Robust Multidimensional Mean-Payoff Games are Undecidable

  • Yaron Velner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


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) ≤ 0 ∨ LimAvg(x 2) ≥ 0 ∧ LimAvg(x 3) ≥ 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 consider arbitrary (infinite-memory) strategies for both players and show that the problem is undecidable.


Simulation Step Winning Strategy Boolean Formula Left Transition Game Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yaron Velner
    • 1
  1. 1.The Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael

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