Analyzing generalized planning under nondeterminism

Abstract

In automated planning, there has been a recent interest in solving a class of problems, where a single solution applies for multiple, possibly infinitely many, instances. This necessitates a generalized notion of plans, such as plans with loops. However, the correctness of such plans is non-trivial to define, making it difficult to provide a clear specification of what we should be looking for. In an influential paper, Levesque proposed a formal specification for analyzing the correctness of such plans. He motivated a logical characterization within the situation calculus that included binary sensing actions. This characterization argued that from each state considered possible initially, the plan should terminate while satisfying the goal. Increasingly, classical plan structures are being applied to stochastic environments such as robotics applications. This raises the question as to what the specification for correctness should look like, since Levesque's account makes the assumption that actions are deterministic.In this work, we aim to generalize Levesque's account to handle actions with nondeterministic outcomes, which may also be accorded probabilities. By appealing to an extension of the situation calculus to handle probabilistic nondeterminism, we will show that Levesque's definition, as well as a notion of goal achievability proposed by Lin and Levesque, have limited appeal under stochastic nondeterminism. In essence, they correspond to one correct execution, which is unlikely to be adequate. Rather, we propose to delineate between goal satisfaction and termination leading to a range of correctness criteria. To better study these criteria, and to position the results in a broader context while still allowing for the generality of the situation calculus, we consider an abstract framework to study the correctness of plans with loops, in domains that are possibly unbounded, and/or stochastic, and/or continuous. Within that framework, we then prove numerous relationships between the criteria, including some impossibility results for categorically satisfying goals. Finally, we show that these notions provide a more granular view than those discussed in the literature, such as strong planning and strong cyclic planning.