Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs

We consider Choiceless Polynomial Time ( C ̃ PT ), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate fixed-point logic with counting ( IFP + C ) from P . This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets used, even in C ̃ PT(Card) , an extension of C ̃ PT with counting.