• DOI:10.1007/978-3-319-08909-6_5
• Corpus ID: 29107313

On Axiomatic Rejection for the Description Logic ALC

@inproceedings{Berger2013OnAR,
  title={On Axiomatic Rejection for the Description Logic ALC},
  author={Gerald Berger and Hans Tompits},
  booktitle={KDPD},
  year={2013}
}

• Gerald Berger, H. Tompits
• Published in KDPD 11 September 2013
• Computer Science

Traditional proof calculi are mainly studied for formalising the notion of valid inference, i.e., they axiomatise the valid sentences of a logic. In contrast, the notion of invalid inference received less attention. Logical calculi which axiomatise invalid sentences are commonly referred to as complementary calculi or rejection systems. Such calculi provide a proof-theoretic account for deriving non-theorems from other non-theorems and are applied, in particular, for specifying proof systems for nonmonotonic logics. In this paper, we present a sound and complete sequent-type rejection system which axiomatises concept non-subsumption for the description logic \(\mathcal {ALC}\). Description logics are well-known knowledge-representation languages formalising ontological reasoning and provide the logical underpinning for semantic-web reasoning. We also discuss the relation of our calculus to a well-known tableau procedure for \(\mathcal {ALC}\). Although usually tableau calculi are syntactic variants of standard sequent-type systems, for \(\mathcal {ALC}\) it turns out that tableaux are rather syntactic counterparts of complementary sequent-type systems. As a consequence, counter models for witnessing concept non-subsumption can easily be obtained from a rejection proof. Finally, by the well-known relationship between \(\mathcal {ALC}\) and multi-modal logic \(\mathbf {K}\), we also obtain a complementary sequent-type system for the latter logic, generalising a similar calculus for standard \(\mathbf {K}\) as introduced by Goranko.