Abstract
Probabilistic coupling is a powerful tool for analyzing pairs of probabilistic processes. Roughly, coupling two processes requires finding an appropriate witness process that models both processes in the same probability space. Couplings are powerful tools proving properties about the relation between two processes, include reasoning about convergence of distributions and stochastic dominance—a probabilistic version of a monotonicity property.
While the mathematical definition of coupling looks rather complex and cumbersome to manipulate, we show that the relational program logic pRHL—the logic underlying the EasyCrypt cryptographic proof assistant—already internalizes a generalization of probabilistic coupling. With this insight, constructing couplings is no harder than constructing logical proofs. We demonstrate how to express and verify classic examples of couplings in pRHL, and we mechanically verify several couplings in EasyCrypt.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barthe, G., Grégoire, B., Zanella-Béguelin, S.: Formal certification of code-based cryptographic proofs. In: ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Savannah, Georgia, pp. 90–101, New York (2009)
Barthe, G., Crespo, J.M., Kunz, C.: Relational verification using product programs. In: International Symposium on Formal Methods (FM), Limerick, Ireland, pp. 200–214 (2011a)
Barthe, G., D’Argenio, P.R., Rezk, T.: Secure information flow by self-composition. Math. Struct. Comput. Sci. 21(06), 1207–1252 (2011b)
Barthe, G., Grégoire, B., Heraud, S., Béguelin, S.Z.: Computer-aided security proofs for the working cryptographer. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 71–90. Springer, Heidelberg (2011)
Benton, N.: Simple relational correctness proofs for static analyses and program transformations. In: ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Venice, Italy, pp. 14–25 (2004)
Deng, Y., Du, W.: Logical, metric, and algorithmic characterisations of probabilistic bisimulation. Technical report CMU-CS-11-110, Carnegie Mellon University, March 2011
Lindvall, T.: Lectures on the coupling method. Courier Corporation, Mineola (2002)
Mufa, C.: Optimal markovian couplings and applications. Acta Mathematica Sinica 10(3), 260–275 (1994)
Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)
Villani, C.: Optimal Transport: Old and New. Springer Science, Heidelberg (2008)
Yang, H.: Relational separation logic. Theor. Comput. Sci. 375(1–3), 308–334 (2007)
Zaks, A., Pnueli, A.: CoVaC: compiler validation by program analysis of the cross-product. In: Cuellar, J., Sere, K. (eds.) FM 2008. LNCS, vol. 5014, pp. 35–51. Springer, Heidelberg (2008)
Acknowledgments
We thank Arthur Azevedo de Amorim and the anonymous reviewers for their close reading and useful suggestions. This work was partially supported by a grant from the Simons Foundation (#360368 to Justin Hsu), NSF grant CNS-1065060, Madrid regional project S2009TIC-1465 PROMETIDOS, Spanish national projects TIN2009-14599 DESAFIOS 10 and TIN2012-39391-C04-01 Strongsoft, and a grant from the Cofund Action AMAROUT II (#291803).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Barthe, G., Espitau, T., Grégoire, B., Hsu, J., Stefanesco, L., Strub, PY. (2015). Relational Reasoning via Probabilistic Coupling. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_27
Download citation
DOI: https://doi.org/10.1007/978-3-662-48899-7_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48898-0
Online ISBN: 978-3-662-48899-7
eBook Packages: Computer ScienceComputer Science (R0)