Open Access

Frederik Harwath, Nicole Schweikardt

• We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MODp , 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+MODp . For example, on the class of all finite structures, for any p 2, arb-invariant FO+MODp 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 Niemist ̈o’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 MOD p - gates of arbitrary fan-in.