# Difference between revisions of "2015 ProbabilisticTerminationSoundne"

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This is integrated in a [[small dependent type system]], to overcome the problem that lexicographic [[ranking function]]s fail when combined with [[randomization]]. </s> | This is integrated in a [[small dependent type system]], to overcome the problem that lexicographic [[ranking function]]s fail when combined with [[randomization]]. </s> | ||

Among others, [[this compositional methodology]] enables the verification of [[probabilistic program]]s outside the [[complete class]] that admits [[ranking]] [[supermartingale]]s. </s> | Among others, [[this compositional methodology]] enables the verification of [[probabilistic program]]s outside the [[complete class]] that admits [[ranking]] [[supermartingale]]s. </s> | ||

+ | |||

+ | === Body === | ||

+ | <B>Theorem 5.6 (Soundness).</B> Let <math>P</math> be a [[probabilistic program]] with [[semantics]]] <math>{(L, X) φ n}</math>, and a [[ranking]] [[supermartingale]] <math>{Y_n}</math> such that <math>Y_n = 0</math> implies <math>T φ ≤ n</math> for all valid schedulers φ. Then, P almost sure terminates, and T ∗ is integrable. | ||

==References== | ==References== | ||

− | + | ||

* 1. R. B. Ash and C. Doléans-Dade. Probability and Measure Theory. Harcourt, 2000. | * 1. R. B. Ash and C. Doléans-Dade. Probability and Measure Theory. Harcourt, 2000. | ||

* 2. Josh Berdine, Aziem Chawdhary, Byron Cook, Dino Distefano, Peter O'Hearn, Variance Analyses from Invariance Analyses, Proceedings of the 34th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, January 17-19, 2007, Nice, France [http://doi.acm.org/10.1145/1190216.1190249 doi:10.1145/1190216.1190249] | * 2. Josh Berdine, Aziem Chawdhary, Byron Cook, Dino Distefano, Peter O'Hearn, Variance Analyses from Invariance Analyses, Proceedings of the 34th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, January 17-19, 2007, Nice, France [http://doi.acm.org/10.1145/1190216.1190249 doi:10.1145/1190216.1190249] | ||

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* 34. Kirack Sohn, Allen Van Gelder, Termination Detection in Logic Programs Using Argument Sizes (extended Abstract), Proceedings of the Tenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, p.216-226, May 29-31, 1991, Denver, Colorado, USA [http://doi.acm.org/10.1145/113413.113433 doi:10.1145/113413.113433] | * 34. Kirack Sohn, Allen Van Gelder, Termination Detection in Logic Programs Using Argument Sizes (extended Abstract), Proceedings of the Tenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, p.216-226, May 29-31, 1991, Denver, Colorado, USA [http://doi.acm.org/10.1145/113413.113433 doi:10.1145/113413.113433] | ||

* 35. D. Williams. Probability with Martingales. Cambridge University Press, 1991. | * 35. D. Williams. Probability with Martingales. Cambridge University Press, 1991. | ||

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__NOTOC__ | __NOTOC__ |

## Revision as of 04:53, 14 June 2019

- (Ferrer Fioriti & Hermanns, 2015) ⇒ Luis María Ferrer Fioriti, and Holger Hermanns. (2015). “Probabilistic Termination: Soundness, Completeness, and Compositionality.” In: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ISBN:978-1-4503-3300-9 doi:10.1145/2676726.2677001

**Subject Headings:**

## Notes

## Cited By

- http://scholar.google.com/scholar?q=%222015%22+Probabilistic+Termination%3A+Soundness%2C+Completeness%2C+and+Compositionality
- http://dl.acm.org/citation.cfm?id=2676726.2677001&preflayout=flat#citedby

## Quotes

### Abstract

We propose a framework to prove almost sure termination for probabilistic programs with real valued variables. It is based on ranking supermartingales, a notion analogous to ranking functions on non-probabilistic programs. The framework is proven sound and complete for a meaningful class of programs involving randomization and bounded nondeterminism. We complement this foundational insigh by a practical proof methodology, based on sound conditions that enable compositional reasoning and are amenable to a direct implementation using modern theorem provers. This is integrated in a small dependent type system, to overcome the problem that lexicographic ranking functions fail when combined with randomization. Among others, this compositional methodology enables the verification of probabilistic programs outside the complete class that admits ranking supermartingales.

### Body

**Theorem 5.6 (Soundness).** Let [math]P[/math] be a probabilistic program with semantics] [math]{(L, X) φ n}[/math], and a ranking supermartingale [math]{Y_n}[/math] such that [math]Y_n = 0[/math] implies [math]T φ ≤ n[/math] for all valid schedulers φ. Then, P almost sure terminates, and T ∗ is integrable.

## References

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- 3. Olivier Bournez, Florent Garnier, Proving Positive Almost-sure Termination, Proceedings of the 16th International Conference on Term Rewriting and Applications, p.323-337, April 19-21, 2005, Nara, Japan doi:10.1007/978-3-540-32033-3_24
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- 6. Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Antonín Kučera, Analyzing Probabilistic Pushdown Automata, Formal Methods in System Design, v.43 n.2, p.124-163, October 2013 doi:10.1007/s10703-012-0166-0
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- 35. D. Williams. Probability with Martingales. Cambridge University Press, 1991.;

Author | volume | Date Value | title | type | journal | titleUrl | doi | note | year | |
---|---|---|---|---|---|---|---|---|---|---|

2015 ProbabilisticTerminationSoundne | Luis María Ferrer Fioriti Holger Hermanns | Probabilistic Termination: Soundness, Completeness, and Compositionality | 10.1145/2676726.2677001 | 2015 |

Author | Luis María Ferrer Fioriti + and Holger Hermanns + |

doi | 10.1145/2676726.2677001 + |

title | Probabilistic Termination: Soundness, Completeness, and Compositionality + |

year | 2015 + |