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, …,am) ∈ Am. 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...
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Recommended Reading
Hanf W. Model-theoretic methods in the study of elementary logic. In: Addison JW et al., editors. The theory of models. Amsterdam: North Holland; 1965. p. 132–45.
Gaifman H. On local and non-local properties. In: Proceedings of the Herbrand Symposium: Logic Colloquium ‘81. Amsterdam: North Holland; 1982.
Libkin L. On the forms of locality over finite models. Proceedings of the 12th IEEE Conference on Logic in Computer Science; 1997. p. 204–15.
Libkin L. On counting logics and local properties. ACM Trans Comput Logic. 2000;1(1):33–59.
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Barceló, P. (2018). Locality of Queries. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_1270
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DOI: https://doi.org/10.1007/978-1-4614-8265-9_1270
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