1,436 research outputs found
A Parity Game Tale of Two Counters
Parity games are simple infinite games played on finite graphs with a winning
condition that is expressive enough to capture nested least and greatest
fixpoints. Through their tight relationship to the modal mu-calculus, they are
used in practice for the model-checking and synthesis problems of the
mu-calculus and related temporal logics like LTL and CTL. Solving parity games
is a compelling complexity theoretic problem, as the problem lies in the
intersection of UP and co-UP and is believed to admit a polynomial-time
solution, motivating researchers to either find such a solution or to find
superpolynomial lower bounds for existing algorithms to improve the
understanding of parity games. We present a parameterized parity game called
the Two Counters game, which provides an exponential lower bound for a wide
range of attractor-based parity game solving algorithms. We are the first to
provide an exponential lower bound to priority promotion with the delayed
promotion policy, and the first to provide such a lower bound to tangle
learning.Comment: In Proceedings GandALF 2019, arXiv:1909.0597
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
Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL
We continue the investigation of parameterized extensions of Linear Temporal
Logic (LTL) that retain the attractive algorithmic properties of LTL: a
polynomial space model checking algorithm and a doubly-exponential time
algorithm for solving games. Alur et al. and Kupferman et al. showed that this
is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have
temporal operators equipped with variables that bound their scope in time.
Later, this was also shown to be true for Parametric LDL (PLDL), which extends
PLTL to be able to express all omega-regular properties.
Here, we generalize PLTL to systems with costs, i.e., we do not bound the
scope of operators in time, but bound the scope in terms of the cost
accumulated during time. Again, we show that model checking and solving games
for specifications in PLTL with costs is not harder than the corresponding
problems for LTL. Finally, we discuss PLDL with costs and extensions to
multiple cost functions.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms
This paper presents a new exponential lower bound for the two most popular
deterministic variants of the strategy improvement algorithms for solving
parity, mean payoff, discounted payoff and simple stochastic games. The first
variant improves every node in each step maximizing the current valuation
locally, whereas the second variant computes the globally optimal improvement
in each step. We outline families of games on which both variants require
exponentially many strategy iterations
Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time
We consider the optimization variant of the realizability problem for Prompt
Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the
prompt eventually operator whose scope is bounded by some parameter. In the
realizability optimization problem, one is interested in computing the minimal
such bound that allows to realize a given specification. It is known that this
problem is solvable in triply-exponential time, but not whether it can be done
in doubly-exponential time, i.e., whether it is just as hard as solving LTL
realizability.
We take a step towards resolving this problem by showing that the optimum can
be approximated within a factor of two in doubly-exponential time. Also, we
report on a proof-of-concept implementation of the algorithm based on bounded
LTL synthesis, which computes the smallest implementation of a given
specification. In our experiments, we observe a tradeoff between the size of
the implementation and the bound it realizes. We investigate this tradeoff in
the general case and prove upper bounds, which reduce the search space for the
algorithm, and matching lower bounds.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
New Deterministic Algorithms for Solving Parity Games
We study parity games in which one of the two players controls only a small
number of nodes and the other player controls the other nodes of the
game. Our main result is a fixed-parameter algorithm that solves bipartite
parity games in time , and general parity games in
time , where is the number of distinct
priorities and is the number of edges. For all games with this
improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and
Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved
deterministic algorithm for graphs with small average degree
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