Kleene fixed-point theorem

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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 is a directed-complete partial order (dcpo) with a least element, and let be a Scott-continuous (and therefore monotone) function. Then has a least fixed point, which is the supremum of the ascending Kleene chain of

The ascending Kleene chain of f is the chain

obtained by iterating f on the least element ⊥ of L. Expressed in a formula, the theorem states that

where 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], page 305. Moreover, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations [2].

Proof[3][edit]

We first have to show that the ascending Kleene chain of exists in . To show that, we prove the following:

Lemma. If is a dcpo with a least element, and is Scott-continuous, then
Proof. We use induction:
  • Assume n = 0. Then since is the least element.
  • Assume n > 0. Then we have to show that . By rearranging we get . By inductive assumption, we know that 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:

From the definition of a dcpo it follows that has a supremum, call it What remains now is to show that is the least fixed-point.

First, we show that is a fixed point, i.e. that . Because is Scott-continuous, , that is . Also, since and because has no influence in determining the supremum we have: . It follows that , making a fixed-point of .

The proof that is in fact the least fixed point can be done by showing that any element in is smaller than any fixed-point of (because by property of supremum, if all elements of a set are smaller than an element of then also is smaller than that same element of ). This is done by induction: Assume is some fixed-point of . We now prove by induction over that . The base of the induction obviously holds: since is the least element of . As the induction hypothesis, we may assume that . We now do the induction step: From the induction hypothesis and the monotonicity of (again, implied by the Scott-continuity of ), we may conclude the following: Now, by the assumption that is a fixed-point of we know that and from that we get

See also[edit]

References[edit]

  1. ^ Alfred Tarski (1955). "A lattice-theoretical fixpoint theorem and its applications". Pacific Journal of Mathematics. 5:2: 285–309.
  2. ^ Patrick Cousot and Radhia Cousot (1979). "Constructive versions of Tarski's fixed point theorems". Pacific Journal of Mathematics. 82:1: 43–57.
  3. ^ Stoltenberg-Hansen, V.; Lindstrom, I.; Griffor, E. R. (1994). Mathematical Theory of Domains by V. Stoltenberg-Hansen. Cambridge University Press. p. 24. doi:10.1017/cbo9781139166386. ISBN 0521383447.