A correspondence theory for terminological logics: preliminary report

TL;DR: It is proved that universal implications can be expressed within TSC, and it is shown that features correspond to deterministic programs in dynamic logic preserves decidability, although violates its finite model property.

Abstract: We show that the terminological logic ACC comprising Boolean operations on concepts and value restrictions is a notational variant of the propositional modal logic K(m)- To demonstrate the utility of the correspondence, we give two of its immediate by-products, Namely, we axiomatize ACC and give a simple proof that subsumption in ACC is PSPACE-complete, replacing the original six-page one. Furthermore, we consider an extension of ACC additionally containing both the identity role and the composition, union, transitive-reflexive closure, range restriction, and inverse of roles. It turns out that this language, called TSL, is a notational variant of the propositional dynamic logic converse-PDL. Using this correspondence, we prove that it suffices to consider finite TSL-models, show that TSL-subsumption is decidable, and obtain an axiomatization of TSL By discovering that features correspond to deterministic programs in dynamic logic, we show that adding them to TSC preserves decidability, although violates its finite model property. Additionally, we describe an algorithm for deciding the coherence of inverse-free TSC-concepts with features. Finally, we prove that universal implications can be expressed within TSC.