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 $$\mathcal {C}^{(1)}$$ .