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Comparative similarity, tree automata, and Diophantine equations
 In Proceedings of LPAR 2005, volume 3835 of LNAI
, 2005
"... Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequ ..."
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Cited by 12 (8 self)
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Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional ’ logic with the binary operator ‘closer to a set τ1 than to a set τ2 ’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTimecomplete for the classes of all finite symmetric and all finite (possibly nonsymmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer ’ operator has the same expressive power as the standard operator> of conditional logic, these results may have interesting implications for conditional logic as well. 1
A logic for concepts and similarity
"... Categorisation of objects into classes is currently supported by (at least) two ‘orthogonal’ methods. In logicbased approaches, classifications are defined through ontologies or knowledge bases which describe the existing relationships among terms. Description logic (DL) has become one of the most ..."
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Cited by 6 (0 self)
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Categorisation of objects into classes is currently supported by (at least) two ‘orthogonal’ methods. In logicbased approaches, classifications are defined through ontologies or knowledge bases which describe the existing relationships among terms. Description logic (DL) has become one of the most successful formalisms for representing such knowledge bases, in particular because theoretically wellfounded and efficient reasoning tools have been readily available. In numerical approaches, classifications are obtained by first computing similarity (or proximity) measures between objects and then categorising them into classes by means of Voronoi tessellations, clustering algorithms, nearest neighbour computations, etc. In many areas such as bioinformatics, computational linguistics or medical informatics, these two methods have been used independently of each other: although both of them are often applied to the same domain (and even by the same researcher), up to now no formal interaction mechanism has been developed. In this paper, we propose a DLbased integration of the two classification methods. Our formalism, called SL + ALCQIO, extends the expressive DL ALCQIO by means of the constructors of the similarity logic SL which allow definitions of concepts in terms of both comparative and absolute similarity. In the combined knowledge base the user should declare the similarity spaces where the new operators are interpreted. Of course, SL + ALCQIO can only be useful if classifications with this logic are supported by automated reasoning tools. We lay theoretical foundations for the development of such tools by showing that reasoning problems for SL + ALCQIO can be decomposed into the corresponding problems for its DLpart ALCQIO and similarity part SL. Then we investigate reasoning in SL and prove that consistency and many other reasoning problems are ExpTimecomplete for this logic. Using this result and a recent complexity result of PrattHartmann for ALCQIO, we prove that reasoning in SL + ALCQIO is
From topology to metric: modal logic and quantification in metric spaces
 Proceedings of AiML– 2006
, 2006
"... abstract. We propose a framework for comparing the expressive power and computational behaviour of modal logics designed for reasoning about qualitative aspects of metric spaces. Within this framework we can compare such wellknown logics as S4 (for the topology induced by the metric), wK4 (for the ..."
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Cited by 5 (4 self)
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abstract. We propose a framework for comparing the expressive power and computational behaviour of modal logics designed for reasoning about qualitative aspects of metric spaces. Within this framework we can compare such wellknown logics as S4 (for the topology induced by the metric), wK4 (for the derivation operator of the topology), variants of conditional logic, as well as logics of comparative similarity. One of the main problems for the new family of logics is to delimit the borders between ‘decidable ’ and ‘undecidable. ’ As a first step in this direction, we consider the modal logic with the operator ‘closer to a set τ0 than to a set τ1 ’ interpreted in metric spaces. This logic contains S4 with the universal modality and corresponds to a very natural language within our framework. We prove that over arbitrary metric spaces this logic is ExpTimecomplete. Recall that over R, Q, and Z, as well as their finite subspaces, this logic is undecidable.
Modal logics for metric spaces: Open problems
 We Will Show Them! Essays in Honour of Dov Gabbay, Volume Two
, 2005
"... The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \ ..."
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Cited by 5 (1 self)
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The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \Delta into the set R *0 (of nonnegative real numbers) satisfying the following
Closer' representation and reasoning
 International Workshop on Description Logics, (DL 2005
, 1999
"... We argue that orthodox tools for defining concepts in the framework of description logic should often be augmented with constructors that could allow definitions in terms of similarity (or closeness). We present a corresponding logical formalism with the binary operator ‘more similar or closer to X ..."
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Cited by 1 (1 self)
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We argue that orthodox tools for defining concepts in the framework of description logic should often be augmented with constructors that could allow definitions in terms of similarity (or closeness). We present a corresponding logical formalism with the binary operator ‘more similar or closer to X than to Y ’ and investigate its computational behaviour in different distance (or similarity) spaces. The concept satisfiability problem turns out to be ExpTimecomplete for many classes of distances spaces no matter whether they are required to be symmetric and/or satisfy the triangle inequality. Moreover, the complexity remains the same if we extend the language with the operators ‘somewhere in the neighbourhood of radius a’ where a is a nonnegative rational number. However, for various natural subspaces of the real line R (and Euclidean spaces of higher dimensions) even the similarity logic with the sole ‘closer ’ operator turns out to be undecidable. This quite unexpected result is proved by reduction of the solvability problem for Diophantine equations (Hilbert’s 10th problem). “There is nothing more basic to thought and language than our sense of similarity; our sorting of things into kinds.” (Quine 1969) 1
Reasoning about Uncertainty in Metric Spaces
"... We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric structures. To represent both aspect of uncertainty, we choose an expected distance function as a measure of uncertainty. A formal logical system ..."
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We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric structures. To represent both aspect of uncertainty, we choose an expected distance function as a measure of uncertainty. A formal logical system is constructed for the reasoning about expected distance. Soundness and completeness are shown for this logic. For reasoning on product metric spaces with uncertainty, a new metric is defined and shown to have good properties. 1