Universal Plans: One Action Sequence to Solve Them All!

This paper introduces the notion of a universal plan, which when executed, is guaranteed to solve all planning problems in a category, regardless of the obstacles, initial state, and goal set. Such plans are specified as a deterministic sequence of actions that are blindly applied without any sensor feedback. Thus, they can be considered as pure exploration in a reinforcement learning context, and we show that with basic memory requirements, they even yield asymptotically optimal plans. Building upon results in number theory and theory of automata, we provide universal plans both for discrete and continuous (motion) planning and prove their (semi)completeness. The concepts are applied and illustrated through simulation studies, and several directions for future research are sketched.

An Internal Model Principle For Robots

When designing a robot's internal system, one often makes assumptions about the structure of the intended environment of the robot. One may even assign meaning to various internal components of the robot in terms of expected environmental correlates. In this paper we want to make the distinction between robot's internal and external worlds clear-cut. Can the robot learn about its environment, relying only on internally available information, including the sensor data? Are there mathematical conditions on the internal robot system which can be internally verified and make the robot's internal system mirror the structure of the environment? We prove that sufficiency is such a mathematical principle, and mathematically describe the emergence of the robot's internal structure isomorphic or bisimulation equivalent to that of the environment. A connection to the free-energy principle is established, when sufficiency is interpreted as a limit case of surprise minimization. As such, we show that surprise minimization leads to having an internal model isomorphic to the environment. This also parallels the Good Regulator Principle which states that controlling a system sufficiently well means having a model of it. Unlike the mentioned theories, ours is discrete, and non-probabilistic.

A Mathematical Characterization of Minimally Sufficient Robot Brains

This paper addresses the lower limits of encoding and processing the information acquired through interactions between an internal system (robot algorithms or software) and an external system (robot body and its environment) in terms of action and observation histories. Both are modeled as transition systems. We want to know the weakest internal system that is sufficient for achieving passive (filtering) and active (planning) tasks. We introduce the notion of an information transition system for the internal system which is a transition system over a space of information states that reflect a robot's or other observer's perspective based on limited sensing, memory, computation, and actuation. An information transition system is viewed as a filter and a policy or plan is viewed as a function that labels the states of this information transition system. Regardless of whether internal systems are obtained by learning algorithms, planning algorithms, or human insight, we want to know the limits of feasibility for given robot hardware and tasks. We establish, in a general setting, that minimal information transition systems exist up to reasonable equivalence assumptions, and are unique under some general conditions. We then apply the theory to generate new insights into several problems, including optimal sensor fusion/filtering, solving basic planning tasks, and finding minimal representations for modeling a system given input-output relations.