We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given $\Delta _0$-formula $\varphi (x)$; the parameter is the size of $\varphi $. This parameterization focusses attention on instances where n is large compared to the size of $\varphi $. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf {AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta _0$ proves the MRDP theorem in a certain weak sense.

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, almost 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 ${\text{FPR}}^{\text{*}}$ 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.