Among the many approaches for reasoning about degrees of belief in the presence of noisy sensing and acting, the logical account proposed by Bacchus, Halpern, and Levesque is perhaps the most expressive. While their formalism is quite general, it is restricted to fluents whose values are drawn from discrete finite domains, as opposed to the continuous domains seen in many robotic applications. In this work, we show how this limitation in that approach can be lifted. By dealing seamlessly with both discrete distributions and continuous densities within a rich theory of action, we provide a very general logical specification of how belief should change after acting and sensing in complex noisy domains.
In this paper we present PDDL+, a planning domain description language for modelling mixed discrete-continuous planning domains. We describe the syntax and modelling style of PDDL+, showing that the language makes convenient the modelling of complex time-dependent effects. We provide a formal semantics for PDDL+ by mapping planning instances into constructs of hybrid automata. Using the syntax of HAs as our semantic model we construct a semantic mapping to labelled transition systems to complete the formal interpretation of PDDL+ planning instances. An advantage of building a mapping from PDDL+ to HA theory is that it forms a bridge between the Planning and Real Time Systems research communities. One consequence is that we can expect to make use of some of the theoretical properties of HAs. For example, for a restricted class of HAs the Reachability problem (which is equivalent to Plan Existence) is decidable. PDDL+ provides an alternative to the continuous durative action model of PDDL2.1, adding a more flexible and robust model of time-dependent behaviour.  Overcoming the disadvantages of equidistant discretization of continuous actions, we introduce an approach that separates time into slices of varying length bordered by certain events. Such events are points in time at which the equations describing the system's behavior--that is, the equations which specify the ongoing processes--change. Between two events the system's parameters stay continuous. A high-level semantics for drawing logical conclusions about dynamic systems with continuous processes is presented, and we have developed an adequate calculus to automate this reasoning process. In doing this, we have combined deduction and numerical calculus, offering logical reasoning about precise, quantitative system information. The scenario of multiple balls moving in 1-dimensional space interacting with a pendulum serves as demonstration example of our method.