  Motivated by the search for a logic for polynomial time, we study rank logic
(FPR) which extends fixed-point logic with counting (FPC) by operators that
determine the rank of matrices over finite fields. While FPR can express most
of the known queries that separate FPC from PTIME, nearly nothing was known
about the limitations of its expressive power.
  In our first main result we show that the extensions of FPC by rank operators
over different prime fields are incomparable. This solves an open question
posed by Dawar and Holm and also implies that rank logic, in its original
definition with a distinct rank operator for every field, fails to capture
polynomial time. In particular we show that the variant of rank logic FPR* with
an operator that uniformly expresses the matrix rank over finite fields is more
expressive than FPR.
  One important step in our proof is to consider solvability logic FPS which is
the analogous extension of FPC by quantifiers which express the solvability
problem for linear equation systems over finite fields. Solvability logic can
easily be embedded into rank logic, but it is open whether it is a strict
fragment. In our second main result we give a partial answer to this question:
in the absence of counting, rank operators are strictly more expressive than
solvability quantifiers.
