Knaster-Tarski fixed point theorem for complete partial order

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. set theory. • He was famous for his sense of humour. ** Alfred Tarski original name Alfred Teitelbaum (1901 - 1983) • Polish logician • Tarski made contributions to algebra, mathematical logic, set theory symbolic 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) Sds[if b then S1 else S2] =cond( B[b], Sds[S1], Sds[S2]) while b do S  if b then (S; while b do S) else skip Sds[ while b do S]= cond( B[b], Sds[ while b do S] o Sds[S], id) Sds[ while b do S]= FIX F where F g = cond( B[b], g o Sds[S],id)