&op=content&tid=4148">er ] [ ef="module.php?name=Lexikon&op=content&tid=4150">era ] [ ef="module.php?name=Lexikon&op=content&tid=4171">es ] [ ef="module.php?name=Lexikon&op=content&tid=4199">et ] [ ef="module.php?name=Lexikon&op=content&tid=4273">excl ] [ ef="module.php?name=Lexikon&op=content&tid=4497">fi ] [ ef="module.php?name=Lexikon&op=content&tid=4520">file ] [ ef="module.php?name=Lexikon&op=content&tid=4595">fix ] [ ef="module.php?name=Lexikon&op=content&tid=4597">fixed point ] [ ef="module.php?name=Lexikon&op=content&tid=4700">fo ] [ ef="module.php?name=Lexikon&op=content&tid=4727">for ] [ ef="module.php?name=Lexikon&op=content&tid=4828">fr ] [ ef="module.php?name=Lexikon&op=content&tid=4829">fractal ] [ ef="module.php?name=Lexikon&op=content&tid=4940">function ] [ ef="module.php?name=Lexikon&op=content&tid=4989">ga ] [ ef="module.php?name=Lexikon&op=content&tid=5057">ge ] [ ef="module.php?name=Lexikon&op=content&tid=5134">gh ] [ ef="module.php?name=Lexikon&op=content&tid=5291">gr ] [ ef="module.php?name=Lexikon&op=content&tid=5403">gu ] [ ef="module.php?name=Lexikon&op=content&tid=5434">h ] [ ef="module.php?name=Lexikon&op=content&tid=5540">hat ] [ ef="module.php?name=Lexikon&op=content&tid=5722">hose ] [ ef="module.php?name=Lexikon&op=content&tid=5768">hr ] [ ef="module.php?name=Lexikon&op=content&tid=5779">ht ] [ ef="module.php?name=Lexikon&op=content&tid=5931">id ] [ ef="module.php?name=Lexikon&op=content&tid=5956">ie ] [ ef="module.php?name=Lexikon&op=content&tid=5986">iff ] [ ef="module.php?name=Lexikon&op=content&tid=6013">il ] [ ef="module.php?name=Lexikon&op=content&tid=6064">in ] [ ef="module.php?name=Lexikon&op=content&tid=6068">inc ] [ ef="module.php?name=Lexikon&op=content&tid=6194">int ] [ ef="module.php?name=Lexikon&op=content&tid=6211">integration ] [ ef="module.php?name=Lexikon&op=content&tid=6413">io ] [ ef="module.php?name=Lexikon&op=content&tid=6482">is ] [ ef="module.php?name=Lexikon&op=content&tid=6542">isomorphic ] [ ef="module.php?name=Lexikon&op=content&tid=6558">it ] [ ef="module.php?name=Lexikon&op=content&tid=6789">ke ] [ ef="module.php?name=Lexikon&op=content&tid=6792">ken ] [ ef="module.php?name=Lexikon&op=content&tid=6918">la ] [ ef="module.php?name=Lexikon&op=content&tid=6939">lambda-calculus ] [ ef="module.php?name=Lexikon&op=content&tid=6950">language ] [ ef="module.php?name=Lexikon&op=content&tid=6989">LaTeX ] [ ef="module.php?name=Lexikon&op=content&tid=7014">lc ] [ ef="module.php?name=Lexikon&op=content&tid=7091">Lex ] [ ef="module.php?name=Lexikon&op=content&tid=7107">li ] [ ef="module.php?name=Lexikon&op=content&tid=7399">ls ] [ ef="module.php?name=Lexikon&op=content&tid=7415">lu ] [ ef="module.php?name=Lexikon&op=content&tid=7437">lv ] [ ef="module.php?name=Lexikon&op=content&tid=7441">ly ] [ ef="module.php?name=Lexikon&op=content&tid=7463">ma ] [ ef="module.php?name=Lexikon&op=content&tid=7582">man ] [ ef="module.php?name=Lexikon&op=content&tid=7710">measure ] [ ef="module.php?name=Lexikon&op=content&tid=7721">meg ] [ ef="module.php?name=Lexikon&op=content&tid=7722">mega- ] [ ef="module.php?name=Lexikon&op=content&tid=8019">mm ] [ ef="module.php?name=Lexikon&op=content&tid=8032">mo ] [ ef="module.php?name=Lexikon&op=content&tid=8040">mod ] [ ef="module.php?name=Lexikon&op=content&tid=8046">mode ] [ ef="module.php?name=Lexikon&op=content&tid=8050">model ] [ ef="module.php?name=Lexikon&op=content&tid=8079">module ] [ ef="module.php?name=Lexikon&op=content&tid=8167">mp ] [ ef="module.php?name=Lexikon&op=content&tid=8228">ms ] [ ef="module.php?name=Lexikon&op=content&tid=8258">mu ] [ ef="module.php?name=Lexikon&op=content&tid=8386">na ] [ ef="module.php?name=Lexikon&op=content&tid=8460">nc ] [ ef="module.php?name=Lexikon&op=content&tid=8472">ne ] [ ef="module.php?name=Lexikon&op=content&tid=8627">ng ] [ ef="module.php?name=Lexikon&op=content&tid=8630">ni ] [ ef="module.php?name=Lexikon&op=content&tid=8660">nl ] [ ef="module.php?name=Lexikon&op=content&tid=8672">nn ] [ ef="module.php?name=Lexikon&op=content&tid=8675">no ] [ ef="module.php?name=Lexikon&op=content&tid=8760">ns ] [ ef="module.php?name=Lexikon&op=content&tid=8820">O ] [ ef="module.php?name=Lexikon&op=content&tid=8838">object ] [ ef="module.php?name=Lexikon&op=content&tid=8964">om ] [ ef="module.php?name=Lexikon&op=content&tid=9014">op ] [ ef="module.php?name=Lexikon&op=content&tid=9071">operator ] [ ef="module.php?name=Lexikon&op=content&tid=9115">ordering ] [ ef="module.php?name=Lexikon&op=content&tid=9204">pa ] [ ef="module.php?name=Lexikon&op=content&tid=9457">pe ] [ ef="module.php?name=Lexikon&op=content&tid=9550">ph ] [ ef="module.php?name=Lexikon&op=content&tid=9651">pl ] [ ef="module.php?name=Lexikon&op=content&tid=9738">ply ] [ ef="module.php?name=Lexikon&op=content&tid=9762">point ] [ ef="module.php?name=Lexikon&op=content&tid=9765">pointed domain ] [ ef="module.php?name=Lexikon&op=content&tid=9908">pr ] [ ef="module.php?name=Lexikon&op=content&tid=9995">process ] [ ef="module.php?name=Lexikon&op=content&tid=10023">program ] [ ef="module.php?name=Lexikon&op=content&tid=10042">programming ] [ ef="module.php?name=Lexikon&op=content&tid=10044">programming language ] [ ef="module.php?name=Lexikon&op=content&tid=10178">pure lambda-calculus ] [ ef="module.php?name=Lexikon&op=content&tid=10244">quarter ] [ ef="module.php?name=Lexikon&op=content&tid=10253">query ] [ ef="module.php?name=Lexikon&op=content&tid=10364">rc ] [ ef="module.php?name=Lexikon&op=content&tid=10385">re ] [ ef="module.php?name=Lexikon&op=content&tid=10433">recursive ] [ ef="module.php?name=Lexikon&op=content&tid=10508">relation ] [ ef="module.php?name=Lexikon&op=content&tid=10754">rl ] [ ef="module.php?name=Lexikon&op=content&tid=10767">ro ] [ ef="module.php?name=Lexikon&op=content&tid=10887">ru ] [ ef="module.php?name=Lexikon&op=content&tid=10918">S ] [ ef="module.php?name=Lexikon&op=content&tid=11010">sc ] [ ef="module.php?name=Lexikon&op=content&tid=11073">Scott domain ] [ ef="module.php?name=Lexikon&op=content&tid=11150">se ] [ ef="module.php?name=Lexikon&op=content&tid=11215">semantics ] [ ef="module.php?name=Lexikon&op=content&tid=11281">set ] [ ef="module.php?name=Lexikon&op=content&tid=11292">set theory ] [ ef="module.php?name=Lexikon&op=content&tid=11376">si ] [ ef="module.php?name=Lexikon&op=content&tid=11651">so ] [ ef="module.php?name=Lexikon&op=content&tid=11725">solution ] [ ef="module.php?name=Lexikon&op=content&tid=11757">space ] [ ef="module.php?name=Lexikon&op=content&tid=11934">st ] [ ef="module.php?name=Lexikon&op=content&tid=11990">state ] [ ef="module.php?name=Lexikon&op=content&tid=12055">stochastic ] [ ef="module.php?name=Lexikon&op=content&tid=12109">struct ] [ ef="module.php?name=Lexikon&op=content&tid=12133">su ] [ ef="module.php?name=Lexikon&op=content&tid=12148">subseteq ] [ ef="module.php?name=Lexikon&op=content&tid=12162">suit ] [ ef="module.php?name=Lexikon&op=content&tid=12246">sy ] [ ef="module.php?name=Lexikon&op=content&tid=12312">system ] [ ef="module.php?name=Lexikon&op=content&tid=12359">T ] [ ef="module.php?name=Lexikon&op=content&tid=12369">table ] [ ef="module.php?name=Lexikon&op=content&tid=12588">th ] [ ef="module.php?name=Lexikon&op=content&tid=12602">theory ] [ ef="module.php?name=Lexikon&op=content&tid=12721">to ] [ ef="module.php?name=Lexikon&op=content&tid=12787">tr ] [ ef="module.php?name=Lexikon&op=content&tid=12896">tt ] [ ef="module.php?name=Lexikon&op=content&tid=12970">type ] [ ef="module.php?name=Lexikon&op=content&tid=12986">ua ] [ ef="module.php?name=Lexikon&op=content&tid=13175">us ] [ ef="module.php?name=Lexikon&op=content&tid=13252">va ] [ ef="module.php?name=Lexikon&op=content&tid=13260">value ] [ ef="module.php?name=Lexikon&op=content&tid=13310">ve ] [ ef="module.php?name=Lexikon&op=content&tid=13366">vi ] [ ef="module.php?name=Lexikon&op=content&tid=13891">X ] [ ef="module.php?name=Lexikon&op=content&tid=14024">Y ]
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