By Alexandre Rademaker
Advent -- historical past -- The Sequent Calculus for ALC -- evaluating SC ALC SC with different ALC Deduction platforms -- A usual Deduction for ALC -- in the direction of an explanation thought for ALCQI -- Proofs and reasons -- A Prototype Theorem Prover -- end
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Extra info for A proof theory for description logics
To illustrate the process, let us consider an application of a quasi-mix rule in the SC∗ ALC -proof fragment below where Πn are proof fragments. The double-line labeled with “perm*; contract*” means the application of rule permutation one or more times followed by one or more applications of contraction rule. Π2 Π1 L Δ1 ⇒ Γ1 α ⇒ Γ2 ∗ +∃R Δ 1 ⇒ Γ1 , Γ2 Π3 (∃R,L α , L α) And its corresponding SCALC -proof: Π2 Π1 Δ1 ⇒ Γ1 Δ1 ⇒ Γ1∗ , ∃R,L α perm*; contract* Δ1 ⇒ Γ1∗ , +∃R Γ2 Π3 Lα ⇒ Γ2 ∃R,L α ⇒ +∃R Γ 2 prom-∃ cut By the proof of Lemma 3, a derivation Π of Δ ⇒ Γ in SCALC with cuts can be transformed in a derivation Π of Δ ⇒ Γ in SC∗ ALC with quasi-mixes (and mixes).
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. That is, ∀R occuring in front of the list of labels in Δ2 , ∃x(a, x) ∈ R I .
Before proceeding to present the procedure to obtain counter-models from SC ALC -proofs, we must introduce Lemma 5 showing that SC ALC is a conservative extension of SCALC . Lemma 5 Consider Δ ⇒ Γ a SCALC sequent. If P is a proof of Δ ⇒ Γ in SC ALC then it is possible to construct a proof P of Δ ⇒ Γ in SCALC . Proof Each application of a frozen-exchange rule correspond to a shift of contexts during the bottom-up proof construction process. To proof Lemma 5 we need a twosteps procedure to: (1) remove all frozen-exchange applications of a given proof (a proof in SC ALC without any frozen-exchange application is naturally translated to a proof in SCALC ); (2) replace the weak rules of SC ALC by their counterparts in SCALC .
A proof theory for description logics by Alexandre Rademaker