A correspondence theory for terminological logics: preliminary report

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 ALC 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 ALC and give a simple proof that subsumption in ALC is PSPACE-complete, replacing the original six-page one. Furthermore, we consider an extension of ALC 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 T SL, is a notational variant of the propositional dynamic logic converse- PDL. Using this correspondence, we prove that it suffices to consider finite T SL-models, show that T SL-subsumption is decidable, and obtain an axiomatization of T SL. By discovering that features correspond to deterministic programs in dynamic logic, we show that adding them to T SL preserves...