Hanf Locality and Invariant Elementary Definability

TL;DR

This paper extends the concept of elementary definability to invariant settings, establishing new locality theorems for boolean queries over locally finite and bounded degree structures with respect to presentations.

Contribution

It introduces invariant elementary definability notions and proves two locality theorems extending Hanf Locality to these invariant contexts.

Findings

Extended Hanf Locality Theorem to invariant boolean queries.

Proved a non-uniform Hanf Threshold Locality Theorem for bounded degree structures.

Established locality results for structures with presentation-based invariance.

Abstract

We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical relations. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call such expansions \emph{presentations} of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect to \emph{elementary, neighborhood-bounded} presentations. The second is a non-uniform version of the Fagin-Stockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, \emph{local} presentations.