1. 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)

2. 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.

3. Elementary Fixed Point Theorem

4. 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

5. 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.

6. 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)