# Stochastic invariants for probabilistic termination

K. Chatterjee, P. Novotny, D. Zikelic, in:, ACM, 2017, pp. 145–160.

Conference Paper | Published | English
Department
Series Title
ACM SIGPLAN Notices
Abstract
Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.
Publishing Year
Date Published
2017-01-01
Volume
52
Issue
1
Page
145 - 160
Conference
POPL: Principles of Programming Languages
Conference Location
Paris, France
Conference Date
2017-01-15 – 2017-01-21
ISSN
IST-REx-ID

### Cite this

Chatterjee K, Novotny P, Zikelic D. Stochastic invariants for probabilistic termination. In: Vol 52. ACM; 2017:145-160. doi:10.1145/3009837.3009873
Chatterjee, K., Novotny, P., & Zikelic, D. (2017). Stochastic invariants for probabilistic termination (Vol. 52, pp. 145–160). Presented at the POPL: Principles of Programming Languages, Paris, France: ACM. https://doi.org/10.1145/3009837.3009873
Chatterjee, Krishnendu, Petr Novotny, and Djordje Zikelic. “Stochastic Invariants for Probabilistic Termination,” 52:145–60. ACM, 2017. https://doi.org/10.1145/3009837.3009873.
K. Chatterjee, P. Novotny, and D. Zikelic, “Stochastic invariants for probabilistic termination,” presented at the POPL: Principles of Programming Languages, Paris, France, 2017, vol. 52, no. 1, pp. 145–160.
Chatterjee K, Novotny P, Zikelic D. 2017. Stochastic invariants for probabilistic termination. POPL: Principles of Programming Languages, ACM SIGPLAN Notices, vol. 52. 145–160.
Chatterjee, Krishnendu, et al. Stochastic Invariants for Probabilistic Termination. Vol. 52, no. 1, ACM, 2017, pp. 145–60, doi:10.1145/3009837.3009873.