Modal Markov Logic for Multiple Agents

Modal Markov Logic for a single agent has previously been proposed as an extension to propositional Markov logic. While the framework allowed reasoning under the principle of maximum entropy for various modal logics, it is not feasible to apply its counting based inference to reason about the beliefs and knowledge of multiple agents due to magnitude of the numbers involved. We propose a modal extension of propositional Markov logic that avoids this problem by coarsening the state space. The problem stems from the fact that in the single-agent setting, the state space is only doubly exponential in the number of propositions in the domain, but the state space can potentially become infinite in the multi-agent setting. In addition, the proposed framework adds only the overhead of deciding satisfiability for the chosen modal logic on the top of the complexity of exact inference in propositional Markov logic. The proposed framework allows one to find a distribution that matches probabilities of formulas obtained from training data (or provided by an expert). Finally, we show how one can compute lower and upper bounds on probabilities of arbitrary formulas.