Comparative Similarity, Tree Automata, and Diophantine Equations

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, 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 τ ₁ than to a set τ ₂’ 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.