On Lk(Q) Types and Boundedness of IFP(Q) on Finite Structures

We show that there is a class C of finite structures and a PTIME quantifier Q such that (1) IFP(Q) is bounded on C but L∞,ωω(Q) ≠ PFP(Q) ≠ IFP(Q) = FO(Q) over C. (2) For all k ≥ 2, IFPk(Q) is bounded but not uniformly bounded over C. (IFPk(Q) denotes the k-variable fragment of IFP(Q)) (3) For all k ≥ 2, IFPk(Q) is not uniformly bounded over C but IFPk(Q) = Lk(Q) over C.

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