Abstract
Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account of various martingale-based methods for over- and underapproximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points—a classic topic in computer science—in the analysis of probabilistic programs. This leads us to two new martingale-based techniques, too. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales.
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Notes
- 1.
We note that the theory of NNRepSupM can also be developed using Markov’s lemma.
- 2.
Precisely it is the restriction of \(\overline{\mathbb {P}}^{\mathrm {reach}}_C\) to I; in what follows we do this identification for \(\overline{\mathbb {P}}^{\mathrm {reach}}_C\), \(\underline{\mathbb {P}}^{\mathrm {reach}}_C\), \(\overline{\mathbb {E}}^{\mathrm {steps}}_C\), and \(\underline{\mathbb {E}}^{\mathrm {steps}}_C\).
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Acknowledgment
We thank Shin-ya Katsumata, Takamasa Okudono and the anonymous referees for useful comments. The authors are supported by JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), the JSPS-INRIA Bilateral Joint Research Project “CRECOGI,” and JSPS KAKENHI Grant Numbers 15KT0012 & 15K11984. Natsuki Urabe is supported by JSPS KAKENHI Grant Number 16J08157.
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Takisaka, T., Oyabu, Y., Urabe, N., Hasuo, I. (2018). Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_28
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