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The paper introduces lifted and/or schematics, which are an abstract representation of the PTP search space. Analyzing the schematic lets us quantify the size of the search space. This is of independent theoretical interest (although the theory is not quite there yet to provide deep insights), and is practically useful in the context of Rao-Blackwellized importance sampling. I believe this type of work is important; we still lack understanding of lifted inference algorithms. The proposed schematics are not really simple enough to provide a whole lot of insight, but it's a move in the right direction.