Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning.Despite this widespread use, a frequently overlooked difficulty associated with polynomial zonotopes is intersection checking. Determining whether the reachable set, represented as a polynomial zonotope, intersects an unsafe set is not straightforward.In fact, we show that this fundamental operation is NP-hard, even for a simple class of polynomial zonotopes.The standard method for intersection checking with polynomial zonotopes is a two-part algorithm that overapproximates a polynomial zonotope with a regular zonotope and then, if the overapproximation error is deemed too large, splits the set and recursively tries again.Beyond the possible need for a large number of splits, we identify two sources of concern related to this algorithm: (1) overapproximating a polynomial zonotope with a zonotope has unbounded error, and (2) after splitting a polynomial zonotope, the overapproximation error can actually increase.Taken together, this implies there may be a possibility that the algorithm does not always terminate.We perform a rigorous analysis of the method and detail necessary conditions for the union of overapproximations to provably converge to the original polynomial zonotope.

翻译：多项式Zonotopes作为一种非凸集合表示，在机器人实时运动规划与控制、非线性系统可达性分析以及强化学习安全防护等领域具有广泛应用。尽管应用广泛，但多项式Zonotopes交集检测这一常被忽视的难点问题亟待解决。判断以多项式Zonotope表示的可达集是否与不安全集相交并非易事。事实上，我们证明即使对于简单类别的多项式Zonotopes，这一基本运算也是NP-hard的。当前标准交集检测算法采用两阶段方法：先用正则Zonotopes对多项式Zonotope进行过逼近，若逼近误差过大则对集合进行分割并递归重试。除可能需要大量分割外，我们发现该算法存在两大隐患：(1) 用Zonotope逼近多项式Zonotope的误差无界，(2) 分割后逼近误差反而可能增大。这意味着算法可能无法保证终止性。我们对该方法进行了严格分析，详述了确保逼近结果序列能够收敛至原始多项式Zonotope的必要条件。