The Region-Connection Calculus (RCC) is a well established formal system for qualitative spatial reasoning. It provides an axiomatization of space which takes regions as primitive, rather than as constructions from sets of points. The paper introduces boolean connection algebras (BCAs), and proves that these structures are equivalent to models of the RCC axioms. BCAs permit a wealth of results from the theory of lattices and boolean algebras to be applied to RCC. This is demonstrated by two theorems which provide constructions for BCAs from suitable distributive lattices. It is already well known that regular connected topological spaces yield models of RCC, but the theorems in this paper substantially generalize this result. Additionally, the lattice theoretic techniques used provide the first proof of this result which does not depend on the existence of points in regions. Keywords: Region-Connection Calculus, Qualitative Spatial Reasoning, Boolean Connection Algebra, Mer...

We explore the relation--algebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the "part -- of" relation, and that the table is closed under first order definable relations iff B is homogeneous. 1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient -- or not present -- to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often l...

This paper is a continuation of [41]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to...

Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa & Sikorski (1963) for relation algebras generated by a contact relation. 1 Introduction Contact relations arise in the context of qualitative geometry and spatial reasoning, going back to the work of de Laguna (1922), Nicod (1924), Whitehead (1929), and, more recently, of Clarke (1981), Cohn et al. (1997), Pratt & Schoop (1998, 1999) and others. They are a generalisation of the "overlap relation" , obtained from a "part of" relation, which for the first time was formalised by Lesniewski (1916), (see also Lesniewski, 1983). One of Lesniewski's main concerns was to build a paradox--free foundation of Mathematics, one pillar of which was mereology 1 or, as it was originally called, the general theory of manifolds or colle...

GQR (Generic Qualitative Reasoner) is a solver for binary qualitative constraint networks. GQR takes a calculus description and one or more constraint networks as input, and tries to solve the networks using the path consistency method and (heuristic) backtracking. In contrast to specialized reasoners, it offers reasoning services for different qualitative calculi, which means that these calculi are not hard-coded into the reasoner. Currently, GQR supports arbitrary binary constraint calculi developed for spatial and temporal reasoning, such as calculi from the RCC family, the intersection calculi, Allen’s interval algebra, cardinal direction calculi, and calculi from the OPRA family. New calculi can be added to the system by specifications in a simple text format or in an XML file format. The tool is designed and implemented with genericity and extensibility in mind, while preserving efficiency and scalability. The user can choose between different data structures and heuristics, and new ones can be easily added to the object-oriented framework. GQR is free software distributed under the terms of the GNU General Public License.

this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structures occur when modal and sufficiency operators arise from the same accessibility relation. In this paper we introduce the classes of sufficiency algebras and that of mixed algebras which include both a modal and a sufficiency operator, and study representation and duality theory for these classes of algebras. We also give examples for classes of first-order definable frames, where such operators are required for a "modal-style" axiomatisation. 2 Why sufficiency and mixed algebras?

We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the Region Connection Calculus [33], and we show how to interpret these relations in the collection of regular open sets in the two-dimensional Euclidean plane. 1 Introduction Mereotopology is an area of qualitative spatial reasoning (QSR) which aims to develop formalisms for reasoning about spatial entities [1, 12, 30, 31]. The structures used in mereotopology consist of three parts: 1. A relational (or mereological) part, 2. An algebraic part, 3. A topological part. The algebraic part is often an atomless Boolean algebra, or, more generally, an orthocomplemented lattice, both without smallest element. Due to the presence of the binary relations "part-of" and "contact" in the relational part of mereotopology, composition based reasoning with binary relations has been of interest to the QSR community, and the expressive power, consistency and complexity o...

In this paper we investigate weak contact relations C on a lattice L, in particular, the relation between various axioms for contact, and their connection to the algebraic structure of the lattice. Furthermore, we will study a notion of orthogonality which is motivated by a weak contact relation in an inner product space. Although this is clearly a spatial application, we will show that, in case L is distributive and C satisfies the orthogonality condition, the only weak contact relation on L is the overlap relation; in particular no RCC model satisfies this condition.

To be presented at the 5th Seminar on Relational Methods in Computer Science Ivo Dntsch School of Information and Software Engineering University of Ulster at Jordanstown Newtownabbey, BT 37 0QB, N.Ireland I.Duentsch@ulst.ac.uk Ewa Orl/owska Institute of Telecommunications Szachowa 1 04--894, Warszawa, Poland orlowska@itl.waw.pl Hui Wang School of Information and Software Engineering University of Ulster at Jordanstown Newtownabbey, BT 37 0QB, N.Ireland H.Wang@ulst.ac.uk Abstract We report the definition of a class of algebras which can serve as an algebraic tool for handling regions that can only be approximated due to inappropriate granularity. We also generalise the concepts of "contact relation" and "part of" to the case of incompletely known regions to arrive at approximate contact algebras. Finally, we present a relational proof system for these structures. Proofs and details will be published elsewhere. 1 Introduction It is rarely the case that spatial regions can be d...

Abstract. In AI a large number of calculi for efficient reasoning about spatial and temporal entities have been developed. The most prominent temporal calculi are the point algebra of linear time and Allen’s interval calculus. Examples of spatial calculi include mereotopological calculi, Frank’s cardinal direction calculus, Freksa’s double cross calculus, Egenhofer and Franzosa’s intersection calculi, and Randell, Cui, and Cohn’s region connection calculi. These calculi are designed for modeling specific aspects of space or time, respectively, to the effect that the class of intended models may vary widely with the calculus at hand. But from a formal point of view these calculi are often closely related to each other. For example, the spatial region connection calculus RCC5 may be considered a coarsening of Allen’s (temporal) interval calculus. And vice versa, intervals can be used to represent spatial objects that feature an internal direction. The central question of this paper is how these calculi as well as their mutual dependencies can be axiomatized by algebraic specifications. This question will be investigated within the framework of the Common Algebraic Specification Language (CASL), a specification language developed by the Common Framework Initiative for algebraic specification and development (COFI). We explain scope and expressiveness of CASL by discussing the specifications of some of the calculi mentioned before. 1