A Semantical Account of Progression in the Presence of Uncertainty

Building on a general theory of action by Reiter and his colleagues, Bacchus et al. give an account for formalizing degrees of belief and noisy actions in the situation calculus. Unfortunately, there is no clear solution to the projection problem for the formalism. And, while the model has epistemic features, it is not obvious what the agent's knowledge base should look like. Also, reasoning about uncertainty essentially resorts to second-order logic. In recent work, Gabaldon and Lakemeyer remedy these shortcomings somewhat, but here too the utility seems to be restricted to queries (with action operators) about the initial theory. In this paper, we propose a fresh amalgamation of a modal fragment of the situation calculus and uncertainty, where the idea will be to update the initial knowledge base, containing both ordinary and (certain kinds of) probabilistic beliefs, when noisy actions are performed. We show that the new semantics has the right properties, and study a special case where updating probabilistic beliefs is computable. Our ideas are closely related to the Lin and Reiter notion of progression.