Kleene fixed-point theorem: Difference between revisions

Revision as of 05:43, 6 October 2019

Revision as of 16:48, 6 November 2019

Line 15:

where <math>\textrm{lfp}</math> denotes the least fixed point.

This result is often attributed to [[Alfred Tarski]], butAlthough [[Tarski's fixed point theorem]]

does not consider how fixed points can be computed by iterating ''f'' from some seed (also, it pertains to [[monotone function]]s on [[complete lattices]]), this result is often attributed to [[Alfred Tarski]] who proves it for additive functions <ref>{{cite journal | author=Alfred Tarski | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044538 | title=A lattice-theoretical fixpoint theorem and its applications | journal = Pacific Journal of Mathematics | volume=5:2 | year=1955 | pages=285&ndash;309}}</ref>, page 305. Moreover, Kleene Fixed-Point Theorem can be extended to [[monotone function]]s using transfinite iterations <ref>{{cite journal | author=Patrick Cousot and Radhia Cousot | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044538 | title=Constructive versions of Tarski's fixed point theorems | journal = Pacific Journal of Mathematics | volume=82:1 | year=1979 | pages=43&ndash;57}}</ref>.

does not consider how fixed points can be computed by iterating ''f'' from some seed (also, it pertains to [[monotone function]]s on [[complete lattices]]).

== Proof<ref>{{Cite book|title=Mathematical Theory of Domains by V. Stoltenberg-Hansen|last=Stoltenberg-Hansen |first=V.| last2=Lindstrom |first2=I.|last3=Griffor|first3=E. R.|publisher=Cambridge University Press |year=1994 |isbn=0521383447|___location=|pages=24|language=en|doi=10.1017/cbo9781139166386|quote=}}</ref> ==