Weak MSO+U with Path Quantifiers over Infinite Trees
  This paper shows that over infinite trees, satisfiability is decidable for weak monadic second-order logic extended by the unbounding quantifier U and quantification over infinite paths. The proof is by reduction to emptiness for a certain automaton model, while emptiness for the automaton model is decided using profinite trees. 
Miko{\l}aj Boja\'nczyk
cs.LO