Safety and Liveness of Quantitative Automata

Authors Udi Boker , Thomas A. Henzinger , Nicolas Mazzocchi , N. Ege Saraç

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Udi Boker
  • Reichman University, Herzliya, Israel
Thomas A. Henzinger
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Nicolas Mazzocchi
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
N. Ege Saraç
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria


We thank Christof Löding for pointing us to some results on {{PSpace}-hard}ess of universality problems and the anonymous reviewers for their helpful comments.

Cite AsGet BibTex

Udi Boker, Thomas A. Henzinger, Nicolas Mazzocchi, and N. Ege Saraç. Safety and Liveness of Quantitative Automata. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The safety-liveness dichotomy is a fundamental concept in formal languages which plays a key role in verification. Recently, this dichotomy has been lifted to quantitative properties, which are arbitrary functions from infinite words to partially-ordered domains. We look into harnessing the dichotomy for the specific classes of quantitative properties expressed by quantitative automata. These automata contain finitely many states and rational-valued transition weights, and their common value functions Inf, Sup, LimInf, LimSup, LimInfAvg, LimSupAvg, and DSum map infinite words into the totally-ordered domain of real numbers. In this automata-theoretic setting, we establish a connection between quantitative safety and topological continuity and provide an alternative characterization of quantitative safety and liveness in terms of their boolean counterparts. For all common value functions, we show how the safety closure of a quantitative automaton can be constructed in PTime, and we provide PSpace-complete checks of whether a given quantitative automaton is safe or live, with the exception of LimInfAvg and LimSupAvg automata, for which the safety check is in ExpSpace. Moreover, for deterministic Sup, LimInf, and LimSup automata, we give PTime decompositions into safe and live automata. These decompositions enable the separation of techniques for safety and liveness verification for quantitative specifications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Quantitative automata
  • quantitative safety
  • quantitative liveness
  • quantitative automata


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