2,773 research outputs found
Countdown games, and simulation on (succinct) one-counter nets
We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS
2016) [HLMT16] for simulation preorder of succinct one-counter nets (i.e.,
one-counter automata with no zero tests where counter increments and decrements
are integers written in binary), by showing that all relations between
bisimulation equivalence and simulation preorder are EXPSPACE-hard for these
nets. We describe a reduction from reachability games whose
EXPSPACE-completeness in the case of succinct one-counter nets was shown by
Hunter [RP 2015], by using other results. We also provide a direct
self-contained EXPSPACE-completeness proof for a special case of such
reachability games, namely for a modification of countdown games that were
shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie [LMCS 2008]; in our
modification the initial counter value is not given but is freely chosen by the
first player. We also present a new simplified proof of the belt theorem that
gives a simple graphic presentation of simulation preorder on one-counter nets
and leads to a polynomial-space algorithm; it is an alternative to the proof
from [HLMT16].Comment: A part of this paper elaborates arxiv-paper 1801.01073 and the
related paper presented at Reachability Problems 201
The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games
We study multiplayer quantitative reachability games played on a finite
directed graph, where the objective of each player is to reach his target set
of vertices as quickly as possible. Instead of the well-known notion of Nash
equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE),
a refinement of NE well-suited in the framework of games played on graphs. It
is known that there always exists an SPE in quantitative reachability games and
that the constrained existence problem is decidable. We here prove that this
problem is PSPACE-complete. To obtain this result, we propose a new algorithm
that iteratively builds a set of constraints characterizing the set of SPE
outcomes in quantitative reachability games. This set of constraints is
obtained by iterating an operator that reinforces the constraints up to
obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be
represented by a finite graph of size at most exponential. A careful inspection
of the computation allows us to establish PSPACE membership
Termination Criteria for Solving Concurrent Safety and Reachability Games
We consider concurrent games played on graphs. At every round of a game, each
player simultaneously and independently selects a move; the moves jointly
determine the transition to a successor state. Two basic objectives are the
safety objective to stay forever in a given set of states, and its dual, the
reachability objective to reach a given set of states. We present in this paper
a strategy improvement algorithm for computing the value of a concurrent safety
game, that is, the maximal probability with which player~1 can enforce the
safety objective. The algorithm yields a sequence of player-1 strategies which
ensure probabilities of winning that converge monotonically to the value of the
safety game.
Our result is significant because the strategy improvement algorithm
provides, for the first time, a way to approximate the value of a concurrent
safety game from below. Since a value iteration algorithm, or a strategy
improvement algorithm for reachability games, can be used to approximate the
same value from above, the combination of both algorithms yields a method for
computing a converging sequence of upper and lower bounds for the values of
concurrent reachability and safety games. Previous methods could approximate
the values of these games only from one direction, and as no rates of
convergence are known, they did not provide a practical way to solve these
games
Optimal Strategies in Infinite-state Stochastic Reachability Games
We consider perfect-information reachability stochastic games for 2 players
on infinite graphs. We identify a subclass of such games, and prove two
interesting properties of it: first, Player Max always has optimal strategies
in games from this subclass, and second, these games are strongly determined.
The subclass is defined by the property that the set of all values can only
have one accumulation point -- 0. Our results nicely mirror recent results for
finitely-branching games, where, on the contrary, Player Min always has optimal
strategies. However, our proof methods are substantially different, because the
roles of the players are not symmetric. We also do not restrict the branching
of the games. Finally, we apply our results in the context of recently studied
One-Counter stochastic games
Model-checking Quantitative Alternating-time Temporal Logic on One-counter Game Models
We consider quantitative extensions of the alternating-time temporal logics
ATL/ATLs called quantitative alternating-time temporal logics (QATL/QATLs) in
which the value of a counter can be compared to constants using equality,
inequality and modulo constraints. We interpret these logics in one-counter
game models which are infinite duration games played on finite control graphs
where each transition can increase or decrease the value of an unbounded
counter. That is, the state-space of these games are, generally, infinite. We
consider the model-checking problem of the logics QATL and QATLs on one-counter
game models with VASS semantics for which we develop algorithms and provide
matching lower bounds. Our algorithms are based on reductions of the
model-checking problems to model-checking games. This approach makes it quite
simple for us to deal with extensions of the logical languages as well as the
infinite state spaces. The framework generalizes on one hand qualitative
problems such as ATL/ATLs model-checking of finite-state systems,
model-checking of the branching-time temporal logics CTL and CTLs on
one-counter processes and the realizability problem of LTL specifications. On
the other hand the model-checking problem for QATL/QATLs generalizes
quantitative problems such as the fixed-initial credit problem for energy games
(in the case of QATL) and energy parity games (in the case of QATLs). Our
results are positive as we show that the generalizations are not too costly
with respect to complexity. As a byproduct we obtain new results on the
complexity of model-checking CTLs in one-counter processes and show that
deciding the winner in one-counter games with LTL objectives is
2ExpSpace-complete.Comment: 22 pages, 12 figure
Kleene Algebras and Semimodules for Energy Problems
With the purpose of unifying a number of approaches to energy problems found
in the literature, we introduce generalized energy automata. These are finite
automata whose edges are labeled with energy functions that define how energy
levels evolve during transitions. Uncovering a close connection between energy
problems and reachability and B\"uchi acceptance for semiring-weighted
automata, we show that these generalized energy problems are decidable. We also
provide complexity results for important special cases
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