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

TL;DR

This paper investigates the locality properties of arb-invariant first-order logic extended with modulo p counting quantifiers, revealing conditions under which such formulas are local or non-local on finite and string structures.

Contribution

It provides a comprehensive analysis of the locality and non-locality of arb-invariant FO+MOD_p, including new results on their Gaifman and Hanf locality with polylogarithmic radius.

Findings

For p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local on all finite structures.

When p is an odd prime power, arb-invariant FO+MOD_p is weakly Gaifman local with polylogarithmic radius.

On string structures, for odd prime powers p, it is both Hanf and Gaifman local with polylogarithmic radius.

Abstract

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.