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A New Perspective on the Mereotopology of RCC8

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A New Perspective on the Mereotopology of RCC8 A New Perspective on the Mereotopology of RCC8 Michael Grüninger1 and Bahar Aameri2 1 University of Toronto, Toronto, ON, Canada [email protected] 2 University of Toronto, Toronto, ON, Canada [email protected] Abstract RCC8 is a set of eight jointly exhaustive and pairwise disjoint binary relations representing mereotopological relationships between ordered pairs of individuals. Although the RCC8 relations were originally presented as defined relations of Region Connection Calculus (RCC), virtually all implementations use the RCC8 Composition Table (CT) rather than the axioms of RCC. This raises the question of which mereotopology actually underlies the RCC8 composition table. In this paper, we characterize the algebraic and mereotopological properties of the RCC8 CT based on the metalogical relationship between the first-order theory that captures the RCC8 CT and Ground Mereotopology (MT) of Casati and Varzi. In particular, we show that the RCC8 theory and MT are relatively interpretable in each other. We further show that a nonconservative extension of the RCC8 theory that captures the intended interpretation of the RCC8 relations is logically synonymous with MT, and that a conservative extension of MT is logically synonymous with the RCC8 theory. We also present a characterization of models of MT up to isomorphism, and explain how such a characterization provides insights for understanding models of the RCC8 theory. 1998 ACM Subject Classification F.4.1 Mathematical Logic, I.2.4 Knowledge Representation Formalisms and Methods Keywords and phrases RCC8, mereotopology, spatial reasoning, ontologies Digital Object Identifier 10.4230/LIPIcs.COSIT.2017.2 1 Introduction Representations of space, and their use in qualitative spatial reasoning, are widely recog- nized as key aspects in commonsense reasoning, with applications ranging from biology to geography. The predominant approach to spatial representation within the applied ontology community has used mereotopologies, which combine topological (expressing connectedness) with mereological (expressing parthood) relations. A variety of first-order mereotopological ontologies have been proposed, the most widespread being the Region Connection Calculus (RCC) [17], the ontology RT [1], and the ontologies introduced by Casati and Varzi [4]. Properties of RCC in particular have been studied extensively; [18, 5] present algebraic representations for the RCC theory, and [9] describes various mereotopological settings that satisfy axioms of RCC. While theoretical work has focused on the first-order theories for mereotopologies, work within the qualitative spatial reasoning community has primarily used a formalism known as RCC8, which is a set of eight jointly exhaustive and pairwise disjoint binary relations representing mereotopological relationships between ordered pairs of individuals. Reasoning © Michael Grüninger and Bahar Aameri; licensed under Creative Commons License CC-BY 13th International Conference on Spatial Information Theory (COSIT 2017). Editors: Eliseo Clementini, Maureen Donnelly, May Yuan, Christian Kray, Paolo Fogliaroni, and Andrea Ballatore; Article No. 2; pp. 2:1–2:13 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 2:2 A New Perspective on the Mereotopology of RCC8 is supported through the use of a composition table, which specifies all possible mereotopolo- gical relationships between pairs of elements; deduction is implemented through constraint propagation algorithms. Although the RCC8 relations were originally presented as defined relations within RCC, the theoretical analyses of RCC have not been helpful in understanding properties of formalisms that use the RCC8 relations. The reason is that virtually all implementations use the RCC8 composition table rather than the axioms of RCC, and the composition table has very different mereotopological properties than RCC. Of particular importance is the widespread use of RCC8 in efforts such as GeoSPARQL, which is an international standard for the representation of geospatial linked data developed by the Open Geospatial Consortium. A characterization of all solutions for a set of RCC8 constraints presumes an understanding of the possible models of some first-order logical theory. In this paper, we investigate algebraic and mereotopological properties of the RCC8 composition table based on the metalogical relationship between the first-order theory that captures the RCC8 composition table and Ground Mereotopology (MT) of Casati and Varzi. After reviewing the basic axiomatizations of the mereotopological theories in Section 2, we discuss the relationship between the RCC8 theory and MT in Section 3. Our key result is that a nonconservative extension of the RCC8 theory, we called RCC8*, is logically synonymous with the MT theory, meaning MT and RCC8* axiomatize the same class of structures. In other words, MT and RCC8* are semantically equivalent, and only differ in signature (i.e., the non-logical symbols). Further, we present a conservative extension of MT which is logically synonymous with the RCC8 theory. We also show that the RCC8 theory and MT are relatively interpretable in each other. Finally, in Section 4, we present a characterization of models of MT up to isomorphism, and explain how such a characterization can be used in characterizing algebraic properties of models of the RCC8 theory. 