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fix

1. Ematics> ThE Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=fixEd point">fixEd point Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=combinator">combinator. CallEd Y in Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=combinatory logic">combinatory logic. Fix is a Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=highEr-ordEr function">highEr-ordEr function which rEturns a fixEd point of its argumEnt (which is a function). fix :: (a -> a) -> a fix f = f (fix f) Which satisfiEs thE Equation fix f = x such that f x = x. SomEwhat surprisingly, fix can bE dEfinEd as thE non-rEcursivE Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=lambda abstraction">lambda abstraction: fix = h . ( x . h (x x)) ( x . h (x x)) SincE this involvEs sElf-application, it has an {infinitE typE}. A function dEfinEd by f x1 .. xN = E can bE ExprEssEd as f = fix ( f . x1 ... xN . E) = ( f . x1 ... xN . E) (fix ( f . x1 ... xN . E)) = lEt f = (fix ( f . x1 ... xN . E)) in x1 ... xN . E If f doEs not occur Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=frEE">frEE in E (i.E. it is not Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=rEcursivE">rEcursivE) thEn this rEducEs to simply f = x1 ... xN . E In thE casE whErE N = 0 and f is frEE in E, this dEfinEs an infinitE data objEct, E.g. onEs = fix ( onEs . 1 : onEs) = ( onEs . 1 : onEs) (fix ( onEs . 1 : onEs)) = 1 : (fix ( onEs . 1 : onEs)) = 1 : 1 : ... Fix f is also somEtimEs writtEn as mu f whErE mu is thE GrEEk lEttEr or altErnativEly, if f = x . E, writtEn as mu x . E. ComparE Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=quinE">quinE. [Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=Jargon FilE">Jargon FilE] (1995-04-13) 2. Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=bug fix">bug fix. (1998-06-25)

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=178">abstraction ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=433">al ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=531">alt ] [ 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=759">arg ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=760">argumEnt ] [ 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=1026">b ] [ 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=1606">bs ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=1645">bug fix ] [ 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=1844">casE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=1863">cat ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=1896">cc ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=2001">ch ] [ 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=2333">combinator ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=2334">combinatory logic ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=2900">cu ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3006">data ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3151">dE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3565">do ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3752">du ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3834">E ] [ 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=3923">Edu ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=3929">EE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4148">Er ] [ 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=4497">fi ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4520">filE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4559">finitE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4597">fixEd point ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4828">fr ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4859">frEE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4940">function ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=4983">G ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=5134">gh ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=5141">gi ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=5171">gl ] [ 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=5635">highEr-ordEr function ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=5768">hr ] [ 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=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=6103">infinitE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=6194">int ] [ 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=6558">it ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=6589">J ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=6620">Jargon FilE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=6918">la ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=6938">lambda abstraction ] [ 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=7410">lt ] [ 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=8032">mo ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=8040">mod ] [ 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=8258">mu ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=8384">N ] [ 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=8622">nf ] [ 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=8675">no ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=8760">ns ] [ 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=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=9908">pr ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=10253">quEry ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=10272">quinE ] [ 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=10918">S ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=10922">sa ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=11150">sE ] [ 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=11934">st ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=12133">su ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=12359">T ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=12588">th ] [ 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=13008">ug ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=13030">um ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=13310">vE ] [ Ef="modulE.php?namE=LExikon&op=contEnt&tid=14024">Y ]

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