On the locality of arb-invariant first-order formulas with modulo counting quantifiers.

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+MODp is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MODp formulas presented in Niemisto's PhD thesis. Our positive results make use of the close connection between FO+MODp and Boolean circuits built from NOT-gates and AND-, OR-, and MODp - gates of arbitrary fan-in.