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 <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=denotAtionAl semAntics">denotAtionAl semAnticsA> in computer science. In <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=denotAtionAl semAntics">denotAtionAl semAnticsA> 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 <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=LATeX">LATeXA> As <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=subseteq">subseteqA>) 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 <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=pure lAmbdA-cAlculus">pure lAmbdA-cAlculusA> All objects Are functions or <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=ApplicAtion">ApplicAtionA>s of functions to other functions. To represent the meAning of such progrAms, we must solve the <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=recursive">recursiveA> equAtion over domAins, D = D -> D which stAtes thAt domAin D is (<A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=isomorphic">isomorphicA> 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 <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=omegA-AlgebrAic">omegA-AlgebrAicA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=consistently complete">consistently completeA> <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=CPO">CPOA>s. There Are domAin-theoretic computAtionAl models in other brAnches of mAthemAtics including <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=dynAmicAl systems">dynAmicAl systemsA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=frActAls">frActAlsA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=meAsure theory">meAsure theoryA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=integrAtion theory">integrAtion theoryA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=probAbility theory">probAbility theoryA>, And <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=stochAstic processes">stochAstic processesA>. See Also <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=AbstrAct interpretAtion">AbstrAct interpretAtionA>, <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=bottom">bottomA>, {pointed domAin}. (1999-12-09) Align="left">In Addition suitAble contents:
[ <A href="module.php?nAme=Lexikon&op=content&tid=31">2A> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=134">=A> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=177">AbstrAct interpretAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=396">AgA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=411">AiA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=433">AlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=464">AlgebrAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=466">AlgebrAicA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=544">AmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=592">AnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=683">AppA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=700">ApplicAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=740">ArA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=743">ArcA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=800">AsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=894">AtA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=996">AvA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1026">bA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1034">bAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1177">bdA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1181">beA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1269">biA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1368">bjA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1444">boA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1501">botA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1503">bottomA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1535">brA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1550">brAnchA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1606">bsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1695">byA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1708">CA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1724">cAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1863">cAtA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2001">chA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2099">ciA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2138">clA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2247">coA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2330">comA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2441">completeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2491">computerA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2545">conA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2594">connectA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2606">consA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2609">consistently completeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2767">CPA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2900">cuA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2976">DA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3151">deA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3282">denotAtionAl semAnticsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3371">diffA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3436">dingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3565">doA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3595">domAinA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3752">duA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3865">ecA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3896">edA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3929">eeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3946">egA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4008">elementA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4148">erA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4150">erAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4171">esA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4199">etA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4273">exclA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4497">fiA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4520">fileA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4595">fixA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4597">fixed pointA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4700">foA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4727">forA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4828">frA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4829">frActAlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4940">functionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4989">gAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5057">geA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5134">ghA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5291">grA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5403">guA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5434">hA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5540">hAtA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5722">hoseA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5768">hrA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5779">htA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5931">idA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5956">ieA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5986">iffA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6013">ilA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6064">inA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6068">incA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6194">intA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6211">integrAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6413">ioA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6482">isA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6542">isomorphicA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6558">itA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6789">keA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6792">kenA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6918">lAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6939">lAmbdA-cAlculusA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6950">lAnguAgeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6989">LATeXA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7014">lcA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7091">LexA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7107">liA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7399">lsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7415">luA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7437">lvA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7441">lyA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7463">mAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7582">mAnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7710">meAsureA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7721">megA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7722">megA-A> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8019">mmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8032">moA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8040">modA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8046">modeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8050">modelA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8079">moduleA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8167">mpA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8228">msA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8258">muA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8386">nAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8460">ncA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8472">neA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8627">ngA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8630">niA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8660">nlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8672">nnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8675">noA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8760">nsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8820">OA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8838">objectA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8964">omA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9014">opA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9071">operAtorA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9115">orderingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9204">pAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9457">peA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9550">phA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9651">plA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9738">plyA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9762">pointA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9765">pointed domAinA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9908">prA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9995">processA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10023">progrAmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10042">progrAmmingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10044">progrAmming lAnguAgeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10178">pure lAmbdA-cAlculusA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10244">quArterA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10253">queryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10364">rcA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10385">reA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10433">recursiveA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10508">relAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10754">rlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10767">roA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10887">ruA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10918">SA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11010">scA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11073">Scott domAinA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11150">seA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11215">semAnticsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11281">setA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11292">set theoryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11376">siA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11651">soA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11725">solutionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11757">spAceA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11934">stA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11990">stAteA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12055">stochAsticA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12109">structA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12133">suA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12148">subseteqA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12162">suitA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12246">syA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12312">systemA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12359">TA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12369">tAbleA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12588">thA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12602">theoryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12721">toA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12787">trA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12896">ttA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12970">typeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12986">uAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13175">usA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13252">vAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13260">vAlueA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13310">veA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13366">viA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13891">XA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=14024">YA> ]
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