ascending Kleene chain - definizione. Che cos'è ascending Kleene chain
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Cosa (chi) è ascending Kleene chain - definizione

Kleene fixed point theorem; Ascending Kleene chain; Kleene fixpoint theorem; Kleene Fixed-Point Theorem
  • interval]] [0,7] with the usual order

Stephen Cole Kleene         
AMERICAN MATHEMATICIAN AND THEORETICAL COMPUTER SCIENTIST
Kleene, Stephen Cole; Stephen Kleene; S. C. Kleene; Kleene, S.C.; Kleene; Stephen C. Kleene
Kleene star         
UNARY OPERATION ON SETS OF STRINGS, USED IN REGULAR EXPRESSIONS FOR "ZERO OR MORE REPETITIONS"
Kleene closure; Kleene plus; Star operation; Σ*; Kleene operator; Kleene operators; Star closure
<text> (Or "Kleene closure", named after Stephen Kleene) The postfix "*" operator used in regular expressions, Extended Backus-Naur Form, and similar formalisms to specify a match for zero or more occurrences of the preceding expression. For example, the regular expression "be*t" would match the string "bt", "bet", "beet", "beeeeet", and so on. (2000-07-26)
Kleene star         
UNARY OPERATION ON SETS OF STRINGS, USED IN REGULAR EXPRESSIONS FOR "ZERO OR MORE REPETITIONS"
Kleene closure; Kleene plus; Star operation; Σ*; Kleene operator; Kleene operators; Star closure
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics,

Wikipedia

Kleene fixed-point theorem

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 ( L , ) {\displaystyle (L,\sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, and let f : L L {\displaystyle f:L\to L} be a Scott-continuous (and therefore monotone) function. Then f {\displaystyle f} has a least fixed point, which is the supremum of the ascending Kleene chain of f . {\displaystyle f.}

The ascending Kleene chain of f is the chain

f ( ) f ( f ( ) ) f n ( ) {\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

lfp ( f ) = sup ( { f n ( ) n N } ) {\displaystyle {\textrm {lfp}}(f)=\sup \left(\left\{f^{n}(\bot )\mid n\in \mathbb {N} \right\}\right)}

where lfp {\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 Moreover, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations.