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Knaster-Tarski fixed point theorem for complete partial order **************** contents **************** Who are Knaster-Tarski? What is elementary fixed point theorem? What is complete partial order? What is Knaster-Tarski fixed point theorem for complete partial order? Why do we have to know it? (applications in CS)

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Who are Knaster-Tarski? ** Bronisław Knaster (1893–1990 ) Polish mathematician He worked on topology, continuum.topologycontinuum set theory.set theory He was famous for his sense of humour. ** Alfred Tarski original name Alfred Teitelbaum (1901 - 1983) Polish logicianlogician Tarski made contributions to algebra,algebra mathematical logicmathematical logic, set theoryset theory symbolic logicsymbolic logic.

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Elementary Fixed Point Theorem

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What is complete partial order? Let (P, ≤) be a partial order –There is, in general, no reason for greatest lower bounds and least upper bounds to exist. –P is a complete partial order if every subset has both greatest lower bounds and least upper bounds

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What is Knaster-Tarski fixed point theorem Let(P,≤) be a complete ordered set and F: P P monotone. Then the set of fixed points of F, Fix(F), is not empty, that is F has a fixed point. Moreover, Fix(F) is a complete ordered subset of P. In particular, it has the least and greatest elements.

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Why do we have to know it? (applications in CS) System of equations in knowledge – base systems( database) Denotational semantics in programming languages ex) S ds [if b then S 1 else S 2 ] =cond( B[b], S ds [S 1 ], S ds [S 2 ]) while b do S if b then (S; while b do S) else skip S ds [ while b do S]= cond( B[b], S ds [ while b do S] o S ds [S], id) S ds [ while b do S]= FIX F where F g = cond( B[b], g o S ds [S],id)

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