EXPLANATION IN RECOMMENDER SYSTEMS 195 ≤sim(C2,Q)+ Wa(sima(C1, Q")+I-sima(C1, C2)) <sim(C1, 2)+ Wasim(C1, 0*)=sim(C1, @") a∈Ao+-A So C2 is dominated by CI as required. It remains to show that if: sim(C1, 0)<sim(C2, 0)+ Wa(l-sima(C1, C2)) a∈A-A then C2 is not dominated by Cl. Let Q* be the complete extension of Q such that a(@*)=Ta(C2) for all a E A-Ao. It can be seen that sima(C2, Q*)=l and sima(C1, @*)=sima(C1, C2)for all aEA-Ag and sim(C1,Q)=sim(C,)+∑ Wasim2(C,Q”) ≤sim(C2,Q)+ Wa(sima(C1,0*)+1-sima(Cl, C2)) sim(C2, 0)+ Wa=sim(C2, 0") It follows as required that C2 is not dominated by C Theorem 2: The recommendation dialogue in iNn can be safely termi nated if and only if the following conditions hold. 1. any case that equals the similarity of the target case to the current query has the same values as the target case for all remaining attributes, 2. all cases that are less similar than the target case are dominated by Proof: By definition, the recommendation dialogue can be safely ter- minated if and only if the current query Q is such that r(Q)=r(e) for all possible extensions Q" of @. Also by definition, CtEr(Q) where Ct is the case currently selected by inN as the target case Suppose now that Conditions I and 2 hold, and let o* be any extension of o. It is clear from Condition I that sim(C, 0*) sim(C1, Q*) for any CEr(Q). On the other hand, it follows from Con dition 2 that for any C r(O), sim(C, 0*)<sim(C, 0*)and so CEXPLANATION IN RECOMMENDER SYSTEMS 195 ≤sim(C2, Q)+
a∈AQ∗−AQ wa(sima(C1, Q∗ )+1−sima(C1, C2)) <sim(C1, Q)+
a∈AQ∗−AQ wasima(C1, Q∗ )=sim(C1, Q∗ ). So C2 is dominated by C1 as required. It remains to show that if: sim(C1, Q)≤sim(C2, Q)+
a∈A−AQ wa(1−sima(C1, C2)) then C2 is not dominated by C1. Let Q∗ be the complete extension of Q such that πa(Q∗) = πa(C2) for all a ∈ A − AQ. It can be seen that sima(C2, Q∗) = 1 and sima(C1, Q∗) = sima(C1, C2) for all a ∈ A − AQ, and so: sim(C1, Q∗ )=sim(C1, Q)+
a∈A−AQ wasima(C1, Q∗ ) ≤sim(C2, Q)+
a∈A−AQ wa(sima(C1, Q∗ )+1−sima(C1, C2)) =sim(C2, Q)+
a∈A−AQ wa =sim(C2, Q∗ ). It follows as required that C2 is not dominated by C1. Theorem 2: The recommendation dialogue in iNN can be safely termi￾nated if and only if the following conditions hold: 1. any case that equals the similarity of the target case to the current query has the same values as the target case for all remaining attributes, 2. all cases that are less similar than the target case are dominated by the target case. Proof: By definition, the recommendation dialogue can be safely ter￾minated if and only if the current query Q is such that r(Q∗)=r(Q) for all possible extensions Q∗ of Q. Also by definition, Ct ∈ r(Q), where Ct is the case currently selected by iNN as the target case. Suppose now that Conditions 1 and 2 hold, and let Q∗ be any extension of Q. It is clear from Condition 1 that sim(C,Q∗) = sim(Ct, Q∗) for any C ∈r(Q). On the other hand, it follows from Con￾dition 2 that for any C /∈ r(Q), sim(C,Q∗) < sim(Ct, Q∗) and so C /∈