Synthesis from LTL Specifications with Mean-Payoff Objectives

  • Aaron Bohy
  • Véronique Bruyère
  • Emmanuel Filiot
  • Jean-François Raskin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7795)


The classical LTL synthesis problem is purely qualitative: the given LTL specification is realized or not by a reactive system. LTL is not expressive enough to formalize the correctness of reactive systems with respect to some quantitative aspects. This paper extends the qualitative LTL synthesis setting to a quantitative setting. The alphabet of actions is extended with a weight function ranging over the integer numbers. The value of an infinite word is the mean-payoff of the weights of its letters. The synthesis problem then amounts to automatically construct (if possible) a reactive system whose executions all satisfy a given LTL formula and have mean-payoff values greater than or equal to some given threshold. The latter problem is called LTL \(_\textsf{MP}\) synthesis and the LTL \(_\textsf{MP}\) realizability problem asks to check whether such a system exists. By reduction to two-player mean-payoff parity games, we first show that LTL \(_\textsf{MP}\) realizability is not more difficult than LTL realizability: it is 2ExpTime-Complete. While infinite memory strategies are required to realize LTL \(_\textsf{MP}\) specifications in general, we show that ε-optimality can be obtained with finite-memory strategies, for any ε > 0. To obtain efficient algorithms in practice, we define a Safraless procedure to decide whether there exists a finite-memory strategy that realizes a given specification for some given threshold. This procedure is based on a reduction to two-player energy safety games which are in turn reduced to safety games. Finally, we show that those safety games can be solved efficiently by exploiting the structure of their state spaces and by using antichains as a symbolic data-structure. All our results extend to multi-dimensional weights. We have implemented an antichain-based procedure and we report on some promising experimental results.


Realizability Problem Linear Temporal Logic Synthesis Problem Winning Strategy Game Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aaron Bohy
    • 1
  • Véronique Bruyère
    • 1
  • Emmanuel Filiot
    • 2
  • Jean-François Raskin
    • 3
  1. 1.Université de MonsBelgium
  2. 2.Université Paris-Est CréteilFrance
  3. 3.Université Libre de BruxellesBelgium

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