# Relational Reasoning via Probabilistic Coupling

@inproceedings{Barthe2015RelationalRV, title={Relational Reasoning via Probabilistic Coupling}, author={Gilles Barthe and Thomas Espitau and Benjamin Gr{\'e}goire and Justin Hsu and L{\'e}o Stefanesco and Pierre-Yves Strub}, booktitle={LPAR}, year={2015} }

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…

## 24 Citations

Coupling proofs are probabilistic product programs

- Computer SciencePOPL 2017
- 2016

An extension of pRHL is defined, called x-pRHL, which explicitly constructs the coupling in a pRH derivation in the form of a probabilistic product program that simulates two correlated runs of the original program.

Probabilistic Couplings for Probabilistic Reasoning

- Computer Science, MathematicsArXiv
- 2017

An approximate coupling proof of privacy for the Sparse Vector mechanism is given, a well-known algorithm from the privacy literature whose privacy proof is notoriously subtle, and the first formalized proof ofPrivacy for SparseVector in apRHL is produced.

Proving uniformity and independence by self-composition and coupling

- Computer Science, MathematicsLPAR
- 2017

It is demonstrated that the program logic pRHL— whose proofs are formal versions of proofs by coupling—can be used for formalizing uniformity and probabilistic independence.

Relational Expectation Properties by Probabilistic Coupling

- Computer Science
- 2016

This work designs a relational program logic EPRHL that is inspired by the logic PRHL, a powerful tool for proving relational properties by reasoning about probabilistic couplings, and enhances PRHL with an orthogonal, compositional reasoning principle based on premetrics.

A Program Logic for Union Bounds

- Computer ScienceICALP
- 2016

A probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory that enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in the logic.

Efficient Local Computation of Differential Bisimulations via Coupling and Up-to Methods

- Computer Science2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

An algorithm for the computation of equivalence relations which can be interpreted as a lifting of probabilistic bisimulation to polynomial differential equations, a ubiquitous model of dynamical systems across science and engineering is developed.

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

- Computer ScienceESOP
- 2018

We extend the simply-typed guarded \(\lambda \)-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations.…

Proving Differential Privacy via Probabilistic Couplings

- Computer Science2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016

This paper develops compositional methods for formally verifying differential privacy for algorithms whose analysis goes beyond the composition theorem, based on deep connections between differential privacy and probabilistic couplings, an established mathematical tool for reasoning about stochastic processes.

Advanced Probabilistic Couplings for Differential Privacy

- Computer ScienceCCS
- 2016

A new formalism extending apRHL, a relational program logic that has been used for proving differential privacy of non-interactive algorithms, and incorporating a HL, a (non-relational) program logic for accuracy properties is addressed, which exemplifies the three classes of algorithms and explores new variants of the Sparse Vector technique.

The next 700 relational program logics

- Computer ScienceProc. ACM Program. Lang.
- 2020

The first framework for defining relational program logics for arbitrary monadic effects is proposed, and it is shown that this generic framework can be used to define relational programLogics for effects as diverse as state, input-output, nondeterminism, and discrete probabilities.

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