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Download e-book for iPad: A proof theory for description logics by Alexandre Rademaker

By Alexandre Rademaker

ISBN-10: 144714001X

ISBN-13: 9781447140016

ISBN-10: 1447140028

ISBN-13: 9781447140023

Advent -- heritage -- The Sequent Calculus for ALC -- evaluating SC ALC SC with different ALC Deduction platforms -- A typical Deduction for ALC -- in the direction of an explanation idea for ALCQI -- Proofs and factors -- A Prototype Theorem Prover -- end

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That is, we show how the axiomatic presentation of ALC can be derived in NDALC . Theorem 4 NDALC is complete regarding the standard semantics of ALC. 4 NDALC Completeness 57 is a derived rule of NDALC , for supposing α implies the existence of a proof (without hypothesis) Π of α. α, without any new hypothesis by means of the following schema: Π. . B), where α ≡ β is an abbreviation for α β and β α, having obvious ≡ elimination and introduction rules, based on elimination and introduction rules.

This proof can be transformed into: P1 P2 Δ1 ⇒ Γ1 , L α, L β Δ2 , L α ⇒ Γ2 P3 ( L α) , Lβ , Lβ Δ 1 , Δ 2 ⇒ Γ 1 , Γ2 Δ2 ⇒ Γ2 Δ1 , Δ2 , Δ2 ⇒ Γ1 , Γ2 , Γ2 perm*; contract* Δ1 , Δ2 ⇒ Γ1 , Γ2 ( L β) This proof contains two mix applications, but both with grade less than g( L (α β)). So by the induction hypothesis, we can obtain a proof which contains no mixes. As mentioned above, the new created mixes are now the last inference rule of proofs which contains no mix. (iii) The outermost logical symbol is ∀.

Rm ,L m Bm )I . The right side of a sequent is interpreted as a disjunction, so that, if empty, its semantics for any interpretation function is the empty set. If we consider the simplified case where all roles (labels) are equal, that is ∀R,L 1 B1 , . . , ∀R,L m Bm ⇒, we only need to provide a new element a without fillers in R, that is, ∃x(a, x) ∈ R I . For the general case, where the most external roles on each concept can be different, the element a cannot have fillers in any of the roles.

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A proof theory for description logics by Alexandre Rademaker

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