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Editor: Stephan Kreutzer

Erich Grädel, Wied Pakusa

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Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik
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unpublished

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
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... estion 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. 1 Introduction "Le roi est mort, vive le roi!" has been the traditional proclamation, in France and other countries, to announce not only the death of the monarch, but also the immediate installment of his successor on the throne. The purpose of this paper is to kill the rank logic FPR, in the form in which it has been proposed in [7], as a candidate for a logic for Ptime. The logic FPR extends fixed-point logic by operators rk p (for every prime p) which compute the rank of definable matrices over the prime field F p with p elements. Although rank logic is well-motivated, as a logic that strictly extends fixed-point logic with counting by the ability to express important properties of linear algebra, most notably the solvability of linear equation systems over finite fields, our results show that the choice of having a separate rank operator for every prime p leads to a significant deficiency of the logic. Indeed, it follows from our main theorem that even the uniform rank problem, of computing the rank of a given matrix over an arbitrary prime, cannot be expressed in FPR and thus separates FPR from Ptime. This also reveals that a more general variant of rank logic, which has already been proposed in [14, 15, 17] and which is based on a rank operator that takes not only the matrix but also the prime p as part of the input, is indeed strictly more powerful than FPR. Our result thus installs this new rank logic, denoted FPR * , as the rightful and distinctly more powerful successor of FPR as a potential candidate for a logic for Ptime. A full version of this paper can be found at [11].

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