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. Suppose {displaystyle (L,sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, and let {displaystyle f:Lto L} be a Scott-continuous (and therefore monotone) function. Then {displaystyle 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, the theorem states that {displaystyle {textrm {lfp}}(f)=sup left(left{f^{n}(bot )mid nin mathbb {N} right}right)} where {displaystyle {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 (also, it pertains to monotone functions on complete lattices), this result is often attributed to Alfred Tarski who proves it for additive functions [1] Moreover, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations.[2] Proof[3] We first have to show that the ascending Kleene chain of {displaystyle f} exists in {displaystyle L} . To show that, we prove the following: Lemma. If {displaystyle L} is a dcpo with a least element, and {displaystyle f:Lto L} is Scott-continuous, then {displaystyle f^{n}(bot )sqsubseteq f^{n+1}(bot ),nin mathbb {N} _{0}} Proof. We use induction: Assume n = 0. Then {displaystyle f^{0}(bot )=bot sqsubseteq f^{1}(bot ),} since {displaystyle bot } is the least element. Assume n > 0. Then we have to show that {displaystyle f^{n}(bot )sqsubseteq f^{n+1}(bot )} . By rearranging we get {displaystyle f(f^{n-1}(bot ))sqsubseteq f(f^{n}(bot ))} . By inductive assumption, we know that {displaystyle f^{n-1}(bot )sqsubseteq f^{n}(bot )} 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: {displaystyle mathbb {M} ={bot ,f(bot ),f(f(bot )),ldots }.} From the definition of a dcpo it follows that {displaystyle mathbb {M} } has a supremum, call it {displaystyle m.} What remains now is to show that {displaystyle m} is the least fixed-point.

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

The proof that {displaystyle 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 {displaystyle 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(D)} is smaller than that same element of {displaystyle L} ). This is done by induction: Assume {displaystyle k} is some fixed-point of {displaystyle f} . We now prove by induction over {displaystyle i} that {displaystyle forall iin mathbb {N} :f^{i}(bot )sqsubseteq k} . The base of the induction {displaystyle (i=0)} obviously holds: {displaystyle f^{0}(bot )=bot sqsubseteq k,} since {displaystyle bot } is the least element of {displaystyle L} . As the induction hypothesis, we may assume that {displaystyle f^{i}(bot )sqsubseteq k} . We now do the induction step: From the induction hypothesis and the monotonicity of {displaystyle f} (again, implied by the Scott-continuity of {displaystyle f} ), we may conclude the following: {displaystyle f^{i}(bot )sqsubseteq k~implies ~f^{i+1}(bot )sqsubseteq f(k).} Now, by the assumption that {displaystyle k} is a fixed-point of {displaystyle f,} we know that {displaystyle f(k)=k,} and from that we get {displaystyle f^{i+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, I.; Griffor, E. R. (1994). Mathematical Theory of Domains by V. Stoltenberg-Hansen. Cambridge University Press. pp. 24. doi:10.1017/cbo9781139166386. ISBN 0521383447. Categories: Order theoryFixed-point theorems