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The condition $\tau(\mathbf{D}_{0}, w_{0,1},\ldots,w_{0,q})=\tau(\mathbf{D}_{1}, w_{1,1},\ldots,w_{1,q})$ rules out the possibility that $\mathcal{A}$ issues the keywords as I pointed in the question, since the output of $\tau$ includes the search pattern matrix defined in page 9. This has the effect of weakening the definition to allow search pattern ...