# Download A proof theory for description logics by Alexandre Rademaker PDF ISBN-10: 144714001X

ISBN-13: 9781447140016

ISBN-10: 1447140028

ISBN-13: 9781447140023

Advent -- historical past -- The Sequent Calculus for ALC -- evaluating SC ALC SC with different ALC Deduction structures -- A normal Deduction for ALC -- in the direction of an explanation thought for ALCQI -- Proofs and motives -- A Prototype Theorem Prover -- end

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It is easy to see that from (1) and (2) S1 ∪ S2 |= (C1 C2 ) L α and S1 ∪ S2 |= (C1 C2 ) L β. 1 we get the desired result S1 ∪ S2 |= (C1 C2 ) L (α β). Rules -i Π1 Again by induction hypothesis, if L α is a derivation with all hypothesis in {C, S} then S |= C L α. 3 to S |= C L (α β). Rule ( -e) By induction hypothesis, if Π1 L (α β), [ L β] [ L α] Π2 Π3 γ and γ are derivations with hypothesis in {C, S}, { L α, S} and { L β, S}, respectively. Then, L (α β), S |= L α γ and S |= L β γ . 3, S |= L (α β) transitivity of set inclusion we can get the desired result S |= C γ .

Rm ,L m Bm ⇒ C1 , . . , Cl , ∃R1 ,L 1 D1 , . . 4) where we group the concepts into four sets Δ1 , Δ2 , Δ3 and Δ4 . A1,n and C1,l are sets of atomic concepts. In Δ2 , B1,m are atomic concepts or disjunctions of concepts (not necessarily atomic). In Δ4 , D1, p are atomic concepts or conjunctions of concepts (not necessarily atomic). 4), one has just to observe that: (1) the -r and -l rules provisos are blocking the decomposition of the conjunctions and disjunctions; and (2) the prom-∀ (prom-∃) rule cannot be applied due to the lack of a universal (existential) quantified concept on the right (left).

That is, ∀R occuring in front of the list of labels in Δ2 , ∃x(a, x) ∈ R I . We must mention that even if one of the concepts in Δ2 is or ⊥, we can always construct I. Case ⇒ Δ4 We can construct a counter-model such that I |= ⇒ Δ4 . From the natural interpretation of a sequent, we know that an interpretation will not satisfy this case when there is at least one element a ∈ (∃R1 ,L 1 D1 . . ∃R p ,L p D p )I . 3 Obtaining Counter-Models From Unsuccessful Proof Trees 47 sequent is interpreted as a conjunction, if empty, its semantics for any interpretation function is the universe set of the interpretation.