Workshop on Logic Programming

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

# On Axiomatic Rejection for the Description Logic $$\mathcal {ALC}$$

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

## Abstract

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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## Copyright information

© Springer International Publishing Switzerland 2014

## Authors and Affiliations

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