Tree Canonization and Transitive Closure

We prove that tree isomorphism is not expressible in the language FO TC COUNT This is surprising since in the presence of ordering the language captures NL whereas tree isomorphism and canonization are in L Lin Our proof uses an Ehrenfeucht Fra ss e game for transitive closure logic with counting Gr a IL As a corresponding upper bound we show that tree canonization is expressible in FO COUNT logn The best previous upper bound had been FO COUNT n DM The lower bound remains true for bounded degree trees and we show that for bounded degree trees counting is not needed in the upper bound These results are the rst separations of the unordered versions of the logical languages for NL AC and ThC Our results were motivated by our conjecture in EI that FO TC COUNT LO NL i e that a one way local ordering su ced to capture NL We disprove this conjecture but we prove that a two way local ordering does su ce i e FO TC COUNT LO NL