### On Representational Issues About Combinations of Classical Theories with Nonmonotonic Rules [chapter]

Jos de Bruijn, Thomas Eiter, Axel Polleres, Hans Tompits
2006 Lecture Notes in Computer Science
In the context of current efforts around Semantic-Web languages, the combination of classical theories in classical first-order logic (and in particular of ontologies in various description logics) with rule languages rooted in logic programming is receiving considerable attention. Existing approaches such as SWRL, dl-programs, and DL+log, differ significantly in the way ontologies interact with (nonmonotonic) rules bases. In this paper, we identify fundamental representational issues which
more » ... to be addressed by such combinations and formulate a number of formal principles which help to characterize and classify existing and possible future approaches to the combination of rules and classical theories. We use the formal principles to explicate the underlying assumptions of current approaches. Finally, we propose a number of settings, based on our analysis of the representational issues and the fundamental principles underlying current approaches. Principle 3.1 (Interaction based on single models). Let KB = Φ, P be a combined knowledge base such that Φ ⊆ L, I an interpretation of L, and B a variable assignment. The classical component of the body of a rule r ∈ P is true in I with respect to B, denoted I, B |= CB(r), iff I, B |= CB + (r) and I, B |= CB − (r). An interpretation M s-satisfies a rule r with respect to I and B, denoted M, B |= I r, iff M, B |= RB(r) and I, B |= CB(r) only if M, B |= H(r). We call M an s-model of r with respect to I iff M, B |= I r, for every variable assignment B. Furthermore, M is an s-model of P with respect to I iff M |= I r, for every rule r ∈ P . Principle 3.2 (Interaction based on entailment). Let KB = Φ, P be a combined knowledge base such that Φ ⊆ L.