Many first-order probabilistic models can be represented much more compactly using aggregation operations such as counting. While traditional statistical relational representations share factors across sets of interchangeable random variables, representations that explicitly model aggregations also exploit interchange-ability of random variables within factors. This is especially useful in decision making settings, where an agent might need to reason about counts of the different types of objects it interacts with. Previous work on counting formulas in statistical relational representations has mostly focused on the problem of exact inference on an existing model. The problem of learning such models is largely unexplored. In this paper, we introduce Counting Markov Logic Networks (C-MLNs), an extension of Markov logic networks that can compactly represent complex counting formulas. We present a structure learning algorithm for C-MLNs; we apply this algorithm to the novel problem of generalizing natural language instructions, and to relational reinforcement learning in the Crossblock domain, in which standard MLN learning algorithms fail to find any useful structure. The C-MLN policies learned from natural language instructions are compact and intuitive, and, despite requiring no instructions on test games, win 20% more Crossblock games than a state-of-the-art algorithm for following natural language instructions.
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