Quantified Linear Temporal Logic over Probabilistic Systems with an Application to Vacuity Checking

Authors Jakob Piribauer , Christel Baier , Nathalie Bertrand , Ocan Sankur

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Author Details

Jakob Piribauer
  • Technische Universität Dresden, Germany
Christel Baier
  • Technische Universität Dresden, Germany
Nathalie Bertrand
  • Université Rennes, Inria, CNRS, IRISA, France
Ocan Sankur
  • Université Rennes, Inria, CNRS, IRISA, France

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Jakob Piribauer, Christel Baier, Nathalie Bertrand, and Ocan Sankur. Quantified Linear Temporal Logic over Probabilistic Systems with an Application to Vacuity Checking. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Quantified linear temporal logic (QLTL) is an ω-regular extension of LTL allowing quantification over propositional variables. We study the model checking problem of QLTL-formulas over Markov chains and Markov decision processes (MDPs) with respect to the number of quantifier alternations of formulas in prenex normal form. For formulas with k{-}1 quantifier alternations, we prove that all qualitative and quantitative model checking problems are k-EXPSPACE-complete over Markov chains and k{+}1-EXPTIME-complete over MDPs. As an application of these results, we generalize vacuity checking for LTL specifications from the non-probabilistic to the probabilistic setting. We show how to check whether an LTL-formula is affected by a subformula, and also study inherent vacuity for probabilistic systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Verification by model checking
  • Quantified linear temporal logic
  • Markov chain
  • Markov decision process
  • vacuity


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