194 D MCSHERRY the recommendation process in inn and how it combines an effec tive approach to reducing the length of recommendation dialogues with a mechanism for ensuring that the dialogue is terminated only when it is certain that the recommendation will be the same no matter how the user chooses to extend her query. We have also presented novel approach to explanation of the recommendation process in which there is no requirement for domain knowledge other than the similarit knowledge and cases already available to the system We have demonstrated our approach in a mixed-initiative recom- mender system called Top Case which can explain the relevance of any question the user is asked in terms of its strategy of eliminating competing cases and ultimately confirming the target case as the rec- ommended case. Top Case can also justify its recommendations on the grounds that any un-elicited preferences of the user cannot affect the outcome. In future research we plan to investigate the potential impact of the system's explanation capabilities and ability to support mixed-initiative interaction on the effectiveness of the recommenda- tion process Appendix A. Theorems I and 2 Lemma 1: For any cases C1, C2, query e, and a E A: sima(C2, 2)< sima(C1, 0)+I-sima(Cl, C2) Proof: By the triangle inequality, 1-sima(C1, 0)s1-sima(Cl, C2)+ 1-sima(C2, 0). The required inequality easily follows Theorem 1: A given case C2 is dominated by another case CI with respect to a query e if and only if: sim(C2,0)+> wa(l-sima(C1, C2))<sim(C1, 0) Proof: If the latter condition holds. then it must also be true that sim(C2,@+> wa(l-sima(Cl, C2))<sim(C1, e) for any extension Q* of Q. It follows from Lemma I that for any extension 0""of Q sim(C2,Q")=sim(C2,0)+> Wasim(C2, 0)194 D. MCSHERRY the recommendation process in iNN and how it combines an effec￾tive approach to reducing the length of recommendation dialogues with a mechanism for ensuring that the dialogue is terminated only when it is certain that the recommendation will be the same no matter how the user chooses to extend her query. We have also presented a novel approach to explanation of the recommendation process in which there is no requirement for domain knowledge other than the similarity knowledge and cases already available to the system. We have demonstrated our approach in a mixed-initiative recom￾mender system called Top Case which can explain the relevance of any question the user is asked in terms of its strategy of eliminating competing cases and ultimately confirming the target case as the rec￾ommended case. Top Case can also justify its recommendations on the grounds that any un-elicited preferences of the user cannot affect the outcome. In future research we plan to investigate the potential impact of the system’s explanation capabilities and ability to support mixed-initiative interaction on the effectiveness of the recommenda￾tion process. Appendix A. Theorems 1 and 2. Lemma 1: For any cases C1, C2, query Q, and a ∈ A: sima(C2, Q) ≤ sima(C1, Q)+1−sima(C1, C2). Proof: By the triangle inequality, 1 − sima(C1, Q) ≤ 1 − sima(C1, C2) + 1−sima(C2, Q). The required inequality easily follows. Theorem 1: A given case C2 is dominated by another case C1 with respect to a query Q if and only if: sim(C2, Q)+
a∈A−AQ wa(1−sima(C1, C2)) <sim(C1, Q). Proof: If the latter condition holds, then it must also be true that: sim(C2, Q)+
a∈AQ∗−AQ wa(1−sima(C1, C2)) <sim(C1, Q) for any extension Q∗ of Q. It follows from Lemma 1 that for any extension Q∗ of Q: sim(C2, Q∗ )=sim(C2, Q)+
a∈AQ∗−AQ wasima(C2, Q∗ )