2 Preliminaries: Mereotopological Theories 2.1 Region Connection Calculus The Region Connection Calculus (RCC) is a first-order theory whose signature contains the single primitive binary relation C(x, y) denoting “x is connected to y”. Parthood is defined in terms of connection alone, being equivalent to the topological notion of enclosure. Repres- entation theorems [18] have shown that the models of RCC are equivalent to mathematical structures known as Boolean contact algebras which consist of a standard Boolean algebra together with a binary relation C that is reflexive, anti-symmetric, and extensional. 2.2 RCC8 RCC8 is a set of eight binary relations representing mereotopological relationships between (ordered) pairs of individuals. These relations and their intended interpretations are illustrated in Figure 1. The RCC8 relations have been proven to be jointly exhaustive and pairwise disjoint (JEPD), that is, every ordered pair of individuals are related by exactly one RCC8 relation. Originally, RCC8 relations were presented as defined relations of RCC (throughout the paper, free variables in a displayed formula are assumed to be universally quantified): DC(x, y) ≡ ¬C(x, y). (1) EC(x, y) ≡ C(x, y) ∧ ¬O(x, y). (2) M. Grüninger and B. Aameri 2:3 Figure 1 Illustration of RCC8 relations – DC(a, b) (a is disconnected from b), EC(a, b) (a is externally connected with b), PO(a, b) (a partially overlaps b), TPP (a, b) (a is a tangential proper part of b), T P P i(a, b) (b is a tangential proper part of a), NTPP (a, b) (a is a nontangential proper part of b), NT P P i(a, b) (b is a nontangential proper part of a), a = b (a is identical with b). DC EC PO TPP NTPP TPPi NTPPi = DC * DC, EC, PO, TPP, DC, EC, PO, TPP, DC, EC, PO, TPP, DC, EC, PO, TPP, DC DC DC NTPP NTPP NTPP NTPP EC DC, EC, PO, DC, EC, PO, TPP, DC, EC, PO, TPP, EC, PO, TPP, PO, TPP, NTPP DC, EC DC EC TPPi, NTPPi TPPi, = NTPP NTPP PO DC, EC, PO, DC, EC, PO, TPPi, * PO, TPP, NTPP PO, TPP, NTPP DC, EC, PO, DC, EC, PO, PO TPPi, NTPPi NTPPi TPPi, NTPPi TPPi, NTPPi TPP DC DC, EC DC, EC, PO, TPP, TPP, NTPP NTPP DC, EC, PO, DC, EC, PO, TPP NTPP TPP, TPPi, = TPPi, NTPPi NTPP DC DC DC, EC, PO, TPP, NTPP NTPP DC, EC, PO, * NTPP NTPP TPP, NTPP TPPi DC, EC, PO, EC, PO, TPPi, PO,TPPi,NTPPi PO, TPP, TPPi, = PO, TPP, NTPP TPPi, NTPPi NTPPi TPPi TPPi, NTPPi NTPPi NTPPi DC, EC, PO, PO,TPPi,NTPPi PO,TPPi,NTPPi PO,TPPi,NTPPi PO, TPP, NTPP, NTPPi NTPPi NTPPi TPPi, NTPPi TPPi, NTPPi, = = DC EC PO TPP NTPP TPPi NTPPi = Figure 2 RCC8 Composition Table. “∗” indicates that all RCC8 relations are possible. PO(x, y) ≡ O(x, y) ∧ ¬P (x, y) ∧ ¬P (y, x). (3) (x = y) ≡ P (x, y) ∧ P (y, x). (4) T P P i(x, y) ≡ TPP (y, x). (5) NT P P i(x, y) ≡ NTPP (y, x). (6) TPP (x, y) ≡ PP (x, y) ∧ ¬NTPP (x, y). (7) NTPP (x, y) ≡ PP (x, y) ∧ ¬(∃z) EC(z, x) ∧ EC(z, y). (8) In the axioms above, C(x, y) denotes “x is connected to y,” P (x, y) denotes “x is a part of y,” O(x, y) denotes “x overlaps y,” PP (x, y) denotes “x is a proper part of y”): O(x, y) ≡ (∃z) P (z, x) ∧ P (z, y). (9) PP (x, y) ≡ P (x, y) ∧ ¬P (y, x). (10) Given its origin within RCC, it is interesting to note that RCC8 is typically used inde- pendently of the RCC theory – the RCC axioms are not considered to be part of the RCC8 formalism, and in most reasoning tasks even the axiomatic descriptions of RCC8 relations are not explicitly used. Instead, the RCC8 Composition Table (CT) is used. The RCC8 CT (illustrated in Figure 2) is an 8 × 8 matrix such that for each ordered pair of RCC8 relations Ri,Rj, the cell CT (Ri,Rj) indicates possible mereotopological relationships between two individuals a and c assuming that Ri(a, b) and Rj(b, c) holds. For example, CT (EC, NT P P ) = {PO,TPP,NTPP }, meaning that if EC(a, b) and NTPP (b, c), then a is related to c by either PO or TPP or NTPP . C O S I T 2 0 1 7 2:4 A New Perspective on the Mereotopology of RCC8 2.3 Combined Mereotopology Even though the RCC8 CT is entailed by the RCC theory, they have very different mereotopo- logical properties. In fact, the RCC8 CT seems to be closely related to Ground Mereotopology (also called MT), which is the weakest theory among the mereotopological theories proposed in [4].
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