A Logic of Only-Believing over Arbitrary Probability Distributions

Qihui Feng, Daxin Liu, Vaishak Belle, Gerhard Lakemeyer

Abstract: When it comes to robotic agents operating in an uncertain world, a major concern in knowledge representation is to better relate high-level logical accounts of beliefs and actions to the low-level probabilistic sensorimotor data. Perhaps the most general formalism for dealing with degrees of belief in formulas, and in particular, with how that should evolve in the presence of noisy sensing and acting is the first-order logical account by Bacchus, Halpern, and Levesque. The main advantage of such a logical account is that it allows a specification of beliefs that can be partial or incomplete, in keeping with whatever information is available about the domain, making it particularly attractive for general-purpose cognitive robotics. Recently, this model was extended to handle continuous probability distributions. However, it is limited to finitely many nullary fluents and defines beliefs and integration axiomatically, the latter making semantic proofs about beliefs and meta-beliefs difficult.In this paper, we revisit the continuous model and cast it in a modal language. We will go beyond nullary fluents and allow fluents of arbitrary arity as is usual in the standard situation calculus. This necessitates a new and general treatment of probabilities on possible worlds, where we define measures on uncountably many worlds that interpret infinitely many fluents. We then show how this leads to a fairly simple definition of knowing, degrees of belief, and only-knowing. Properties thereof will also be analyzed. In this paper, we focus on the static setting and conclude with some thoughts about extending this account to actions as the next step and what challenges might arise.