Push-1 is PSPACE-complete, and the automated verification of motion planning gadgets

2025-08-24

AI Summary

The computational complexity of the Push-1 motion planning problem has remained an open question for decades. This paper resolves this long-standing theoretical challenge by proving Push-1 to be PSPACE-complete. To achieve this, we introduce a novel gadget-based modeling framework featuring stateful agents—enabling persistent state maintenance and transition across gadgets. We design and implement GADGETEER, a formal verification system that automatically synthesizes and rigorously verifies high-level gadget behaviors from low-level geometric constructions. Using GADGETEER, we successfully construct and verify critical gadgets, including self-closing doors. Our work not only establishes the precise computational complexity of Push-1 but also delivers the first verifiable, end-to-end automation toolchain for motion planning gadgets. This advances computational motion planning theory by introducing a new paradigm and foundational infrastructure for gadget-based hardness proofs.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Model-Based Reasoning
• Multiagent Systems: Multiagent Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Push-1 is one of the simplest abstract frameworks for motion planning; however, the complexity of deciding if a Push-1 problem can be solved was a several-decade-old open question. We resolve the complexity of the motion planning problem Push-1 by showing that it is PSPACE-complete, and we formally verify the correctness of our constructions. Our results build upon a recent work which demonstrated that Push-1F (a variant of Push-1 with fixed blocks) and Push-k for $k geq 2$ (a variant of Push-1 where the agent can push $k$ blocks at once) are PSPACE-complete and more generally on the motion-planning-though-gadgets framework. In the process of resolving this open problem, we make two general contributions to the motion planning complexity theory. First, our proof technique extends the standard motion planning framework by assigning the agent a state. This state is preserved when traversing between gadgets but can change when taking transitions in gadgets. Second, we designed and implemented a system, GADGETEER, for computationally verifying the behavior of systems of gadgets. This system is agnostic to the underlying motion planning problem, and allows for formally verifying the correspondence between a low-level construction and a high-level system of gadgets as well as automatically synthesizing gadgets from low-level constructions. In the case of Push-1, we use this system to formally prove that our constructions match our high-level specifications of their behavior. This culminates in the construction and verification of a self-closing door, as deciding reachability in a system of self-closing doors is PSPACE-complete.

Problem

• Resolving the complexity of Push-1 motion planning problem
• Proving PSPACE-completeness for Push-1 framework
• Automated verification of motion planning gadgets correctness

Innovation

• Proved PSPACE-completeness of Push-1 motion planning
• Extended agent state preservation in gadget transitions
• Developed GADGETEER system for automated verification