Rank logic is dead, long live rank logic!
Erich Gr\"adel, Wied Pakusa

TL;DR
This paper investigates the limitations of rank logic (FPR) in capturing polynomial time, demonstrating that extensions over different fields are incomparable and that a uniform rank operator increases expressiveness beyond FPR.
Contribution
It proves that rank logic extensions over different fields are incomparable and shows that a uniform rank operator enhances expressiveness beyond FPR, solving open questions in the field.
Findings
Extensions of FPC by rank operators over different fields are incomparable.
Rank logic with a uniform rank operator is more expressive than FPR.
Rank operators are strictly more expressive than solvability quantifiers without counting.
Abstract
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…
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