Hanf normal form for first-order logic with unary counting quantifiers

We study the existence of Hanf normal forms for extensions FO(Q) of first-order logic by sets Q ⊆ P(ℕ) of unary counting quantifiers. A formula is in Hanf normal form if it is a Boolean combination of formulas ζ(x̅) describing the isomorphism type of a local neighbourhood around its free variables x̅ and statements of the form “the number of witnesses y of ψ(y) belongs to (Q+k)” where Q ϵ Q, k ϵ ℕ, and ψ describes the isomorphism type of a local neighbourhood around its unique free variable y. We show that a formula from FO(Q) can be transformed into a formula in Hanf normal form that is equivalent on all structures of degree ≤ d if, and only if, all counting quantifiers occurring in the formula are ultimately periodic. This transformation can be carried out in worst-case optimal 3-fold exponential time. In particular, this yields an algorithmic version of Nurmonen's extension of Hanf's theorem for first-order logic with modulocounting quantifiers. As an immediate consequence, we obtain that on finite structures of degree ≤ d, model checking of first-order logic with modulo-counting quantifiers is fixed-parameter tractable. Categories and Subject Descriptors F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic-Computational Logic, Model Theory General Terms Theory,