
ConstraintBased Synthesis of Coupling Proofs
Proof by coupling is a classical technique for proving properties about ...
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Synthesizing Coupling Proofs of Differential Privacy
Differential privacy has emerged as a promising probabilistic formulatio...
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Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus
We extend the simplytyped guarded λcalculus with discrete probabilitie...
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Proving Expected Sensitivity of Probabilistic Programs
Program sensitivity, also known as Lipschitz continuity, describes how s...
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A Probabilistic Separation Logic
Probabilistic independence is a fundamental tool for reasoning about ran...
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Trace Abstraction Modulo Probability
We propose trace abstraction modulo probability, a proof technique for v...
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Certifying Certainty and Uncertainty in Approximate Membership Query Structures – Extended Version
Approximate Membership Query structures (AMQs) rely on randomisation for...
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Probabilistic Couplings for Probabilistic Reasoning
This thesis explores proofs by coupling from the perspective of formal verification. Long employed in probability theory and theoretical computer science, these proofs construct couplings between the output distributions of two probabilistic processes. Couplings can imply various guarantees comparing two runs of a probabilistic computation. We first show that proofs in the program logic pRHL describe couplings. We formalize couplings that establish various probabilistic properties, including distribution equivalence, convergence, and stochastic domination. Then we give a proofsasprograms interpretation: a coupling proof encodes a probabilistic product program, whose properties imply relational properties of the original programs. We design the logic xpRHL to construct the product, with extensions to model shift coupling and path coupling. We then propose an approximate version of probabilistic coupling and a corresponding proof techniqueproof by approximate couplinginspired by the logic apRHL, a version of pRHL for building approximate liftings. Drawing on ideas from existing privacy proofs, we extend apRHL with novel proof rules for constructing new approximate couplings. We give an approximate coupling proof of privacy for the Sparse Vector mechanism, a wellknown algorithm from the privacy literature whose privacy proof is notoriously subtle, and produce the first formalized proof of privacy for Sparse Vector in apRHL. Finally, we propose several more sophisticated constructions for approximate couplings: a principle for showing accuracydependent privacy, a generalization of the advanced composition theorem, and an optimal approximate coupling relating two subsets. We also show equivalences between our approximate couplings and other existing definitions. These ingredients support the first formalized proof of privacy for the Between Thresholds mechanism.
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