Topological elementary equivalence of closed semi-algebraic sets in the real plane

Abstract. We consider two-dimensional spatial databases defined in terms of polynomial inequalities and focus on the potential of programming languages for such databases to express queries related to topological connectivity. It is known that the topological connectivity test is not first-order expressible. One approach to obtain a language in which connectivity queries can be expressed would be to extend FO+Poly with a generalized (or Lindström) quantifier expressing that two points belong to the same connected component of a given database. For the expression of topological connectivity, extensions of first-order languages with recursion have been studied (in analogy with the classical relational model). Two such languages are spatial Datalog and FO+Poly+While. Although both languages allow the expression of non-terminating programs, their (proven for FO+Poly+While and conjectured for spatial Datalog) computational completeness makes them interesting objects of study. Previously, spatial Datalog programs have been studied for more restrictive forms of connectivity (e.g., piece-wise linear connectivity) and these programs were proved to correctly test connectivity on restricted classes of spatial databases (e.g., linear databases) only. In this paper, we present a spatial Datalog program that correctly tests topological connectivity of arbitrary compact (i.e., closed and bounded) spatial databases. In particular, it is guaranteed to terminate on this class of databases. This program is based on a first-order description of a known topological property of spatial databases, namely that locally they are conical. We also give a very natural implementation of topological connectivity in FO+Poly+While, that is based on a first-order implementation of the curve selection lemma, and that works correctly on arbitrary spatial databases inputs. Finally, we raise the question whether topological connectivity of arbitrary spatial databases can also be expressed in spatial Datalog.

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“…In the next sections, we will use the following property. It can be proven similarly as was done for closed spatial databases [12]. Property 2.…”

Section: Spatial Databases Are Locally Conicalmentioning

confidence: 73%

Expressing topological connectivity of spatial databases

Geerts

Kuijpers

2000

Research Issues in Structured and Semistructured Database Programming

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“…One of the problems in particular that received attention in recent years is that of determining the exact power of FO in expressing topological properties of spatial databases [3], [8], [11], [16], [21]. One such well-known property, which is central in this research, is that inside a small enough ball around each point, a semi-algebraic set has the topology of a cone.…”

Section: Introductionmentioning

confidence: 99%

“…A radius at which 608 F. Geerts this behavior shows is called a box cone radius in the point for the set. The existence of such a radius was already proven for semi-algebraic sets in R 2 , but the methods of the proof do not generalize to arbitrary dimensions [16].…”

Section: Introductionmentioning

confidence: 99%

Expressing the box cone radius in the relational calculus with real polynomial constraints

Geerts

2003

Discrete and Computational Geometry

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“…The first-order topological properties of plain sets are precisely those expressible in CL. We have the following theorem [14], which plays a crucial role in the proof of Theorem 2: An illustration of this theorem is given in Figure 3. For the full proof of this theorem we refer to the paper [14], but we give an idea of the proof here.…”

mentioning

confidence: 99%

“…We have the following theorem [14], which plays a crucial role in the proof of Theorem 2: An illustration of this theorem is given in Figure 3. For the full proof of this theorem we refer to the paper [14], but we give an idea of the proof here. The proof is based on a transformation of plain sets into a normal form called flower normal form.…”

mentioning

confidence: 99%

Logical aspects of spatial databases

Kuijpers¹,

Bussche²

2011

Finite and Algorithmic Model Theory

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In most general terms, a spatial dataset is any set S ⊂ R n for some n. Equivalently, we can view such a set as an n-ary relation S over R (using Cartesian coordinates). Viewing R as a structure R = (R, 0, 1, +, ·, <) over the language of ordered fields, we can then use first-order logic to express properties of spatial datasets.For example, the sentenceexpresses that S ⊂ R 2 lies on a straight line.Since the structure on R in this paper remains the same, we abbreviate (R, S) |= φ as S |= φ.

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Topological elementary equivalence of closed semi-algebraic sets in the real plane

Cited by 18 publications

References 24 publications

Expressing topological connectivity of spatial databases

Expressing topological connectivity of spatial databases

Expressing the box cone radius in the relational calculus with real polynomial constraints

Logical aspects of spatial databases

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