We build a general theory for characterizing the computational complexity of motion planning of robot(s) through a graph of "gadgets", where each gadget has its own state defining a set of allowed traversals which in turn modify the gadget's state. We study two families of such gadgets, one which naturally leads to motion planning problems with polynomially bounded solutions, and another which leads to polynomially unbounded (potentially exponential) solutions. We also study a range of competitive game-theoretic scenarios, from one player controlling one robot to teams of players each controlling their own robot and racing to achieve their team's goal. Under small restrictions on these gadgets, we fully characterize the complexity of bounded 1-player motion planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs. PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete), and we partially characterize the complexity of unbounded 2-player motion planning (P vs. EXPTIME-complete), bounded 2-team motion planning (P vs. NEXPTIME-complete), and unbounded 2-team motion planning (P vs. undecidable). These results can be seen as an alternative to Constraint Logic (which has already proved useful as a basis for hardness reductions), providing a wide variety of agent-based gadgets, any one of which suffices to prove a problem hard.

中文翻译：

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迈向运动规划复杂性的一般理论：表征哪些小工具使游戏变得困难

我们建立了一个通用理论，通过“小工具”图来表征机器人运动规划的计算复杂性，其中每个小工具都有自己的状态，定义了一组允许的遍历，这些遍历又会修改小工具的状态。我们研究了两类这样的小工具，一种自然会导致多项式有界解决方案的运动规划问题，另一种会导致多项式无界（潜在指数）解决方案。我们还研究了一系列竞争性博弈论场景，从一名玩家控制一个机器人到各组玩家各自控制自己的机器人并竞相实现团队目标。在对这些小工具的小限制下，我们充分表征了有界 1 人运动规划（NL 与 NP 完全）、无界 1 人运动规划（NL 与 NP 完全）的复杂性。PSPACE-complete）和有界 2 人运动规划（P vs. PSPACE-complete），我们部分描述了无界 2 人运动规划（P vs. EXPTIME-complete）、有界 2 人运动规划（ P vs. NEXPTIME-complete），以及无限的 2-team 运动规划（P vs. undecidable）。这些结果可以看作是约束逻辑的替代方案（已经证明它可以作为降低硬度的基础），提供了各种各样的基于代理的小工具，其中任何一个都足以证明一个问题。