In this section, we interpret L(LBP T ) over models based on a 2D Euclidean space R2 (Definition7.53).
Definition 9.13(2D Euclidean Model). A 2D Euclidean model M is a tuple (R2, d, I, σ), where (R2, d) is a 2D Euclidean space, I is an interpretation function which
maps each individual name to a non-empty set of elements of R2, and σ ∈ R ≥0
is a margin of error. The notion of M |= φ (φ is true in model M ) is defined as follows:
M |= BP T (a, b) iff ∀pa ∈ I (a)∃pb ∈ I (b) : d (pa, pb) ∈ [0 , σ];
M |= N EAR(a, b) iff ∃pa∈ I(a)∃pb ∈ I(b) : d(pa, pb) ∈ [0, 2σ];
M |= F AR(a, b) iff ∀pa∈ I(a)∀pb ∈ I(b) : d(pa, pb) ∈ (4σ, ∞);
M |= ¬φ iff M 6|= φ;
M |= φ ∧ ψ iff M |= φ and M |= ψ,
where a, b are individual names, φ, ψ are formulas in L(LBP T ).
For 2D Euclidean models, the soundness theorem of LBPT can be easily proved, whilst proving the completeness theorem and decidability theorem is more dif- ficult, very similar to proving those for LNFS. This is left for future work.
Validating Matches using
Qualitative Spatial Logic
In Chapters7-9, three new qualitative spatial logics, LNF, LNFS and LBPT, are introduced. This chapter explains the use of them for detecting problematic matches between spatial features. Similar to description logic reasoning, the use of these logics follows the rationale of the data integration framework de- scribed in Section4.2, where errors are located by logical contradictions. Logi- cal contradictions are detected using qualitative spatial logic, and matches are checked with respect to location information.
This chapter consists of two sections. Section 10.1 explains how LNF, LNFS, LBPT are used to validate object matches. The use of LBPT has been described briefly in MatchMaps Step 5 in Section4.3. Section10.2describes several heuris- tics allowing domain experts to remove several similar wrong matches at a time to restore consistency.
10.1
Validating Matches using LNF, LNFS and LBPT
In this section, we explain the use of qualitative spatial logics LNF, LNFS and LBPT for validating object matches, i.e. sameAs and partOf matches between spatial features. For spatial features a and b from different datasets, sameAs(a, b) states that a and b refer to the same object in the real world; partOf (a, b) states that the object represented by a is part of the object represented by b in the real world. Note that partOf here does not mean ‘proper part of’. sameAs(a, b) is seen as the conjunction of partOf (a, b) and partOf (b, a).To validate matches between spatial features with respect to location informa- tion, we verify consistency of their corresponding BEQ and BP T relations be- tween the geometries of spatial features against relative location information (N EAR or F AR) in each input dataset, as explained below.
Let A, B be two sets of spatial features, S be a set of object matches between A and B. For any spatial feature o, let g(o) denote its geometry. For any pair of spatial features a ∈ A, b ∈ B, we assume that if sameAs(a, b) is true, then BEQ(g(a), g(b)) holds; if partOf (a, b) is true, then BP T (g(a), g(b)) holds, where BEQ and BP T are defined using an appropriate level of tolerance σ. BEQ and BP T relations are generated from sameAs and partOf matches in S as re- tractable assumptions. We also generate N EAR and F AR relations as facts for geometries of spatial objects in the same dataset.
We reason about BEQ and BP T relations together with N EAR and F AR facts using axioms of LNF, LNFS or LBPT. If for each spatial feature o in input datasets, g(o) is a point, then we apply LNF, otherwise, we apply LNFS or LBPT. As shown in Chapters8and9, LNFS cannot deal with BP T relations, whilst LBPT is more expressive and can reason about both BEQ and BP T relations (for any pair of geometries a, b, BEQ(a, b) is defined as BP T (a, b) and BP T (b, a)). In the current version of MatchMaps, LBPT axioms and the axiom BEQ(a, b) ↔
BP T (a, b) ∧ BP T (b, a) are implemented in a dedicated LBPT reasoner integrated with an assumption-based truth maintenance system (ATMS) [de Kleer, 1986], as this project focuses on matching spatial features with polygonal geometries. The implementation of the LBPT reasoner with an ATMS is explained in [Du et al., 2015b]. If every spatial feature has a point geometry, then an additional axiom N EAR(a, b) ∧ N EAR(b, c) → ¬F AR(a, c) (LNF Axiom 7) needs to be added to the reasoner.
FIGURE10.1: Examples of using LNFS and LBPT for validating matches
The LBPT reasoner is used to check the consistency of BEQ and BP T rela- tions together with N EAR and F AR facts. If any contradiction exists, all the minimal sets of statements for deriving it are calculated. If a minimal set of statements contains more than one retractable assumption, a domain expert is needed to decide the correctness of the retractable assumptions and remove the wrong one(s) to restore consistency. Location information is visualized and pro- vided to domain experts for making such decisions, as shown in Fig.10.1, where a1, b1, c1, d1(red) are from OSGB data and a2, b2, c2, d2(blue) are from OSM data.
In the left example, by LNFS Axiom 6 (or by LBPT Axiom 6 and BEQ(a, b) ↔ BP T (a, b) ∧ BP T (b, a)), a minimal set of statements for deriving an inconsis- tency consists of BEQ(a1, a2), BEQ(b1, b2), N EAR(a1, b1), F AR(a2, b2). It is clear
that BEQ(b1, b2) is wrong. In the right example, BP T (d2, d1) is wrong, because
consequence, the sameAs and partOf matches corresponding to BEQ(b1, b2) and
BP T (d2, d1) respectively are also incorrect and removed by domain experts.
The spatial logics LNF, LNFS and LBPT are generally applicable to reason with spatial features whose locations are represented at different levels of accuracy or granularity in different datasets. Locations of spatial features can be rep- resented using vector data (coordinates) or raster data (images). Sometimes, for spatial features in different datasets, measuring whether their locations are buffered equal directly is difficult or impossible, for example, when locations are represented as images without knowing their coordinates. In such cases, spatial features may be matched by comparing shapes in images or using lexi- cal information. No matter how the matches between spatial features are gen- erated, the LNF/LNFS/LBPT reasoning can be used to verify consistency of matches, regarding relative locations (N EAR/F AR facts) between spatial fea- tures in the same dataset, which are often reliable and easy to capture.