Series: Penn State Logic Seminar Date: Tuesday, April 1, 2003 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Stephen G. Simpson, Penn State, Mathematics Title: Some Results Concerning Muchnik Degrees, part 3 Abstract: Let P and Q be sets of reals. P is said to be Muchnik reducible to Q if every member of Q Turing-computes a member of P. A Muchnik degree is an equivalence class of sets of reals under mutual Muchnik reducibility. It is easy to see that the Muchnik degrees form a distributive lattice under the partial ordering induced by Muchnik reducibility. Call this lattice L. We study not only L but also its distributive sublattice L_0 consisting of the Muchnik degrees of nonempty Pi^0_1 subsets of the closed unit interval [0,1]. In part 1 we introduce L and L_0. We also present Simpson's result that any two nonempty Pi^0_1 subsets of [0,1] belonging to the top element of L_0 are Turing degree isomorphic. In parts 2 and 3 we discuss some further topics concerning L_0: 1-random reals, applications of the hyperimmune-free basis theorem, applications of Arslanov's completeness criterion, the Sigma^0_3/Pi^0_1 lemma, 2-random reals, DNR functions, embedding the r. e. degrees.