A Weakness Measure for GR(1) Formulae

  • Davide Giacomo CavezzaEmail author
  • Dalal Alrajeh
  • András György
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10951)


In spite of the theoretical and algorithmic developments for system synthesis in recent years, little effort has been dedicated to quantifying the quality of the specifications used for synthesis. When dealing with unrealizable specifications, finding the weakest environment assumptions that would ensure realizability is typically a desirable property; in such context the weakness of the assumptions is a major quality parameter. The question of whether one assumption is weaker than another is commonly interpreted using implication or, equivalently, language inclusion. However, this interpretation does not provide any further insight into the weakness of assumptions when implication does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it fails to provide a sufficiently refined notion of weakness in case of GR(1) formulae, a subset of linear temporal logic formulae which is of particular interest in controller synthesis. In this paper we propose a more refined measure of weakness based on the Hausdorff dimension, a concept that captures the notion of size of the omega-language satisfying a linear temporal logic formula. We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae. We evaluate our proposed weakness measure in the context of computing GR(1) assumptions refinements.



The support of the EPSRC HiPEDS Centre for Doctoral Training (EP/L016796/1) is gratefully acknowledged. We also thank our reviewers for their insightful comments and suggestions.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Davide Giacomo Cavezza
    • 1
    Email author
  • Dalal Alrajeh
    • 1
  • András György
    • 1
  1. 1.Imperial College LondonLondonUK

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