Interactive First-Order Probabilistic Logic

Being able to compactly represent large state spaces is crucial in solving a vast majority of practical stochastic planning problems. This requirement is even more stringent in the context of multi-agent systems, in which the world to be modeled also includes the mental state of other agents. This leads to a hierarchy of beliefs that results in a continuous, unbounded set of possible interactive states, as in the case of Interactive POMDPs. In this paper, we describe a novel representation for interactive belief hierarchies that combines first-order logic and probability. The semantics of this new formalism is based on recursively partitioning the belief space at each level of the hierarchy; in particular, the partitions of the belief simplex at one level constitute the vertices of the simplex at the next higher level. Since in general a set of probabilistic statements only partially specifies a probability distribution over the space of interest, we adopt the maximum entropy principle in order to convert it to a full specification.