First-Order Logic with Counting: At Least, Weak Hanf Normal Forms Always Exist and Can Be Computed!

We introduce the logic FOCN(P) which extends first-order logic by counting and by numerical predicates from a set P, and which can be viewed as a natural generalisation of various counting logics that have been studied in the literature. We obtain a locality result showing that every FOCN(P)-formula can be transformed into a formula in Hanf normal form that is equivalent on all finite structures of degree at most d. A formula is in Hanf normal form if it is a Boolean combination of formulas describing the neighbourhood around its tuple of free variables and arithmetic sentences with predicates from P over atomic statements describing the number of realisations of a type with a single centre. The transformation into Hanf normal form can be achieved in time elementary in $d$ and the size of the input formula. From this locality result, we infer the following applications: () The Hanf-locality rank of first-order formulas of bounded quantifier alternation depth only grows polynomially with the formula size. () The model checking problem for the fragment FOC(P) of FOCN(P) on structures of bounded degree is fixed-parameter tractable (with elementary parameter dependence). (*) The query evaluation problem for fixed queries from FOC(P) over fully dynamic databases of degree at most d can be solved efficiently: there is a dynamic algorithm that can enumerate the tuples in the query result with constant delay, and that allows to compute the size of the query result and to test if a given tuple belongs to the query result within constant time after every database update.