103,990 research outputs found
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
Limit Your Consumption! Finding Bounds in Average-energy Games
Energy games are infinite two-player games played in weighted arenas with
quantitative objectives that restrict the consumption of a resource modeled by
the weights, e.g., a battery that is charged and drained. Typically, upper
and/or lower bounds on the battery capacity are part of the problem
description. Here, we consider the problem of determining upper bounds on the
average accumulated energy or on the capacity while satisfying a given lower
bound, i.e., we do not determine whether a given bound is sufficient to meet
the specification, but if there exists a sufficient bound to meet it.
In the classical setting with positive and negative weights, we show that the
problem of determining the existence of a sufficient bound on the long-run
average accumulated energy can be solved in doubly-exponential time. Then, we
consider recharge games: here, all weights are negative, but there are recharge
edges that recharge the energy to some fixed capacity. We show that bounding
the long-run average energy in such games is complete for exponential time.
Then, we consider the existential version of the problem, which turns out to be
solvable in polynomial time: here, we ask whether there is a recharge capacity
that allows the system player to win the game.
We conclude by studying tradeoffs between the memory needed to implement
strategies and the bounds they realize. We give an example showing that memory
can be traded for bounds and vice versa. Also, we show that increasing the
capacity allows to lower the average accumulated energy.Comment: In Proceedings QAPL'16, arXiv:1610.0769
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
Average-energy games
Two-player quantitative zero-sum games provide a natural framework to
synthesize controllers with performance guarantees for reactive systems within
an uncontrollable environment. Classical settings include mean-payoff games,
where the objective is to optimize the long-run average gain per action, and
energy games, where the system has to avoid running out of energy.
We study average-energy games, where the goal is to optimize the long-run
average of the accumulated energy. We show that this objective arises naturally
in several applications, and that it yields interesting connections with
previous concepts in the literature. We prove that deciding the winner in such
games is in NP inter coNP and at least as hard as solving mean-payoff games,
and we establish that memoryless strategies suffice to win. We also consider
the case where the system has to minimize the average-energy while maintaining
the accumulated energy within predefined bounds at all times: this corresponds
to operating with a finite-capacity storage for energy. We give results for
one-player and two-player games, and establish complexity bounds and memory
requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
Non-Zero Sum Games for Reactive Synthesis
In this invited contribution, we summarize new solution concepts useful for
the synthesis of reactive systems that we have introduced in several recent
publications. These solution concepts are developed in the context of non-zero
sum games played on graphs. They are part of the contributions obtained in the
inVEST project funded by the European Research Council.Comment: LATA'16 invited pape
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
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