Qihui Feng: A Logic of Belief over Arbitrary Probability Distribution
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 is the first-order logical account by Bacchus, Halpern, and Levesque. The main advantage of this 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 distributions and joint distributions of discrete and continuous random variables. 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 our recent work, we revisit the continuous model and cast it in a modal language. We will go beyond absolutely continuous distributions. Also, we allow fluents of arbitrary arity as is usual in the standard situation calculus. These necessitate a new and general treatment of probabilities on possible worlds, where we define measures on uncountably many worlds that interpret fluents of arbitrary arity. We then show how this leads to a fairly simple definition of knowing, degrees of belief, and also only-knowing. Properties thereof will also be analyzed.