# Reasoning about Actions with Sensing under Qualitative and Probabilistic Uncertainty

*Luca Iocchi‚ Thomas Lukasiewicz‚ Daniele Nardi and Riccardo Rosati*

### Abstract

We focus on the aspect of sensing in reasoning about actions under qualitative and probabilistic uncertainty. We first define the action language E for reasoning about actions with sensing, which has a semantics based on the autoepistemic description logic ALCK_NF, and which is given a formal semantics via a system of deterministic transitions between epistemic states. As an important feature, the main computational tasks in E can be done in linear and quadratic time. We then introduce the action language E+ for reasoning about actions with sensing under qualitative and probabilistic uncertainty, which is an extension of E by actions with nondeterministic and probabilistic effects, and which is given a formal semantics in a system of deterministic, nondeterministic, and probabilistic transitions between epistemic states. We also define the notion of a belief graph, which represents the belief state of an agent after a sequence of deterministic, nondeterministic, and probabilistic actions, and which compactly represents a set of unnormalized probability distributions. Using belief graphs, we then introduce the notion of a conditional plan and its goodness for reasoning about actions under qualitative and probabilistic uncertainty. We formulate the problems of optimal and threshold conditional planning under qualitative and probabilistic uncertainty, and show that they are both uncomputable in general. We then give two algorithms for conditional planning in our framework. The first one is always sound, and it is also complete for the special case in which the relevant transitions between epistemic states are cycle-free. The second algorithm is a sound and complete solution to the problem of finite-horizon conditional planning in our framework. Under suitable assumptions, it computes every optimal finite-horizon conditional plan in polynomial time. We also describe an application of our formalism in a robotic-soccer scenario, which underlines its usefulness in realistic applications.