Papers by Mikhail Sheremet
Lecture Notes in Computer Science, 2005
The notion of comparative similarity 'X is more similar or closer to Y than to Z' has been invest... more 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, similarity-based 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 ExpTime-complete for the classes of all finite symmetric and all finite (possibly non-symmetric) 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.
Annals of Pure and Applied Logic, 2010
We propose and investigate a uniform modal logic framework for reasoning about topology and relat... more We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski's S4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S4 and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.
Combining description and similarity logics
Categorisation of objects into classes is currently supported by (at least) two &... more Categorisation of objects into classes is currently supported by (at least) two 'ortho- gonal' methods. In logic-based approaches, classiflcations are deflned 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 know- ledge bases, in particular because theoretically well-founded and e-cient reasoning tools have
Journal of Logic and Computation, 2007
Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' meth... more Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' methods. In logic-based 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 well-founded 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 DL-based 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 DL-part ALCQIO and similarity part SL. Then we investigate reasoning in SL and prove that consistency and many other reasoning problems are ExpTime-complete for this logic. Using this result and a recent complexity result of Pratt-Hartmann for ALCQIO, we prove that reasoning in SL + ALCQIO is NExpTime-complete. As the 'closer' operator of SL has the same expressive power as the standard implication > of conditional logic, these results may have interesting consequences for conditional logic as well.
Aiml, 2006
We propose a framework for comparing the expressive power and computational behaviour of modal lo... more 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 well-known 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 τ 0 than to a set τ 1 ' 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 ExpTime-complete. Recall that over R, Q, and Z, as well as their finite subspaces, this logic is undecidable.
We argue that orthodox tools for defining concepts in the framework of description logic should o... more 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 ExpTime-complete 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 non-negative 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
Lecture Notes in Computer Science, 2005
The notion of comparative similarity 'X is more similar or closer to Y than to Z' has been invest... more 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, similarity-based 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 ExpTime-complete for the classes of all finite symmetric and all finite (possibly non-symmetric) 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.
Journal of Logic and Computation, 2007
Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' meth... more Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' methods. In logic-based 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 well-founded and efficient reasoning tools have been readily available.
Annals of Pure and Applied Logic, 2010
In 1944, McKinsey and Tarski proved that S4 is the logic of the topological interior and closure ... more In 1944, McKinsey and Tarski proved that S4 is the logic of the topological interior and closure operators of any separable dense-in-itself metric space. Thus, the logic of topological interior and closure over arbitrary metric spaces coincides with the logic of the real line, the real plane, and any separable dense-in-itself metric space; it is finitely axiomatisable and PSpace-complete. Because of this result S4 has become a logic of prime importance in Qualitative Spatial Representation and Reasoning in Artificial Intelligence. And in Logic this result has triggered the investigation of a number of variants and extensions of S4 designed for reasoning about qualitative aspects of metric spaces.
We argue that orthodox tools for defining concepts in the framework of description logic should o... more 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 ExpTime-complete 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 non-negative 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).
We propose a framework for comparing the expressive power and computational behaviour of modal lo... more 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 well-known 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 τ 0 than to a set τ 1 ' 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 ExpTime-complete. Recall that over R, Q, and Z, as well as their finite subspaces, this logic is undecidable.
Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' meth... more Categorisation of objects into classes is currently supported by (at least) two 'orthogonal' methods. In logic-based 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 well-founded and efficient reasoning tools have been readily available.
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Papers by Mikhail Sheremet