RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

Erich Grädel,Wied Pakusa

doi:10.1017/jsl.2018.33

Abstract

AbstractMotivated 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. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power.In our first main result we show that the extensions ofFPCby 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 thanFPR.One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby 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.

Highlights

One important step in our proof is to consider solvability logic FPS which is the analogous extension of fixed-point logic with counting (FPC) by quantifiers which express the solvability problem for linear equation systems over finite fields
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
Together with the standard argument that inflationary fixed-points can be evaluated in polynomial time and the fact that the matrix rank over any field can be determined in polynomial time, this ensures that all the logics which we introduce in the following have polynomial-time data complexity

Summary

Introduction

“Le roi est mort, vive le roi!” has been the traditional proclamation, in France and other countries, to announce the death of the monarch, and the immediate installment of his successor on the throne. 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. Atserias, Bulatov and Dawar [2] proved that FPC cannot express the solvability of linear equation systems over any finite Abelian group It follows, that other problems from the field of linear algebra are not definable in FPC. (b) The extensions of fixed-point logic by rank operators over different prime fields are incomparable (Theorem 1), cf [14, 8, 15] We obtain these classes of structures Kq by generalising the well-known construction of Cai, Fürer and Immerman [4]. This last result separates solvability quantifiers and rank operators in the absence of counting

Logics with linear-algebraic operators

Separation results over different classes of fields

Discussion