This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language ℒ with a distinguished unary predicate c(x), function-symbols +, · and - and constants 0 and 1 is defined. An interpretation ℜ for ℒ is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as 'region x is connected' and the function-symbols and (...) constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation ������ base on the real closed plane which turns out to be isomorphic to ℜ. A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation. (shrink)

In recent years, there has been renewed interest in the development of formal languages for describing mereological (part-whole) and topological relationships between objects in space. Typically, the non-logical primitives of these languages are properties and relations such as `x is connected' or `x is a part of y', and the entities over which their variables range are, accordingly, not points, but regions: spatial entities other than regions are admitted, if at all, only as logical constructs of regions. This paper considers (...) two first-order mereotopological languages, and investigates their expressive power. It turns out that these languages, notwithstanding the simplicity of their primitives, are surprisingly expressive. In particular, it is shown that infinitary versions of these languages are adequate to express (in a sense made precise below) all topological relations over the domain of polygons in the closed plane. (shrink)

In recent years, there has been renewed interest in the development of formal languages for describing mereological (part-whole) and topological relationships between objects in space. Typically, the non-logical primitives of these languages are properties and relations such as ‘xis connected’ or ‘xis a part ofy’, and the entities over which their variables range are, accordingly, notpoints, butregions: spatial entities other than regions are admitted, if at all, only as logical constructs of regions. This paper considers two first-order mereotopological languages, and (...) investigates their expressive power. It turns out that these languages, notwithstanding the simplicity of their primitives, are surprisingly expressive. In particular, it is shown that infinitary versions of these languages are adequate to express (in a sense made precise below) all topological relations over the domain of polygons in the closed plane. (shrink)

A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a mereotopological formalism originally introduced by Whitehead, which (...) employs a single primitive binary relation C(x, y) (read: "x is in contact with y"). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of ℝ³, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit. (shrink)