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17
Boolean Connection Algebras: A New Approach to the RegionConnection Calculus
 Artificial Intelligence
, 1999
"... The RegionConnection 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 prove ..."
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The RegionConnection 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: RegionConnection Calculus, Qualitative Spatial Reasoning, Boolean Connection Algebra, Mer...
Mereotopological Connection
 Journal of Philosophical Logic
, 2003
"... Abstract. The paper outlines a modeltheoretic framework for investigating and comparing a variety of mereotopological theories. In the first part we consider different ways of characterizing a mereotopology with respect to (i) the intended interpretation of the connection primitive, and (ii) the co ..."
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Cited by 15 (3 self)
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Abstract. The paper outlines a modeltheoretic framework for investigating and comparing a variety of mereotopological theories. In the first part we consider different ways of characterizing a mereotopology with respect to (i) the intended interpretation of the connection primitive, and (ii) the composition of the admissible domains of quantification (e.g., whether or not they include boundary elements). The second part extends this study by considering two further dimensions along which different patterns of topological connection can be classified—the strength of the connection and its multiplicity. 1.
Beyond modalities: Sufficiency and mixed algebras
 In E. Orłowska & A. Szałas (Eds.), Relational Methods in Computer Science Applications, 277– 299
, 2000
"... 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 ..."
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Cited by 14 (12 self)
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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 firstorder definable frames, where such operators are required for a "modalstyle" axiomatisation. 2 Why sufficiency and mixed algebras?
Spatial Logics with Connectedness Predicates
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2010
"... We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of thes ..."
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We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of these logics and show that the connectedness constraints can increase complexity from NP to PSpace, ExpTime and, if component counting is allowed, to NExpTime.
AXIOMS, ALGEBRAS, AND TOPOLOGY
"... This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations. ..."
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Cited by 9 (0 self)
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This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations.
Two proof systems for Peirce algebras
 In Proc. RelMiCS7, vol. 3051 of LNCS
, 2004
"... Abstract. This paper develops and compares two tableauxstyle proof systems for Peirce algebras. One is a tableau refutation proof system, the other is a proof system in the style of RasiowaSikorski. 1 ..."
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Abstract. This paper develops and compares two tableauxstyle proof systems for Peirce algebras. One is a tableau refutation proof system, the other is a proof system in the style of RasiowaSikorski. 1
Construction of Boolean Contact Algebras
 AI Communications
, 2004
"... We consider Boolean algebras endowed with a contact relation which are abstractions of Boolean algebras of regular closed sets together with Whitehead's connection relation [17], in which two nonempty regular closed sets are connected if they have a nonempty intersection. These are standard e ..."
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We consider Boolean algebras endowed with a contact relation which are abstractions of Boolean algebras of regular closed sets together with Whitehead's connection relation [17], in which two nonempty regular closed sets are connected if they have a nonempty intersection. These are standard examples for structures used in qualitative reasoning, mereotopology, and proximity theory. We exhibit various methods how such algebras can be constructed and give several nonstandard examples, the most striking one being a countable model of the Region Connection Calculus in which every proper region has infinitely many holes. 1
Weak contact structures
 RELATIONAL METHODS IN COMPUTER SCIENCE, LNCS NO
, 2006
"... 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 ..."
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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.
An algebraic and logical approach to the approximation of regions, in
 Proc. 5th Seminar on Relational Methods in Computer Science
, 2000
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Structures with manyvalued information and their relational proof theory
 In Proc. of 30th IEEE International Symposium on MultipleValued Logic
, 2000
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