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Consider an agent traversing a graph of “gadgets”, where each gadget has local state that changes with each traversal by the agent according to specified rules. Prior work has studied the computational complexity of deciding whether the agent can reach a specified location, a problem we call reachability. This paper introduces new goals for the agent, aiming to characterize when the computational complexity of these problems is the same or differs from that of reachability. First we characterize the complexity of universal traversal—where the goal is to traverse every gadget at least once—for DAG gadgets (partially), one-state gadgets, and reversible deterministic gadgets. Then we study the complexity of reconfiguration—where the goal is to bring the system of gadgets to a specified state. We prove many cases PSPACE-complete, and show in some cases that reconfiguration is strictly harder than reachability, while in other cases, reachability is strictly harder than reconfiguration.

We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.

We prove PSPACE-completeness of the well-studied pushing-block puzzle Push-1F, a theoretical abstraction of many video games (first posed in 1999). We also prove PSPACE-completeness of two versions of the recently studied block-moving puzzle game with gravity, Block Dude - a video game dating back to 1994 - featuring either liftable blocks or pushable blocks. Two of our reductions are built on a new framework for "checkable" gadgets, extending the motion-planning-through-gadgets framework to support gadgets that can be misused, provided those misuses can be detected later.

© 2020 Information Processing Society of Japan. We prove PSPACE-completeness of all but one problem in a large space of pulling-block problems where the goal is for the agent to reach a target destination. The problems are parameterized by whether pulling is optional, the number of blocks which can be pulled simultaneously, whether there are fixed blocks or thin walls, and whether there is gravity. We show NP-hardness for the remaining problem, Pull?-1FG (optional pulling, strength 1, fixed blocks, with gravity).

We prove PSPACE-completeness of the well-studied pushing-block puzzle Push-1F, a theoretical abstraction of many video games (first posed in 1999). We also prove PSPACE-completeness of two versions of the recently studied block-moving puzzle game with gravity, Block Dude - a video game dating back to 1994 - featuring either liftable blocks or pushable blocks. Two of our reductions are built on a new framework for "checkable" gadgets, extending the motion-planning-through-gadgets framework to support gadgets that can be misused, provided those misuses can be detected later.