14 research outputs found
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
Symblicit algorithms for optimal strategy synthesis in monotonic Markov decision processes
When treating Markov decision processes (MDPs) with large state spaces, using
explicit representations quickly becomes unfeasible. Lately, Wimmer et al. have
proposed a so-called symblicit algorithm for the synthesis of optimal
strategies in MDPs, in the quantitative setting of expected mean-payoff. This
algorithm, based on the strategy iteration algorithm of Howard and Veinott,
efficiently combines symbolic and explicit data structures, and uses binary
decision diagrams as symbolic representation. The aim of this paper is to show
that the new data structure of pseudo-antichains (an extension of antichains)
provides another interesting alternative, especially for the class of monotonic
MDPs. We design efficient pseudo-antichain based symblicit algorithms (with
open source implementations) for two quantitative settings: the expected
mean-payoff and the stochastic shortest path. For two practical applications
coming from automated planning and LTL synthesis, we report promising
experimental results w.r.t. both the run time and the memory consumption.Comment: In Proceedings SYNT 2014, arXiv:1407.493
Robust Multidimensional Mean-Payoff Games are Undecidable
Mean-payoff games play a central role in quantitative synthesis and
verification. In a single-dimensional game a weight is assigned to every
transition and the objective of the protagonist is to assure a non-negative
limit-average weight. In the multidimensional setting, a weight vector is
assigned to every transition and the objective of the protagonist is to satisfy
a boolean condition over the limit-average weight of each dimension, e.g.,
\LimAvg(x_1) \leq 0 \vee \LimAvg(x_2)\geq 0 \wedge \LimAvg(x_3) \geq 0. We
recently proved that when one of the players is restricted to finite-memory
strategies then the decidability of determining the winner is inter-reducible
with Hilbert's Tenth problem over rationals (a fundamental long-standing open
problem). In this work we allow arbitrary (infinite-memory) strategies for both
players and we show that the problem is undecidable
Strategy Synthesis for Multi-dimensional Quantitative Objectives
Multi-dimensional mean-payoff and energy games provide the mathematical
foundation for the quantitative study of reactive systems, and play a central
role in the emerging quantitative theory of verification and synthesis. In this
work, we study the strategy synthesis problem for games with such
multi-dimensional objectives along with a parity condition, a canonical way to
express -regular conditions. While in general, the winning strategies
in such games may require infinite memory, for synthesis the most relevant
problem is the construction of a finite-memory winning strategy (if one
exists). Our main contributions are as follows. First, we show a tight
exponential bound (matching upper and lower bounds) on the memory required for
finite-memory winning strategies in both multi-dimensional mean-payoff and
energy games along with parity objectives. This significantly improves the
triple exponential upper bound for multi energy games (without parity) that
could be derived from results in literature for games on VASS (vector addition
systems with states). Second, we present an optimal symbolic and incremental
algorithm to compute a finite-memory winning strategy (if one exists) in such
games. Finally, we give a complete characterization of when finite memory of
strategies can be traded off for randomness. In particular, we show that for
one-dimension mean-payoff parity games, randomized memoryless strategies are as
powerful as their pure finite-memory counterparts.Comment: Conference version published in CONCUR 2012, LNCS 7454. Journal
version published in Acta Informatica, volume 51, issue 3-4, Springer, 201
Qualitative Analysis of Concurrent Mean-payoff Games
We consider concurrent games played by two-players on a finite-state graph,
where in every round the players simultaneously choose a move, and the current
state along with the joint moves determine the successor state. We study a
fundamental objective, namely, mean-payoff objective, where a reward is
associated to each transition, and the goal of player 1 is to maximize the
long-run average of the rewards, and the objective of player 2 is strictly the
opposite. The path constraint for player 1 could be qualitative, i.e., the
mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative,
i.e., a given threshold between the minimal and maximal reward. We consider the
computation of the almost-sure (resp. positive) winning sets, where player 1
can ensure that the path constraint is satisfied with probability 1 (resp.
positive probability). Our main results for qualitative path constraints are as
follows: (1) we establish qualitative determinacy results that show that for
every state either player 1 has a strategy to ensure almost-sure (resp.
positive) winning against all player-2 strategies, or player 2 has a spoiling
strategy to falsify almost-sure (resp. positive) winning against all player-1
strategies; (2) we present optimal strategy complexity results that precisely
characterize the classes of strategies required for almost-sure and positive
winning for both players; and (3) we present quadratic time algorithms to
compute the almost-sure and the positive winning sets, matching the best known
bound of algorithms for much simpler problems (such as reachability
objectives). For quantitative constraints we show that a polynomial time
solution for the almost-sure or the positive winning set would imply a solution
to a long-standing open problem (the value problem for turn-based deterministic
mean-payoff games) that is not known to be solvable in polynomial time
Hyperplane Separation Technique for Multidimensional Mean-Payoff Games
We consider both finite-state game graphs and recursive game graphs (or
pushdown game graphs), that can model the control flow of sequential programs
with recursion, with multi-dimensional mean-payoff objectives. In pushdown
games two types of strategies are relevant: global strategies, that depend on
the entire global history; and modular strategies, that have only local memory
and thus do not depend on the context of invocation. We present solutions to
several fundamental algorithmic questions and our main contributions are as
follows: (1) We show that finite-state multi-dimensional mean-payoff games can
be solved in polynomial time if the number of dimensions and the maximal
absolute value of the weight is fixed; whereas if the number of dimensions is
arbitrary, then problem is already known to be coNP-complete. (2) We show that
pushdown graphs with multi-dimensional mean-payoff objectives can be solved in
polynomial time. (3) For pushdown games under global strategies both single and
multi-dimensional mean-payoff objectives problems are known to be undecidable,
and we show that under modular strategies the multi-dimensional problem is also
undecidable (whereas under modular strategies the single dimensional problem is
NP-complete). We show that if the number of modules, the number of exits, and
the maximal absolute value of the weight is fixed, then pushdown games under
modular strategies with single dimensional mean-payoff objectives can be solved
in polynomial time, and if either of the number of exits or the number of
modules is not bounded, then the problem is NP-hard. (4) Finally we show that a
fixed parameter tractable algorithm for finite-state multi-dimensional
mean-payoff games or pushdown games under modular strategies with
single-dimensional mean-payoff objectives would imply the solution of the
long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282