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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