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