Piecewise testable tree languages
  This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma_1 sentences. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma_1 sentences if and only if its syntactic monoid is J-trivial. 
Miko{\l}aj Boja\'nczyk (University of Warsaw), Luc Segoufin (INRIA),   Howard Straubing (Boston college)
cs.FL