By Alexandre Rademaker

ISBN-10: 144714001X

ISBN-13: 9781447140016

ISBN-10: 1447140028

ISBN-13: 9781447140023

Creation -- 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 idea for ALCQI -- Proofs and motives -- A Prototype Theorem Prover -- end

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Proof Theory. Number 81 in Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1975) Chapter 4 Comparing SCALC With Other ALC Deduction Systems Abstract The structural subsumption algorithm is restricted to a quite inexpressive language. Simple Tableaux based algorithms generally fails to provide short proofs. On the other hand, the latter has a useful property, it returns a counter-model from an unsuccessful proof. A counter-model, that is, an interpretation that falsifies the premise, is a quite useful object to a knowledge-base engineer.

1, we call each pair (Δk , Γk ) a context in the sequent. 1) Δ1 , [Δ2 ]1 , . . , [Δn ]n−1 ⇒ Γ1 , [Γ2 ]1 , . . , [Γn ]n−1 SC[] ALC does not have permutation, contraction or the cut rule from SCALC . Reading bottom-up, the weak rules of SC[] ALC save the context of the proof before removing a concept from the lefthand (antecedent) or righthand side (succedent) of the sequent and the frozen-exchange changes the contexts during a proof construction. Considering that in SC[] ALC the sequents are constructed by two sets (not lists) of concepts, weak rules are still necessary only to allow the application of promotional rules.

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 [1] compare the syntactic structure of two normalized concept descriptions in order to verify if the first one is subsumed by the second one. In order to compare deductions in SCALC with deductions in SSA, we just have to observe that each step taken by a bottom-up construction of a SCALC proof corresponds to a step of the SSA algorithm towards the concepts matching.

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