Two-Variable Logic with Counting and Trees

Abstract

We consider the two-variable logic with counting quantifiers (C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) interpreted over finite structures that contain two forests of ranked trees. This logic is strictly more expressive than standard C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and it is no longer a fragment of the first order logic. In particular, it can express that a structure is a ranked tree, a cycle or a connected graph of bounded degree. It is also strictly more expressive than the first-order logic with two variables and two successor relations of two finite linear orders. We give a decision procedure for the satisfiability problem for this logic. The procedure runs in NEXPTIME, which is optimal since the satisfiability problem for plain C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is NEXPTIME-complete.