An optimal property of the hyperplane system in a finite cubing

Abstract

Motivated by navigation and control problems in robotics, Ghrist and Peterson introduced a class of non-positively curved (NPC) cubical complexes arising as configuration spaces of reconfigurable systems, best regarded as discretized state space representations of embodied agents such as a multi-jointed robotic arm. In current real world applications, agents are increasingly required to respond autonomously to sensory input in order for them to contend with a priori unknown obstacles to navigation. In particular, the configuration spaces in question may not be known in advance. This motivates the following problem formulation: Given a NPC cubical complex $\mathcal {C}$ and a point-separating collection $\varSigma$ of Boolean queries on its 0-skeleton, $\mathcal {C}^{(0)}$, find an efficient algorithm for learning $\mathcal {C}$ from the outputs provided by $\varSigma$ along an appropriately chosen path in $\mathcal {C}$. In this note, we tackle the problem of identifying $\mathcal {C}$ when it is known that $\mathcal {C}$ is CAT(0). We show that the collection of canonical hyperplanes of $\mathcal {C}$ is the unique solution of a sub-modular minmax problem over the space of point-separating systems of Boolean queries on $\mathcal {C}^{(0)}$, which may also be formulated in terms of the quadratic form associated with the graph Laplacian of $\mathcal {C}^{(1)}$.