Beth Definability in Expressive Description Logics

Balder ten Cate, Enrico Franconi, Inanc Seylan

The Beth definability property, a well-known property from classical logic, is investigated in the context of description logics (DLs): if a general L-TBox implicitly defines an L-concept in terms of a given signature, where L is a DL, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in DLs. In this paper a complete classification of Beth definability is provided for extensions of the basic DL ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableau-based algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in first-order logic then we show how to compute them in EXPTIME.