By Alexandre Rademaker
Creation -- historical past -- The Sequent Calculus for ALC -- evaluating SC ALC SC with different ALC Deduction platforms -- A common Deduction for ALC -- in the direction of an explanation idea for ALCQI -- Proofs and causes -- A Prototype Theorem Prover -- end
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Extra resources for A proof theory for description logics
On the other hand, a subsumption Φ Ψ has no concept associate to it. It states, instead, a truth-value statement, depending on whether the interpretation of Φ is included in the corresponding interpretation of Ψ . In terms of a logical system, DL has no concept internalizing . 2 The NDALC System 53 As we will see on the next section, this imposes quite particular features on the form of the normal proofs in NDALC . L 2 β only depends on the assumption L 1 α and no other In the rule -i, L 1 α hypothesis.
3 are obtained from a SCALC without cut-rule, we are actually proving the completeness of SCALC without the cut-rule. Given that, the results can also be considered an alternative method of cut-elimination for A. 1007/978-1-4471-4002-3_4, © The Author(s) 2012 37 4 Comparing SCALC with Other ALC Deduction Systems 38 the SCALC presented in Sect. 4, where we followed Gentzen’s original proof for cut elimination. 2 Comparing SALC With the Structural Subsumption Algorithm The structural subsumption algorithms (SSA) presented in  compare the syntactic structure of two normalized concept descriptions in order to verify if the first one is subsumed by the second one.
Moreover, weak rules will be used with the unique purpose of enabling promotion rules applications. 44 4 Comparing SCALC with Other ALC Deduction Systems A more insightful definition of the last item above would be possible if we replace the weak rules in SC ALC by the weak ∗ rule below. [Δ ], [Δ, Δ1 ]k , Δ ⇒ Γ, [Γ1 , Γ ]k , [Γ ] [Δ ], Δ, Δ1 ⇒ Γ1 , Γ, [Γ ] weak ∗ Lemma 6 The weak ∗ rule is a derived rule in SC ALC . Proof To prove Lemma 6, given a derivation fragment Π with a weak ∗ rule application, we show how to replace it by successive weak-l and weak-r applications.