Locality of Queries Definable in Invariant First-Order Logic with Arbitrary Built-in Predicates

We consider first-order formulas over relational structures which may use arbitrary numerical predicates. We require that the validity of the formula is independent of the particular interpretation of the numerical predicates and refer to such formulas as Arb-invariant first-order. Our main result shows a Gaifman locality theorem: two tuples of a structure with n elements, having the same neighborhood up to distance (log n)ω(1), cannot be distinguished by Arb-invariant first-order formulas. When restricting attention to word structures, we can achieve the same quantitative strength for Hanf locality. In both cases we show that our bounds are tight. Our proof exploits the close connection between Arb-invariant first-order formulas and the complexity class AC0, and hinges on the tight lower bounds for parity on constant-depth circuits.