Progression with probabilities in the Situation Calculus: Representation and succinctness

Progression in the Situation Calculus is perhaps one of the most extensively studied cases of updating logical theories over a sequence of actions. While it generally requires second-order logic, several useful first-order and tractable cases have been identified. Recently, there has been an interest in studying the progression of probabilistic knowledge bases expressed using degrees of belief on first-order formulas. However, although a few results exist, they do not provide much clarity about how this progression can be computed or represented in a feasible manner. In this paper, we address this problem for the first time. We first examine the progression of a probabilistic knowledge base (PKB) in a world-level representation; in particular, we show that such a representation is closed under progression for any local-effect actions with quantifier-free contexts. We also propose a more succinct representation of the probabilistic knowledge base, i.e. factored-representation PKB. For this type of PKB, we study the conditions for progression to remain succinct.