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

受机器人技术中导航和控制问题的影响，Ghrist和Peterson引入了一类非正曲面（NPC）立方复合体，它们是可重配置系统的配置空间，最好被视为具体化主体（例如多关节机器人）的离散状态空间表示臂。在当前的现实世界应用中，越来越多的代理商要求自主响应感官输入，以使他们应对先验未知的导航障碍。特别地，所讨论的配置空间可能事先未知。这激发了以下问题的表述：给定一个NPC三次复数\（\ mathcal {C} \） 和一个点分隔 的布尔查询集合\（\ varSigma \） 0 -skeleton， \（\ mathcal {C} ^ {（0）} \） ，发现用于学习的高效算法 \（\ mathcal {C} \） 从所提供的输出 \（\ varSigma \） 沿适当选择的 \（\ mathcal {C} \）中的路径。在本说明中，我们解决了在已知\（\ mathcal {C} \）是CAT（0）的情况下识别\（\ mathcal {C} \）的问题。我们证明\（\ mathcal {C} \）的规范超平面的集合是在\（\ mathcal {C} ^ {上的布尔查询的点分离系统的空间上，子模minmax问题的唯一解。（0）} \），也可以根据与\（\ mathcal {C} ^ {（1）} \）的图Laplacian的二次形式来表示。