We give a PSPACE-completeness reduction from QBF (quantified Boolean formulas) to the Dyson Telescopes puzzle where opposing telescopes can overlap in at least two spaces. The reduction does not use tail ends of telescopes or initially partially extended telescopes. If two opposing telescopes can overlap in at most one space, we can solve the puzzle in polynomial time by a reduction to graph reachability.
The complexity of many motion-planning problems has been studied extensively in the literature. This work has recently focused on very simple combinatorial puzzles (one-player games) that nonetheless exhibit the theoretical difficulty of general motion planning; see, e.g., [1]. Two main examples of this pursuit are a suite of pushing-block puzzles, culminating in [2; 3], and a suite of problems involving sliding-block puzzles [4]. In pushing-block puzzles, an agent must navigate an environment and push blocks in order to reach a goal configuration, while avoiding collisions. The variations of pushing blocks began with several versions that appeared in video games (the most classic being Sokoban), and continued to consider simpler and simpler puzzles with the goal of finding a polynomially solvable puzzle. Nonetheless, all reasonable pushing-block puzzles turned out to be NP-hard, and many turned out to be PSPACE-complete, with no problems known to be in NP, except in one trivial case where solution paths are forced to be short. Similarly, sliding-block puzzles are usually PSPACE-complete, even in very simple models.
In this paper we consider a motion-planning puzzle, the Dyson Telescopes puzzle. It takes the form of an enjoyable computer game [5], invented and developed by the Dyson company to advertise a vacuum cleaner called “Telescope” that is retractable like an astronomical telescope. The puzzle is perhaps most closely related to sliding blocks, in the sense that the agent is outside the environment. At any time, the agent can extend or retract one of several “telescopes”, each of which has a specified, fixed length in extended form. Erickson [6] posed the complexity of the problem in 2003. The complexity remained open despite fairly extensive pursuit—it seemed nearly impossible to build gadgets that required multiple entrances. Thus we hoped that it would be the first “interesting” yet polynomially solvable motion-planning puzzle.
We prove that the Dyson Telescopes puzzle is indeed polynomially solvable in a fairly natural situation in which the extended forms of opposing telescopes (two telescopes on the same row or column, pointing towards each other) overlap in at most one space. However, some of Dyson’s puzzles do not satisfy this restriction. We prove that this small flexibility in the general form of the problem in fact makes the problem PSPACE-complete.
