Locality of Queries

Synonyms

Gaifman-locality; Hanf-locality; Locality

Definition

Letσ be a relational signature without constant symbols. Given a σ-structure $\mathcal{A}$, its Gaifman graph, denoted by $G\left(\mathcal{A}\right)$, has A (the domain of $\mathcal{A}$) as the set of nodes. There is an edge (a₁, a₂) in $G\left(\mathcal{A}\right)$ if there is a relation symbol R in σ such that for some tuple t in the interpretation of this relation in $\mathcal{A}$, both a₁, a₂ occur in t. The distance d(a₁, a₂) is the distance in the Gaifman graph, with d(a, a) = 0. If ā and $\overline{b}$ are tuples of elements, then d($\overline{a},\overline{b}$) stands for the minimum of d(a, b), where a ∈ ā and b ∈ $\overline{b}$.

Let $\mathcal{A}$ be a σ-structure, and ā = (a₁, …,a_m) ∈ A^m. The radius r ball around ā is the set ${B}_r^{\mathcal{A}}\left(\overline{a}\right)\kern0.5em =\kern0.5em \left\{b\in A\left|d\left(\overline{a},b\right)\le r\right.\right\}$. The r-neighborhoo...