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

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LIPIcs.CSL.2013.363.pdf
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## Cite As

Frederik Harwath and Nicole Schweikardt. On the locality of arb-invariant first-order logic with modulo counting quantifiers. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 363-379, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)
https://doi.org/10.4230/LIPIcs.CSL.2013.363

## Abstract

We study Gaifman 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 Niemistö'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.
##### Keywords
• finite model theory
• Gaifman and Hanf locality
• first-order logic with modulo counting quantifiers
• order-invariant and arb-invariant formulas
• lower

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