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Computing Expected Runtimes for Constant Probability Programs

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Automated Deduction – CADE 27 (CADE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11716))

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Abstract

We introduce the class of constant probability (CP) programs and show that classical results from probability theory directly yield a simple decision procedure for (positive) almost sure termination of programs in this class. Moreover, asymptotically tight bounds on their expected runtime can always be computed easily. Based on this, we present an algorithm to infer the exact expected runtime of any CP program.

Supported by the DFG Research Training Group 2236 UnRAVeL and the London Mathematical Society (Grant 41662, Research in Pairs).

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Notes

  1. 1.

    If \(p' = 0\) and \(k = 0\) then \(\mu _\mathcal {P}\ge 0\).

  2. 2.

    If \(m=0\) then \(\chi _\mathcal {P}(\lambda )=(p_0-1) \cdot \lambda ^k+p_{-1} \cdot \lambda ^{k-1}+\ldots +p_{-k}\), and if \(k=0\) then \(\chi _\mathcal {P}(\lambda )=p_m \cdot \lambda ^{m}+\ldots +p_1 \cdot \lambda +(p_0-1)\). Note that \(p_0 \ne 1\) since \(\mathcal {P}\) is PAST and in Definition 12 we required that \(m> 0\) implies \(p_m > 0\) and \(k > 0\) implies \(p_{-k} > 0\). Hence, the characteristic polynomial has exactly the degree \(k + m\), even if \(m = 0\) or \(k = 0\).

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Acknowledgments

We would like to thank Nicos Georgiou and Vladislav Vysotskiy for drawing our attention to Wald’s Lemma and to the work of Frank Spitzer on random walks, and Benjamin Lucien Kaminski and Christoph Matheja for many helpful discussions. Furthermore, we thank Tom Küspert who helped with the implementation of our technique in our tool KoAT.

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Authors and Affiliations

  1. LuFG Informatik 2, RWTH Aachen University, Aachen, Germany

    Jürgen Giesl & Marcel Hark

  2. Department of Mathematics, University of Sussex, Brighton, UK

    Peter Giesl

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  1. Jürgen Giesl
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  2. Peter Giesl
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Correspondence to Marcel Hark .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Giesl, J., Giesl, P., Hark, M. (2019). Computing Expected Runtimes for Constant Probability Programs. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_16

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