Workshop on Logic Programming

WLP 2013: Declarative Programming and Knowledge Management pp 65-82 | Cite as

On Axiomatic Rejection for the Description Logic \(\mathcal {ALC}\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8439)


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.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut Für Informationssysteme 184/3Technische Universität WienViennaAustria

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