Introduction Wagner Hierarchy Muller k-Acceptors Labeled Trees and Forests Main Results Proof Sketch of 1 First-Order Theories Fine Hierarchy of ω-Regular k -Partitions Victor Selivanov A.P. Ershov Institute of Informatics Systems Siberian Division Russian Academy of Sciences CiE-2011, Sofia, June 28, 2011 Victor Selivanov Fine Hierarchy of ω-Regular k-Partitions