TY - GEN
T1 - Sound and Complete Certificates for Quantitative Termination Analysis of Probabilistic Programs
AU - Chatterjee, Krishnendu
AU - Goharshady, Amir Kafshdar
AU - Meggendorfer, Tobias
AU - Žikelić, Đorđe
N1 - Publisher Copyright:
© 2022, The Author(s).
PY - 2022
Y1 - 2022
N2 - We consider the quantitative problem of obtaining lower-bounds on the probability of termination of a given non-deterministic probabilistic program. Specifically, given a non-termination threshold p∈ [ 0, 1 ], we aim for certificates proving that the program terminates with probability at least 1 - p. The basic idea of our approach is to find a terminating stochastic invariant, i.e. a subset SI of program states such that (i) the probability of the program ever leaving SI is no more than p, and (ii) almost-surely, the program either leaves SI or terminates. While stochastic invariants are already well-known, we provide the first proof that the idea above is not only sound, but also complete for quantitative termination analysis. We then introduce a novel sound and complete characterization of stochastic invariants that enables template-based approaches for easy synthesis of quantitative termination certificates, especially in affine or polynomial forms. Finally, by combining this idea with the existing martingale-based methods that are relatively complete for qualitative termination analysis, we obtain the first automated, sound, and relatively complete algorithm for quantitative termination analysis. Notably, our completeness guarantees for quantitative termination analysis are as strong as the best-known methods for the qualitative variant. Our prototype implementation demonstrates the effectiveness of our approach on various probabilistic programs. We also demonstrate that our algorithm certifies lower bounds on termination probability for probabilistic programs that are beyond the reach of previous methods.
AB - We consider the quantitative problem of obtaining lower-bounds on the probability of termination of a given non-deterministic probabilistic program. Specifically, given a non-termination threshold p∈ [ 0, 1 ], we aim for certificates proving that the program terminates with probability at least 1 - p. The basic idea of our approach is to find a terminating stochastic invariant, i.e. a subset SI of program states such that (i) the probability of the program ever leaving SI is no more than p, and (ii) almost-surely, the program either leaves SI or terminates. While stochastic invariants are already well-known, we provide the first proof that the idea above is not only sound, but also complete for quantitative termination analysis. We then introduce a novel sound and complete characterization of stochastic invariants that enables template-based approaches for easy synthesis of quantitative termination certificates, especially in affine or polynomial forms. Finally, by combining this idea with the existing martingale-based methods that are relatively complete for qualitative termination analysis, we obtain the first automated, sound, and relatively complete algorithm for quantitative termination analysis. Notably, our completeness guarantees for quantitative termination analysis are as strong as the best-known methods for the qualitative variant. Our prototype implementation demonstrates the effectiveness of our approach on various probabilistic programs. We also demonstrate that our algorithm certifies lower bounds on termination probability for probabilistic programs that are beyond the reach of previous methods.
UR - http://www.scopus.com/inward/record.url?scp=85135872735&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-13185-1_4
DO - 10.1007/978-3-031-13185-1_4
M3 - Conference contribution
AN - SCOPUS:85135872735
SN - 9783031131844
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 55
EP - 78
BT - Computer Aided Verification - 34th International Conference, CAV 2022, Proceedings
A2 - Shoham, Sharon
A2 - Vizel, Yakir
PB - Springer Science and Business Media Deutschland GmbH
T2 - 34th International Conference on Computer Aided Verification, CAV 2022
Y2 - 7 August 2022 through 10 August 2022
ER -