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}
}
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… 
Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic
TLDR
This paper introduces sequent-type calculi for two variants of default logic, viz., for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default Logic, due to Gelfond, Lifschitz, Przymusinski, and Truszczynski, which are introduced to address certain representational shortcomings of standard default logic.
Sequent-Type Proof Systems for Three-Valued Default Logic
TLDR
This master's thesis introduces sequent-type calculi for a variant of default logic employing Lukasiewicz's three-valued logic as the underlying base logic, addressing some representational shortcomings of standard default logic.
Refutation Systems: An Overview and Some Applications to Philosophical Logics
TLDR
An overview with comprehensive references on the historical development of the theory of refutation systems is provided and some of their applications to philosophical logics are discussed.

References

SHOWING 1-10 OF 25 REFERENCES
Sequent calculi for propositional nonmonotonic logics
A uniform proof-theoretic reconstruction of the major nonmonotonic logics is introduced. It consists of analytic sequent calculi where the details of nonmonotonic assumption making are modelled by an
A Correspondence Theory for Terminological Logics: Preliminary Report
TLDR
It is proved that universal implications can be expressed within TSC, and it is shown that features correspond to deterministic programs in dynamic logic preserves decidability, although violates its finite model property.
A Proof Theory for Description Logics
TLDR
A Proof Theory for Description Logics introduces Sequent Calculi and Natural Deduction for some DLs (ALC, ALCQ) and argues that such systems can improve the extraction of computational content from DLs proofs for explanation purposes.
An Overview of Tableau Algorithms for Description Logics
TLDR
This article focuses on three features that play an important rôle in description logics (number restrictions, terminological axioms, and role constructors), and shows how they can be taken into account by tableau algorithms.
Automated Reasoning with Analytic Tableaux and Related Methods
  • R. Dyckhoff
  • Computer Science
    Lecture Notes in Computer Science
  • 2000
TLDR
Tableau Algorithms for Description Logics and Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory.
Gentzen-Type Refutation Systems for Three-Valued Logics with an Application to Disproving Strong Equivalence
TLDR
Gentzen-type refutation systems for two basic three-valued logics are introduced and an application of one of these calculi is discussed for disproving strong equivalence between answer-set programs.
Contraction-Free Sequent Calculi for Intuitionistic Logic
TLDR
An equivalent calculus is described, exploiting the Dershowtiz-Manna theorem on multiset orderings, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.
Proof Theory of Modal Logic
Preface. Part I: Standard Proof Systems. 1. A Contraction-free Sequent Calculus for S4 J. Hudelmaier. 2. Transfer of Sequent Calculus Strategies to Resolution for S4 G. Mints, et al. 3. A Linear
Refutation systems in modal logic
TLDR
Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics and purely syntactic decision procedures for them are obtained.
...
...