45 research outputs found
Trains, Games, and Complexity: 0/1/2-Player Motion Planning through Input/Output Gadgets
We analyze the computational complexity of motion planning through local
"input/output" gadgets with separate entrances and exits, and a subset of
allowed traversals from entrances to exits, each of which changes the state of
the gadget and thereby the allowed traversals. We study such gadgets in the 0-,
1-, and 2-player settings, in particular extending past
motion-planning-through-gadgets work to 0-player games for the first time, by
considering "branchless" connections between gadgets that route every gadget's
exit to a unique gadget's entrance. Our complexity results include containment
in L, NL, P, NP, and PSPACE; as well as hardness for NL, P, NP, and PSPACE. We
apply these results to show PSPACE-completeness for certain mechanics in
Factorio, [the Sequence], and a restricted version of Trainyard, improving
prior results. This work strengthens prior results on switching graphs and
reachability switching games.Comment: 37 pages, 36 figure
Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) without Flat Angles
A foundational result in origami mathematics is Kawasaki and Justin's simple,
efficient characterization of flat foldability for unassigned single-vertex
crease patterns (where each crease can fold mountain or valley) on flat
material. This result was later generalized to cones of material, where the
angles glued at the single vertex may not sum to . Here we
generalize these results to when the material forms a complex (instead of a
manifold), and thus the angles are glued at the single vertex in the structure
of an arbitrary planar graph (instead of a cycle). Like the earlier
characterizations, we require all creases to fold mountain or valley, not
remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly
for flat material and strongly for complexes). Equivalently, we efficiently
characterize which combinatorially embedded planar graphs with prescribed edge
lengths can fold flat, when all angles must be mountain or valley (not unfolded
flat). Our algorithm runs in time, improving on the previous
best algorithm of .Comment: 17 pages, 8 figures, to appear in Proceedings of the 38th
International Symposium on Computational Geometr
Lower Bounds on Retroactive Data Structures
We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in O(T(n,m)) time per operation, but any partially retroactive version of that data structure requires T(n,m)?m^{1-o(1)} worst-case time per operation, where n is the size of the data structure at any time and m is the number of operations. Any data structure with operations running in O(T(n,m)) time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in O(T(n,m)?m) time per operation, so our lower bound is tight up to an m^o(1) factor common in fine-grained complexity
Traversability, Reconfiguration, and Reachability in the Gadget Framework
Consider an agent traversing a graph of "gadgets", each with local state that
changes with each traversal by the agent. We characterize the complexity of
universal traversal, where the goal is to traverse every gadget at least once,
for DAG gadgets, one-state gadgets, and reversible deterministic gadgets. We
also study the complexity of reconfiguration, where the goal is to bring the
system of gadgets to a specified state, proving many cases PSPACE-complete, and
showing in some cases that reconfiguration can be strictly harder than
reachability (where the goal is for the agent to reach a specified location),
while in other cases, reachability is strictly harder than reconfiguration.Comment: Full version of article appearing in WALCOM 2022. 23 pages, 14
figure
PSPACE-Completeness of Reversible Deterministic Systems
We prove PSPACE-completeness of several reversible, fully deterministic
systems. At the core, we develop a framework for such proofs (building on a
result of Tsukiji and Hagiwara and a framework for motion planning through
gadgets), showing that any system that can implement three basic gadgets is
PSPACE-complete. We then apply this framework to four different systems,
showing its versatility. First, we prove that Deterministic Constraint Logic is
PSPACE-complete, fixing an error in a previous argument from 2008. Second, we
give a new PSPACE-hardness proof for the reversible `billiard ball' model of
Fredkin and Toffoli from 40 years ago, newly establishing hardness when only
two balls move at once. Third, we prove PSPACE-completeness of zero-player
motion planning with any reversible deterministic interacting -tunnel gadget
and a `rotate clockwise' gadget (a zero-player analog of branching hallways).
Fourth, we give simpler proofs that zero-player motion planning is
PSPACE-complete with just a single gadget, the 3-spinner. These results should
in turn make it even easier to prove PSPACE-hardness of other reversible
deterministic systems.Comment: 20 pages, 15 figure
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
Complexity of Simple Folding of Mixed Orthogonal Crease Patterns
Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of
the simple foldability problem -- deciding which crease patterns can be folded
flat by a sequence of (some model of) simple folds. We give new efficient
algorithms for mixed crease patterns, where some creases are assigned
mountain/valley while others are unassigned, for all 1D cases and for 2D
rectangular paper with orthogonal one-layer simple folds. By contrast, we show
strong NP-completeness for mixed orthogonal crease patterns on 2D rectangular
paper with some-layers simple folds, complementing a previous result for
all-layers simple folds. We also prove strong NP-completeness for finite simple
folds (no matter the number of layers) of unassigned orthogonal crease patterns
on arbitrary paper, complementing a previous result for assigned crease
patterns, and contrasting with a previous positive result for infinite
all-layers simple folds. In total, we obtain a characterization of polynomial
vs. NP-hard for all cases -- finite/infinite one/some/all-layers simple folds
of assigned/unassigned/mixed orthogonal crease patterns on
1D/rectangular/arbitrary paper -- except the unsolved case of infinite
all-layers simple folds of assigned orthogonal crease patterns on arbitrary
paper.Comment: 20 pages, 13 figures. Presented at TJCDCGGG 2021. Accepted to Thai
Journal of Mathematic
Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines
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.Comment: 22 pages, 19 figures. Presented at SAND 202