FirstOrder Logic with Counting: At Least, Weak Hanf Normal Forms Always Exist and Can Be Computed!
Abstract
We introduce the logic FOCN(P) which extends firstorder 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 Hanflocality rank of firstorder 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 fixedparameter 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.01122
 Bibcode:
 2017arXiv170301122K
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 41 pages