Ptime Canonization for Two Variables with Counting

We consider innnitary logic with two variable symbols and counting quantiiers, C 2 , and its intersection with Ptime on nite relational structures. In particular we exhibit a Ptime canonization procedure for nite relational structures which provides unique representatives up to equivalence in C 2. As a consequence we obtain a recursive presentation for the class of all those queries on arbitrary nite relational structures which are both Ptime and deenable in C 2. The proof renders a succinct normal form representation of this non-trivial semantically deened fragment of Ptime. Through specializations of the proof techniques similar results apply with respect to the logic L 2 , innnitary logic with two variable symbols, itself.