‘Closer’ representation and reasoning M. Sheremet, 1 D. Tishkovsky, 2 F. Wolter, 2 and M. Zakharyaschev 1 1 Department of Computer Science 2 Department of Computer Science King’s College London University of Liverpool Strand, London WC2R 2LS, U.K. Liverpool L69 3BX, U.K. {mikhail,mz}@dcs.kcl.ac.uk {dmitry,frank}@csc.liv.ac.uk Abstract 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 dif- ferent 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 ex- tend the language with the operators ‘somewhere in the neighbourhood of radius a’ where a is a non-negative rational number. However, for var- ious 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 reduc- tion 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 Introduction How do we define concepts? In description logic, we do this by establishing relationships between con- cepts, for example, Mother ≡ Woman ⊓∃hasChild.Person The main tool for analysing and using such definitions is reasoning.