790 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
The Complexity of All-switches Strategy Improvement
Strategy improvement is a widely-used and well-studied class of algorithms
for solving graph-based infinite games. These algorithms are parameterized by a
switching rule, and one of the most natural rules is "all switches" which
switches as many edges as possible in each iteration. Continuing a recent line
of work, we study all-switches strategy improvement from the perspective of
computational complexity. We consider two natural decision problems, both of
which have as input a game , a starting strategy , and an edge . The
problems are: 1.) The edge switch problem, namely, is the edge ever
switched by all-switches strategy improvement when it is started from on
game ? 2.) The optimal strategy problem, namely, is the edge used in the
final strategy that is found by strategy improvement when it is started from
on game ? We show -completeness of the edge switch
problem and optimal strategy problem for the following settings: Parity games
with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski;
mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff
games and simple stochastic games with their standard strategy improvement
algorithms. We also show -completeness of an analogous problem
to edge switch for the bottom-antipodal algorithm for finding the sink of an
Acyclic Unique Sink Orientation on a cube
The Complexity of the Simplex Method
The simplex method is a well-studied and widely-used pivoting method for
solving linear programs. When Dantzig originally formulated the simplex method,
he gave a natural pivot rule that pivots into the basis a variable with the
most violated reduced cost. In their seminal work, Klee and Minty showed that
this pivot rule takes exponential time in the worst case. We prove two main
results on the simplex method. Firstly, we show that it is PSPACE-complete to
find the solution that is computed by the simplex method using Dantzig's pivot
rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a
specific variable to enter the basis is PSPACE-complete. We use the known
connection between Markov decision processes (MDPs) and linear programming, and
an equivalence between Dantzig's pivot rule and a natural variant of policy
iteration for average-reward MDPs. We construct MDPs and show
PSPACE-completeness results for single-switch policy iteration, which in turn
imply our main results for the simplex method
Quantitative games with interval objectives
Traditionally quantitative games such as mean-payoff games and discount sum
games have two players -- one trying to maximize the payoff, the other trying
to minimize it. The associated decision problem, "Can Eve (the maximizer)
achieve, for example, a positive payoff?" can be thought of as one player
trying to attain a payoff in the interval . In this paper we
consider the more general problem of determining if a player can attain a
payoff in a finite union of arbitrary intervals for various payoff functions
(liminf, mean-payoff, discount sum, total sum). In particular this includes the
interesting exact-value problem, "Can Eve achieve a payoff of exactly (e.g.)
0?"Comment: Full version of CONCUR submissio
Regular Boardgames
We propose a new General Game Playing (GGP) language called Regular
Boardgames (RBG), which is based on the theory of regular languages. The
objective of RBG is to join key properties as expressiveness, efficiency, and
naturalness of the description in one GGP formalism, compensating certain
drawbacks of the existing languages. This often makes RBG more suitable for
various research and practical developments in GGP. While dedicated mostly for
describing board games, RBG is universal for the class of all finite
deterministic turn-based games with perfect information. We establish
foundations of RBG, and analyze it theoretically and experimentally, focusing
on the efficiency of reasoning. Regular Boardgames is the first GGP language
that allows efficient encoding and playing games with complex rules and with
large branching factor (e.g.\ amazons, arimaa, large chess variants, go,
international checkers, paper soccer).Comment: AAAI 201
The Complexity of Codiagnosability for Discrete Event and Timed Systems
In this paper we study the fault codiagnosis problem for discrete event
systems given by finite automata (FA) and timed systems given by timed automata
(TA). We provide a uniform characterization of codiagnosability for FA and TA
which extends the necessary and sufficient condition that characterizes
diagnosability. We also settle the complexity of the codiagnosability problems
both for FA and TA and show that codiagnosability is PSPACE-complete in both
cases. For FA this improves on the previously known bound (EXPTIME) and for TA
it is a new result. Finally we address the codiagnosis problem for TA under
bounded resources and show it is 2EXPTIME-complete.Comment: 24 pages
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
Percentile Queries in Multi-Dimensional Markov Decision Processes
Markov decision processes (MDPs) with multi-dimensional weights are useful to
analyze systems with multiple objectives that may be conflicting and require
the analysis of trade-offs. We study the complexity of percentile queries in
such MDPs and give algorithms to synthesize strategies that enforce such
constraints. Given a multi-dimensional weighted MDP and a quantitative payoff
function , thresholds (one per dimension), and probability thresholds
, we show how to compute a single strategy to enforce that for all
dimensions , the probability of outcomes satisfying is at least . We consider classical quantitative payoffs from
the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum,
discounted sum). Our work extends to the quantitative case the multi-objective
model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape
Trainyard is NP-Hard
Recently, due to the widespread diffusion of smart-phones, mobile puzzle
games have experienced a huge increase in their popularity. A successful puzzle
has to be both captivating and challenging, and it has been suggested that this
features are somehow related to their computational complexity \cite{Eppstein}.
Indeed, many puzzle games --such as Mah-Jongg, Sokoban, Candy Crush, and 2048,
to name a few-- are known to be NP-hard \cite{CondonFLS97,
culberson1999sokoban, GualaLN14, Mehta14a}. In this paper we consider
Trainyard: a popular mobile puzzle game whose goal is to get colored trains
from their initial stations to suitable destination stations. We prove that the
problem of determining whether there exists a solution to a given Trainyard
level is NP-hard. We also \href{http://trainyard.isnphard.com}{provide} an
implementation of our hardness reduction
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