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40
Expressivity in Polygonal, Plane Mereotopology
 JOURNAL OF SYMBOLIC LOGIC
, 1998
"... In recent years, there has been renewed interest in the development of formal languages for describing mereological (partwhole) and topological relationships between objects in space. Typically, the nonlogical primitives of these languages are properties and relations such as `x is connected' or `x ..."
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Cited by 17 (2 self)
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In recent years, there has been renewed interest in the development of formal languages for describing mereological (partwhole) and topological relationships between objects in space. Typically, the nonlogical 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 ørstorder 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 inønitary 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.
The completeness of the isomorphism relation for countable Boolean algebras
 Trans. Amer. Math. Soc
"... Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is o ..."
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Cited by 10 (1 self)
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Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete firstorder theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.
Qualitative Spatial Representation Languages with Convexity
 Spatial Cognition and Computation
, 1999
"... this paper is to demonstrate how sharp that knifeedge is, and thus to establish some limits on what such qualitative spatial description languages might be like. Specifically, we show that, once we can represent the property of convexity and the partwhole relationmodest assumptions by any stand ..."
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Cited by 10 (0 self)
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this paper is to demonstrate how sharp that knifeedge is, and thus to establish some limits on what such qualitative spatial description languages might be like. Specifically, we show that, once we can represent the property of convexity and the partwhole relationmodest assumptions by any standards ntuples of real polygons are completely determined by the sets of formulas they satisfy upto the fixing of three reference points. To be sure, we are not the first to express skepticism about the possibility of such languages, but this is, as far as we are aware, the first time such skepticism has been put on so firm a mathematical footing.
Arrow’s Theorem, Countably Many Agents, and More Visible Invisible Dictators
"... For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow’s conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a co ..."
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Cited by 9 (4 self)
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For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow’s conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain “oracle ” resolves Mihara’s impossibility result (1997) about computability of social welfare functions.
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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Cited by 7 (2 self)
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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.
On Poset Boolean Algebras
 ORDER
"... Let (P, <=) be a partially ordered set. We define the poset Boolean algebra of P, and denote it by F (P). The set of generators of F(P) is {x p : p P}, and the set of relations is q}. We say that a Boolean algebra B is wellgenerated, if B has a sublattice G such that G generates B and G is wellfou ..."
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Cited by 5 (3 self)
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Let (P, <=) be a partially ordered set. We define the poset Boolean algebra of P, and denote it by F (P). The set of generators of F(P) is {x p : p P}, and the set of relations is q}. We say that a Boolean algebra B is wellgenerated, if B has a sublattice G such that G generates B and G is wellfounded. A wellgenerated algebra is superatomic...
Topology, connectedness, and modal logic
 ADVANCES IN MODAL LOGIC
, 2008
"... This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of ..."
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Cited by 5 (3 self)
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This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (lowdimensional) Euclidean spaces.
Radical classes of latticeordered groups vs. classes of compact spaces, Order19
, 2002
"... Abstract. For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓgroups G such that for each g ∈ G, the Yosida space Y(g)of g belongs to T.Conversely,ifR is a class of ℓgroups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g ∈ G ∈ R. ..."
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Cited by 4 (4 self)
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Abstract. For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓgroups G such that for each g ∈ G, the Yosida space Y(g)of g belongs to T.Conversely,ifR is a class of ℓgroups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g ∈ G ∈ R. The correspondences T ↦ → Y(T) and R ↦ → T(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of ℓgroups whose Yosida spaces are zerodimensional. There is a thorough discussion of hyperprojectable ℓgroups, followed by presentations on Y(e.d.),wheree.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal κ, the class Y(discκ), where discκ stands for the class of all compact κdisconnected spaces. Sample results follow. Every strongly projectable ℓgroup lies in Y(e.d.). Theℓgroup G lies in Y(e.d.) if and only if for each g ∈ GY(g)is zerodimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(discκ). Finally, there is a discussion of Y(F), with F standing for the class of compact Fspaces. It is shown that an Archimedean ℓgroup G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G = P ⊥ + Q ⊥.
The role of connections as minimal norms in normative systems
 In Legal Knowledge and Information Systems: Proceedings of Jurix 2002
, 2002
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