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Definability with a predicate for a semi-linear set. (English) Zbl 1045.03032

The paper concerns the first-order linear constraint language \(\text{FO}+\text{LIN}\) and the first-order polynomial constraint query language \(\text{FO}+ \text{POLY}\). The sentences of these languages generate semilinear and semi-algebraic sets, respectively. The authors focus on the expressive power of \(\text{FO}+ \text{LIN}\) with an extra predicate symbol ranging over the semilinear sets. They consider five collections of semilinear sets in the Euclidean plane: Co-Linear (all points in each set are collinear), Is.Line (each set is a line), Cont.Line (each set contains a line), Lin.Reach (each set contains a given line segment), and Lin.Meet (each set contains two lines which intersect in a given point).
In Section 2, a brief review of basic notions is presented including some elementary undefinability results. Section 3 contains the notions from non-standard analysis needed as a tool for proving further undefinability results. Section 4 concerns the definability or undefinability of queries related to Lin.Reach in \(\text{FO}+ \text{LIN}\). In Section 5 universal and existential second-order extensions of \(\text{FO}+ \text{LIN}\) are introduced.
These languages are between \(\text{FO}+ \text{LIN}\) and \(\text{FO}+ \text{POLY}\). The definability problem for queries related to Cont.Line in \(\text{FO}+ \text{LIN}\) is solved in Section 6. Section 7 analyses the \(\text{FO}+ \text{LIN}\)-definability problem for the property that any two points are connected by a polygonal path with at most \(n\) segments for various \(n\). Section 8 discusses extensions of results to higher dimensions.
The results contribute to the foundations of spatial databases in which spatial information is modelled by constraint sets.

MSC:

03B70 Logic in computer science
68P15 Database theory
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References:

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