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On the Hardness of Almost–Sure Termination

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9234))

Abstract

This paper considers the computational hardness of computing expected outcomes and deciding (universal) (positive) almost–sure termination of probabilistic programs. It is shown that computing lower and upper bounds of expected outcomes is \(\varSigma _1^0\)– and \(\varSigma _2^0\)–complete, respectively. Deciding (universal) almost–sure termination as well as deciding whether the expected outcome of a program equals a given rational value is shown to be \(\varPi ^0_2\)–complete. Finally, it is shown that deciding (universal) positive almost–sure termination is \(\varSigma _2^0\)–complete (\(\varPi _3^0\)–complete).

This research is funded by the Excellence Initiative of the German federal and state governments and by the EU FP7 MEALS project.

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Notes

  1. 1.

    Fully probabilistic programs may contain probabilistic but no non–deterministic choices.

  2. 2.

    The program P cheers as it was able to prove the termantion of Q on input \(g_Q(i)\).

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Acknowledgements

We would like to thank Luis María Ferrer Fioriti (Saarland University) and Federico Olmedo (RWTH Aachen) for the fruitful discussions on the topics of this paper. Furthermore, we are very grateful for the valuable and constructive comments we received from the anonymous referees on an earlier version of this paper.

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Authors and Affiliations

  1. Software Modeling and Verification Group, RWTH Aachen University, Aachen, Germany

    Benjamin Lucien Kaminski & Joost-Pieter Katoen

Authors
  1. Benjamin Lucien Kaminski
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  2. Joost-Pieter Katoen
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Correspondence to Benjamin Lucien Kaminski or Joost-Pieter Katoen .

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Editors and Affiliations

  1. Università di Roma "Tor Vergata", Rome, Italy

    Giuseppe F Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Kaminski, B.L., Katoen, JP. (2015). On the Hardness of Almost–Sure Termination. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_24

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  • DOI: https://doi.org/10.1007/978-3-662-48057-1_24

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48056-4

  • Online ISBN: 978-3-662-48057-1

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