View source for Kleene fixed-point theorem

{{Short description|Theorem in order theory and lattice theory}}
{{otheruses4|Kleene's fixed-point theorem in lattice theory|the fixed-point theorem in computability theory|Kleene's recursion theorem}}
[[File:Kleene fixpoint svg.svg|thumb|Computation of the least fixpoint of ''f''(''x'') = {{sfrac|1|10}}''x''<sup>2</sup>+[[arctangent|atan]](''x'')+1 using Kleene's theorem in the real [[interval (mathematics)|interval]] [0,7] with the usual order]]
In the [[mathematics|mathematical]] areas of [[order theory|order]] and [[lattice theory]], the '''Kleene fixed-point theorem''', named after American mathematician [[Stephen Cole Kleene]], states the following:

:'''Kleene Fixed-Point Theorem.''' Suppose <math>(L, \sqsubseteq)</math> is a [[complete partial order|directed-complete partial order]] (dcpo) with a least element, and let <math>f: L \to L</math> be a [[Scott continuity|Scott-continuous]] (and therefore [[monotone function|monotone]]) [[function (mathematics)|function]]. Then <math>f</math> has a [[least fixed point]], which is the [[supremum]] of the ascending Kleene chain of <math>f.</math>

The '''ascending Kleene chain''' of ''f'' is the [[chain (order theory)|chain]]

:<math>\bot  \sqsubseteq  f(\bot) \sqsubseteq f(f(\bot)) \sqsubseteq \cdots \sqsubseteq f^n(\bot) \sqsubseteq \cdots</math>

obtained by [[iterated function|iterating]] ''f'' on the [[least element]] ⊥ of ''L''. Expressed in a formula, the theorem states that

:<math>\textrm{lfp}(f) = \sup \left(\left\{f^n(\bot) \mid n\in\mathbb{N}\right\}\right)</math>

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

Although [[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 | year=1955 | issue=2 | pages=285&ndash;309| doi=10.2140/pjm.1955.5.285 }}, page 305.</ref> Moreover, the Kleene fixed-point theorem can be extended to [[monotone function]]s using transfinite iterations.<ref>{{cite journal | author=Patrick Cousot and Radhia Cousot | url=https://projecteuclid.org/euclid.pjm/1102785059 | title=Constructive versions of Tarski's fixed point theorems | journal = Pacific Journal of Mathematics | volume=82 | year=1979 | issue=1 | pages=43&ndash;57| doi=10.2140/pjm.1979.82.43 }}</ref>

== Proof ==
Source:<ref>{{Cite book|title=Mathematical Theory of Domains by V. Stoltenberg-Hansen|last1=Stoltenberg-Hansen|first1=V.|last2=Lindstrom|first2=I.|last3=Griffor|first3=E. R.|publisher=Cambridge University Press|year=1994|isbn=0521383447|pages=[https://archive.org/details/mathematicaltheo0000stol/page/24 24]|language=en|doi=10.1017/cbo9781139166386|url=https://archive.org/details/mathematicaltheo0000stol/page/24}}</ref>

We first have to show that the ascending Kleene chain of <math>f</math> exists in <math>L</math>. To show that, we prove the following:

:'''Lemma.''' If <math>L</math> is a dcpo with a least element, and <math>f: L \to L</math> is Scott-continuous, then <math>f^n(\bot) \sqsubseteq f^{n+1}(\bot), n \in \mathbb{N}_0</math>

:'''Proof.''' We use induction: 
:* Assume n = 0. Then <math>f^0(\bot) = \bot \sqsubseteq f^1(\bot),</math> since <math>\bot</math> is the least element.
:* Assume n > 0. Then we have to show that <math>f^n(\bot) \sqsubseteq f^{n+1}(\bot)</math>. By rearranging we get <math>f(f^{n-1}(\bot)) \sqsubseteq f(f^n(\bot))</math>. By inductive assumption, we know that <math>f^{n-1}(\bot) \sqsubseteq f^n(\bot)</math> holds, and because f is monotone (property of Scott-continuous functions), the result holds as well.

As a corollary of the Lemma we have the following directed ω-chain:

:<math>\mathbb{M} = \{ \bot, f(\bot), f(f(\bot)), \ldots\}.</math>

From the definition of a dcpo it follows that <math>\mathbb{M}</math> has a supremum, call it <math>m.</math> What remains now is to show that <math>m</math> is the least fixed-point.

First, we show that <math>m</math> is a fixed point, i.e. that <math>f(m) = m</math>. Because <math>f</math> is [[Scott continuity|Scott-continuous]], <math>f(\sup(\mathbb{M})) = \sup(f(\mathbb{M}))</math>, that is <math>f(m) = \sup(f(\mathbb{M}))</math>. Also, since <math>\mathbb{M} = f(\mathbb{M})\cup\{\bot\}</math> and because <math>\bot</math> has no influence in determining the supremum we have: <math>\sup(f(\mathbb{M})) = \sup(\mathbb{M})</math>. It follows that <math>f(m) = m</math>, making <math>m</math> a fixed-point of <math>f</math>.

The proof that <math>m</math> is in fact the ''least'' fixed point can be done by showing that any element in <math>\mathbb{M}</math> is smaller than any fixed-point of <math>f</math> (because by property of [[supremum]], if all elements of a set <math>D \subseteq L</math> are smaller than an element of <math>L</math> then also <math>\sup(D)</math> is smaller than that same element of <math>L</math>). This is done by induction: Assume <math>k</math> is some fixed-point of <math>f</math>. We now prove by induction over <math>i</math> that <math>\forall i \in \mathbb{N}: f^i(\bot) \sqsubseteq k</math>. The base of the induction <math>(i = 0)</math> obviously holds: <math>f^0(\bot) = \bot \sqsubseteq k,</math> since <math>\bot</math> is the least element of <math>L</math>. As the induction hypothesis, we may assume that <math>f^i(\bot) \sqsubseteq k</math>. We now do the induction step: From the induction hypothesis and the monotonicity of <math>f</math> (again, implied by the Scott-continuity of <math>f</math>), we may conclude the following: <math>f^i(\bot) \sqsubseteq k ~\implies~ f^{i+1}(\bot) \sqsubseteq f(k).</math> Now, by the assumption that <math>k</math> is a fixed-point of <math>f,</math> we know that <math>f(k) = k,</math> and from that we get <math>f^{i+1}(\bot) \sqsubseteq k.</math>

== See also ==
* Other [[fixed-point theorem]]s

== References ==
{{Reflist}}
[[Category:Order theory]]
[[Category:Fixed-point theorems]]