# Théorème du point fixe de Kleene Kleene fixed-point theorem This article is about Kleene's fixed-point theorem in lattice theory. For the fixed-point theorem in computability theory, see Kleene's recursion theorem. Computation of the least fixpoint of f(X) = 1 / 10 x2+atan(X)+1 using Kleene's theorem in the real interval [0,7] with the usual order In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem. Supposer {style d'affichage (L,sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, et laissez {style d'affichage f:Lto L} be a Scott-continuous (and therefore monotone) fonction. Alors {style d'affichage f} has a least fixed point, which is the supremum of the ascending Kleene chain of {displaystyle f.} The ascending Kleene chain of f is the chain {displaystyle bot sqsubseteq f(bot )sqsubseteq f(F(bot ))sqsubseteq cdots sqsubseteq f^{n}(bot )sqsubseteq cdots } obtained by iterating f on the least element ⊥ of L. Expressed in a formula, le théorème dit que {style d'affichage {textrm {lfp}}(F)=sup left(la gauche{f ^{n}(bot )mid nin mathbb {N} droit}droit)} où {style d'affichage {textrm {lfp}}} 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 (aussi, it pertains to monotone functions on complete lattices), this result is often attributed to Alfred Tarski who proves it for additive functions  En outre, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations. Preuve We first have to show that the ascending Kleene chain of {style d'affichage f} exists in {displaystyle L} . To show that, we prove the following: Lemme. Si {displaystyle L} is a dcpo with a least element, et {style d'affichage f:Lto L} is Scott-continuous, alors {style d'affichage f^{n}(bot )sqsubseteq f^{n+1}(bot ),nin mathbb {N} _{0}} Preuve. We use induction: Assume n = 0. Alors {style d'affichage f^{0}(bot )=bot sqsubseteq f^{1}(bot ),} puisque {displaystyle bot } is the least element. Assume n > 0. Then we have to show that {style d'affichage f^{n}(bot )sqsubseteq f^{n+1}(bot )} . By rearranging we get {style d'affichage f(f ^{n-1}(bot ))sqsubseteq f(f ^{n}(bot ))} . By inductive assumption, we know that {style d'affichage f^{n-1}(bot )sqsubseteq f^{n}(bot )} détient, 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: {style d'affichage mathbb {M} ={bot ,F(bot ),F(F(bot )),ldots }.} From the definition of a dcpo it follows that {style d'affichage mathbb {M} } has a supremum, appeler {displaystyle m.} What remains now is to show that {style d'affichage m} is the least fixed-point.

Première, nous montrons que {style d'affichage m} is a fixed point, c'est à dire. ce {style d'affichage f(m)=m} . Car {style d'affichage f} is Scott-continuous, {style d'affichage f(sup(mathbb {M} ))= sup(F(mathbb {M} ))} , C'est {style d'affichage f(m)= sup(F(mathbb {M} ))} . Aussi, puisque {style d'affichage mathbb {M} =f(mathbb {M} )Coupe {bot }} and because {displaystyle bot } has no influence in determining the supremum we have: {displaystyle sup(F(mathbb {M} ))= sup(mathbb {M} )} . Il s'ensuit que {style d'affichage f(m)=m} , fabrication {style d'affichage m} a fixed-point of {style d'affichage f} .

The proof that {style d'affichage m} is in fact the least fixed point can be done by showing that any element in {style d'affichage mathbb {M} } is smaller than any fixed-point of {style d'affichage f} (because by property of supremum, if all elements of a set {displaystyle Dsubseteq L} are smaller than an element of {displaystyle L} then also {displaystyle sup(ré)} is smaller than that same element of {displaystyle L} ). This is done by induction: Présumer {style d'affichage k} is some fixed-point of {style d'affichage f} . We now prove by induction over {style d'affichage i} ce {displaystyle forall iin mathbb {N} :f ^{je}(bot )sqsubseteq k} . The base of the induction {style d'affichage (je=0)} obviously holds: {style d'affichage f^{0}(bot )=bot sqsubseteq k,} puisque {displaystyle bot } is the least element of {displaystyle L} . As the induction hypothesis, we may assume that {style d'affichage f^{je}(bot )sqsubseteq k} . We now do the induction step: From the induction hypothesis and the monotonicity of {style d'affichage f} (encore, implied by the Scott-continuity of {style d'affichage f} ), we may conclude the following: {style d'affichage f^{je}(bot )sqsubseteq k~implies ~f^{je+1}(bot )sqsubseteq f(k).} À présent, by the assumption that {style d'affichage k} is a fixed-point of {style d'affichage f,} we know that {style d'affichage f(k)=k,} and from that we get {style d'affichage f^{je+1}(bot )sqsubseteq k.} See also Other fixed-point theorems References ^ Alfred Tarski (1955). "A lattice-theoretical fixpoint theorem and its applications". Pacific Journal of Mathematics. 5:2: 285–309., page 305. ^ Patrick Cousot and Radhia Cousot (1979). "Constructive versions of Tarski's fixed point theorems". Pacific Journal of Mathematics. 82:1: 43–57. ^ Stoltenberg-Hansen, V; Lindstrom, JE.; Griffor, E. R. (1994). Mathematical Theory of Domains by V. Stoltenberg-Hansen. la presse de l'Universite de Cambridge. pp. 24. est ce que je:10.1017/cbo9781139166386. ISBN 0521383447. Catégories: Order theoryFixed-point theorems

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