Reasoning about Continuous Uncertainty in the Situation Calculus

Among the many approaches for reasoning about degrees of belief inthe presence of noisy sensing and acting, the logical accountproposed by Bacchus, Halpern, and Levesque is perhaps the most expressive.While their formalism is quite general, it is restricted to fluentswhose values are drawn from discrete countable domains, as opposed tothe continuous domains seen in many robotic applications. In thispaper, we show how this limitation in their approach can be lifted.By dealing seamlessly with both discrete distributions and continuousdensities within a rich theory of action, we provide a very generallogical specification of how belief should change after acting andsensing in complex noisy domains.