199 research outputs found
On the complexity of heterogeneous multidimensional quantitative games
In this paper, we study two-player zero-sum turn-based games played on a
finite multidimensional weighted graph. In recent papers all dimensions use the
same measure, whereas here we allow to combine different measures. Such
heterogeneous multidimensional quantitative games provide a general and natural
model for the study of reactive system synthesis. We focus on classical
measures like the Inf, Sup, LimInf, and LimSup of the weights seen along the
play, as well as on the window mean-payoff (WMP) measure. This new measure is a
natural strengthening of the mean-payoff measure. We allow objectives defined
as Boolean combinations of heterogeneous constraints. While multidimensional
games with Boolean combinations of mean-payoff constraints are undecidable, we
show that the problem becomes EXPTIME-complete for DNF/CNF Boolean combinations
of heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} and that
exponential memory strategies are sufficient for both players to win. We
provide a detailed study of the complexity and the memory requirements when the
Boolean combination of the measures is replaced by an intersection.
EXPTIME-completeness and exponential memory strategies still hold for the
intersection of measures in {WMP, Inf, Sup, LimInf, LimSup}, and we get
PSPACE-completeness when WMP measure is no longer considered. To avoid
EXPTIME-or PSPACE-hardness, we impose at most one occurrence of WMP measure and
fix the number of Sup measures, and we propose several refinements (on the
number of occurrences of the other measures) for which we get polynomial
algorithms and lower memory requirements. For all the considered classes of
games, we also study parameterized complexity
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
Expectations or Guarantees? I Want It All! A crossroad between games and MDPs
When reasoning about the strategic capabilities of an agent, it is important
to consider the nature of its adversaries. In the particular context of
controller synthesis for quantitative specifications, the usual problem is to
devise a strategy for a reactive system which yields some desired performance,
taking into account the possible impact of the environment of the system. There
are at least two ways to look at this environment. In the classical analysis of
two-player quantitative games, the environment is purely antagonistic and the
problem is to provide strict performance guarantees. In Markov decision
processes, the environment is seen as purely stochastic: the aim is then to
optimize the expected payoff, with no guarantee on individual outcomes.
In this expository work, we report on recent results introducing the beyond
worst-case synthesis problem, which is to construct strategies that guarantee
some quantitative requirement in the worst-case while providing an higher
expected value against a particular stochastic model of the environment given
as input. This problem is relevant to produce system controllers that provide
nice expected performance in the everyday situation while ensuring a strict
(but relaxed) performance threshold even in the event of very bad (while
unlikely) circumstances. It has been studied for both the mean-payoff and the
shortest path quantitative measures.Comment: In Proceedings SR 2014, arXiv:1404.041
Words derivated from Sturmian words
AbstractA return word of a factor of a Sturmian word starts at an occurrence of that factor and ends exactly before its next occurrence. Derivated words encode the unique decomposition of a word in terms of return words. Vuillon has proved that each factor of a Sturmian word has exactly two return words. We determine these two return words, as well as their first occurrence, for the prefixes of characteristic Sturmian words. We then characterize words derivated from a characteristic Sturmian word and give their precise form. Finally, we apply our results to obtain a new proof of the characterization of characteristic Sturmian words which are fixed points of morphisms
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