53 research outputs found
PSPACE-completeness of Pulling Blocks to Reach a Goal
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).Comment: Full version of JCDCGGG2019 paper, 22 pages, 25 figure
Push-Pull Block Puzzles are Hard
This paper proves that push-pull block puzzles in 3D are PSPACE-complete to
solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve,
settling an open question by Zubaran and Ritt. Push-pull block puzzles are a
type of recreational motion planning problem, similar to Sokoban, that involve
moving a `robot' on a square grid with obstacles. The obstacles
cannot be traversed by the robot, but some can be pushed and pulled by the
robot into adjacent squares. Thin walls prevent movement between two adjacent
squares. This work follows in a long line of algorithms and complexity work on
similar problems. The 2D push-pull block puzzle shows up in the video games
Pukoban as well as The Legend of Zelda: A Link to the Past, giving another
proof of hardness for the latter. This variant of block-pushing puzzles is of
particular interest because of its connections to reversibility, since any
action (e.g., push or pull) can be inverted by another valid action (e.g., pull
or push).Comment: Full version of CIAC 2017 paper. 17 page
Recursed Is Not Recursive: A Jarring Result
Recursed is a 2D puzzle platform video game featuring treasure chests that,
when jumped into, instantiate a room that can later be exited (similar to
function calls), optionally generating a jar that returns back to that room
(similar to continuations). We prove that Recursed is RE-complete and thus
undecidable (not recursive) by a reduction from the Post Correspondence
Problem. Our reduction is "practical": the reduction from PCP results in fully
playable levels that abide by all constraints governing levels (including the
15x20 room size) designed for the main game. Our reduction is also "efficient":
a Turing machine can be simulated by a Recursed level whose size is linear in
the encoding size of the Turing machine and whose solution length is polynomial
in the running time of the Turing machine.Comment: Submitted to MFCS2020, 21 page
Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets
A door gadget has two states and three tunnels that can be traversed by an
agent (player, robot, etc.): the "open" and "close" tunnel sets the gadget's
state to open and closed, respectively, while the "traverse" tunnel can be
traversed if and only if the door is in the open state. We prove that it is
PSPACE-complete to decide whether an agent can move from one location to
another through a planar assembly of such door gadgets, removing the
traditional need for crossover gadgets and thereby simplifying past
PSPACE-hardness proofs of Lemmings and Nintendo games Super Mario Bros., Legend
of Zelda, and Donkey Kong Country. Our result holds in all but one of the
possible local planar embedding of the open, close, and traverse tunnels within
a door gadget; in the one remaining case, we prove NP-hardness.
We also introduce and analyze a simpler type of door gadget, called the
self-closing door. This gadget has two states and only two tunnels, similar to
the "open" and "traverse" tunnels of doors, except that traversing the traverse
tunnel also closes the door. In a variant called the symmetric self-closing
door, the "open" tunnel can be traversed if and only if the door is closed. We
prove that it is PSPACE-complete to decide whether an agent can move from one
location to another through a planar assembly of either type of self-closing
door. Then we apply this framework to prove new PSPACE-hardness results for
eight different 3D Mario games and Sokobond.Comment: Accepted to FUN2020, 35 pages, 41 figure
Hierarchical Shape Construction and Complexity for Slidable Polyominoes under Uniform External Forces
Advances in technology have given us the ability to create and manipulate robots for numerous applications at the molecular scale. At this size, fabrication tool limitations motivate the use of simple robots. The individual control of these simple objects can be infeasible. We investigate a model of robot motion planning, based on global external signals, known as the tilt model. Given a board and initial placement of polyominoes, the board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked.
We propose a new hierarchy of shapes and design a single configuration that is strongly universal for any w × h bounded shape within this hierarchy (it can be reconfigured to construct any w × h bounded shape in the hierarchy). This class of shapes constitutes the most general set of buildable shapes in the literature, with most previous work consisting of just the first-level of our hierarchy. We accompany this result with a O(n4 log n)-time algorithm for deciding if a given hole-free shape is a member of the hierarchy. For our second result, we resolve a long-standing open problem within the field: We show that deciding if a given position may be covered by a tile for a given initial board configuration is PSPACEcomplete, even when all movable pieces are 1 × 1 tiles with no glues. We achieve this result by a reduction from Non-deterministic Constraint Logic for a one-player unbounded game
Algorithmic Assembly of Nanoscale Structures
The development of nanotechnology has become one of the most significant endeavors of our time. A natural objective of this field is discovering how to engineer nanoscale structures. Limitations of current top-down techniques inspire investigation into bottom-up approaches to reach this objective. A fundamental precondition for a bottom-up approach is the ability to control the behavior of nanoscale particles. Many abstract representations have been developed to model systems of particles and to research methods for controlling their behavior. This thesis develops theories on two such approaches for building complex structures: the self-assembly of simple particles, and the use of simple robot swarms. The concepts for these two approaches are straightforward. Self-assembly is the process by which simple particles, following the rules of some behavior-governing system, naturally coalesce into a more complex form. The other method of bottom-up assembly involves controlling nanoscale particles through explicit directions and assembling them into a desired form. Regarding the self-assembly of nanoscale structures, we present two construction methods in a variant of a popular theoretical model known as the 2-Handed Tile Self-Assembly Model. The first technique achieves shape construction at only a constant scale factor, while the second result uses only a constant number of unique particle types. Regarding the use of robot swarms for construction, we first develop a novel technique for reconfiguring a swarm of globally-controlled robots into a desired shape even when the robots can only move maximally in a commanded direction. We then expand on this work by formally defining an entire hierarchy of shapes which can be built in this manner and we provide a technique for doing so
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