Kleene fixed-point theorem

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. Supponiamo {stile di visualizzazione (l,sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, e lascia {stile di visualizzazione f:Lto L} be a Scott-continuous (and therefore monotone) funzione. Quindi {stile di visualizzazione f} has a least fixed point, which is the supremum of the ascending Kleene chain of {stile di visualizzazione 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, il teorema lo afferma {stile di visualizzazione {textrm {lfp}}(f)=sup left(sinistra{f^{n}(bot )mid nin mathbb {N} Giusto}Giusto)} dove {stile di visualizzazione {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 (anche, it pertains to monotone functions on complete lattices), this result is often attributed to Alfred Tarski who proves it for additive functions [1] Inoltre, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations.[2] Prova[3] We first have to show that the ascending Kleene chain of {stile di visualizzazione f} exists in {stile di visualizzazione L} . To show that, we prove the following: Lemma. Se {stile di visualizzazione L} is a dcpo with a least element, e {stile di visualizzazione f:Lto L} is Scott-continuous, poi {stile di visualizzazione f^{n}(bot )sqsubseteq f^{n+1}(bot ),nin mathbb {N} _{0}} Prova. We use induction: Assume n = 0. Quindi {stile di visualizzazione f^{0}(bot )=bot sqsubseteq f^{1}(bot ),} da {displaystyle bot } is the least element. Assume n > 0. Then we have to show that {stile di visualizzazione f^{n}(bot )sqsubseteq f^{n+1}(bot )} . By rearranging we get {stile di visualizzazione f(f^{n-1}(bot ))sqsubseteq f(f^{n}(bot ))} . By inductive assumption, we know that {stile di visualizzazione f^{n-1}(bot )sqsubseteq f^{n}(bot )} tiene, 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: {displaystyle mathbb {M} ={bot ,f(bot ),f(f(bot )),ldot }.} From the definition of a dcpo it follows that {displaystyle mathbb {M} } has a supremum, chiamalo {displaystyle m.} What remains now is to show that {stile di visualizzazione m} is the least fixed-point.

Primo, lo mostriamo {stile di visualizzazione m} is a fixed point, cioè. Quello {stile di visualizzazione f(m)= m} . Perché {stile di visualizzazione f} is Scott-continuous, {stile di visualizzazione f(sup(mathbb {M} ))= sup(f(mathbb {M} ))} , questo è {stile di visualizzazione f(m)= sup(f(mathbb {M} ))} . Anche, da {displaystyle mathbb {M} =f(mathbb {M} )tazza {bot }} and because {displaystyle bot } has no influence in determining the supremum we have: {displaystyle sup(f(mathbb {M} ))= sup(mathbb {M} )} . Ne consegue che {stile di visualizzazione f(m)= m} , fabbricazione {stile di visualizzazione m} a fixed-point of {stile di visualizzazione f} .

The proof that {stile di visualizzazione m} is in fact the least fixed point can be done by showing that any element in {displaystyle mathbb {M} } is smaller than any fixed-point of {stile di visualizzazione f} (because by property of supremum, if all elements of a set {displaystyle Dsubseteq L} are smaller than an element of {stile di visualizzazione L} then also {displaystyle sup(D)} is smaller than that same element of {stile di visualizzazione L} ). This is done by induction: Assumere {stile di visualizzazione k} is some fixed-point of {stile di visualizzazione f} . We now prove by induction over {stile di visualizzazione i} Quello {displaystyle forall iin mathbb {N} :f^{io}(bot )sqsubseteq k} . The base of the induction {stile di visualizzazione (io=0)} obviously holds: {stile di visualizzazione f^{0}(bot )=bot sqsubseteq k,} da {displaystyle bot } is the least element of {stile di visualizzazione L} . As the induction hypothesis, we may assume that {stile di visualizzazione f^{io}(bot )sqsubseteq k} . We now do the induction step: From the induction hypothesis and the monotonicity of {stile di visualizzazione f} (ancora, implied by the Scott-continuity of {stile di visualizzazione f} ), we may conclude the following: {stile di visualizzazione f^{io}(bot )sqsubseteq k~implies ~f^{io+1}(bot )sqsubseteq f(K).} Adesso, by the assumption that {stile di visualizzazione k} is a fixed-point of {stile di visualizzazione f,} we know that {stile di visualizzazione f(K)= k,} and from that we get {stile di visualizzazione f^{io+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., pagina 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, IO.; Griffor, e. R. (1994). Mathematical Theory of Domains by V. Stoltenberg-Hansen. Cambridge University Press. pp. 24. doi:10.1017/cbo9781139166386. ISBN 0521383447. Categorie: Order theoryFixed-point theorems