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We examine a novel setting in which two parties have partial knowledge of the elements that make up a Markov Decision Process (MDP) and must cooperate to compute and execute an optimal policy for the problem constructed from those elements. This situation arises when one party wants to give a robot some task, but does not wish to divulge those details to a second party-while the second party possesses sensitive data about the robot's dynamics (information needed for planning). Both parties want the robot to perform the task successfully, but neither is willing to disclose any more information than is absolutely necessary. We utilize techniques from secure multi-party computation, combining primitives and algorithms to construct protocols that can compute an optimal policy while ensuring that the policy remains opaque by being split across both parties. To execute a split policy, we also give a protocol that enables the robot to determine what actions to trigger, while the second party guards against attempts to probe for information inconsistent with the policy's prescribed execution. In order to improve scalability, we find that basis functions and constraint sampling methods are useful in forming effective approximate MDPs. We report simulation results examining performance and precision, and assess the scaling properties of our Python implementation. We also describe a hardware proof-of-feasibility implementation using inexpensive physical robots, which, being a small-scale instance, can be solved directly. 

Typically to a roboticist, a plan is the outcome of other work, a synthesized object that realizes ends defined by some problem; plans qua plans are seldom treated as first-class objects of study. Plans designate functionality: a plan can be viewed as defining a robot’s behavior throughout its execution. This informs and reveals many other aspects of the robot’s design, including: necessary sensors and action choices, history, state, task structure, and how to define progress. Interrogating sets of plans helps in comprehending the ways in which differing executions influence the interrelationships between these various aspects. Revisiting Erdmann’s theory of action-based sensors, a classical approach for characterizing fundamental information requirements, we show how plans (in their role of designating behavior) influence sensing requirements. Using an algorithm for enumerating plans, we examine how some plans for which no action-based sensor exists can be transformed into sets of sensors through the identification and handling of features that preclude the existence of action-based sensors. We are not aware of those obstructing features having been previously identified. Action-based sensors may be treated as standalone reactive plans; we relate them to the set of all possible plans through a lattice structure. This lattice reveals a boundary between plans with action-based sensors and those without. Some plans, specifically those that are not reactive plans and require some notion of internal state, can never have associated action-based sensors. Even so, action-based sensors can serve as a framework to explore and interpret how such plans make use of state. 

One important class of applications entails a robot scrutinizing, monitoring, or recording the evolution of an uncertain time-extended process. This sort of situation leads to an interesting family of active perception problems that can be cast as planning problems in which the robot is limited in what it sees and must, thus, choose what to pay attention to. The distinguishing characteristic of this setting is that the robot has influence over what it captures via its sensors, but exercises no causal authority over the process evolving in the world. As such, the robot’s objective is to observe the underlying process and to produce a “chronicle” of occurrent events, subject to a goal specification of the sorts of event sequences that may be of interest. This paper examines variants of such problems in which the robot aims to collect sets of observations to meet a rich specification of their sequential structure. We study this class of problems by modeling a stochastic process via a variant of a hidden Markov model and specify the event sequences of interest as a regular language, developing a vocabulary of “mutators” that enable sophisticated requirements to be expressed. Under different suppositions on the information gleaned about the event model, we formulate and solve different planning problems. The core underlying idea is the construction of a product between the event model and a specification automaton. Using this product, we compute a policy that minimizes the expected number of steps to reach a goal state. We introduce a general algorithm for this problem as well as several more efficient algorithms for important special cases. The paper reports and compares performance metrics by drawing on some small case studies analyzed in depth via simulation. Specifically, we study the effect of the robot’s observation model on the average time required for the robot to record a desired story. We also compare our algorithm with a baseline greedy algorithm, showing that our algorithm outperforms the greedy algorithm in terms of the average time to record a desired story. In addition, experiments show that the algorithms tailored to specialized variants of the problem are rather more efficient than the general algorithm.