We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas presented in Niemist\"o's PhD thesis. Our positive results make use of the close connection between FO+MOD_p and Boolean circuits built from NOT-gates and AND-, OR-, and MOD_p- gates of arbitrary fan-in.

@article{HarwathS16,
  title = {On the locality of arb-invariant first-order formulas with modulo counting quantifiers},
  author = {Frederik Harwath and Nicole Schweikardt},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 12, Issue 4},
  year = {2017},
  doi = {10.2168/lmcs-12(4:8)2016}
}