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Properties such as provable security and correctness for randomized programs are naturally expressed relationally as approximate equivalences. As a result, a number of relational program logics have been developed to reason about such approximate equivalences of probabilistic programs. However, existing approximate relational logics are mostly restricted to first-order programs without general state. In this paper we develop Approxis, a higher-order approximate relational separation logic for reasoning about approximate equivalence of programs written in an expressive ML-like language with discrete probabilistic sampling, higher-order functions, and higher-order state. The Approxis logic recasts the concept of error credits in the relational setting to reason about relational approximation, which allows for expressive notions of modularity and composition, a range of new approximate relational rules, and an internalization of a standard limiting argument for showing exact probabilistic equivalences by approximation. We also use Approxis to develop a logical relation model that quantifies over error credits, which can be used to prove exact contextual equivalence. We demonstrate the flexibility of our approach on a range of examples, including the PRP/PRF switching lemma, IND$-CPA security of an encryption scheme, and a collection of rejection samplers. All of the results have been mechanized in the Coq proof assistant and the Iris separation logic framework. 

Probabilistic programs often trade accuracy for efficiency, and thus may, with a small probability, return an incorrect result. It is important to obtain precise bounds for the probability of these errors, but existing verification approaches have limitations that lead to error probability bounds that are excessively coarse, or only apply to first-order programs. In this paper we present Eris, a higher-order separation logic for proving error probability bounds for probabilistic programs written in an expressive higher-order language. Our key novelty is the introduction of error credits, a separation logic resource that tracks an upper bound on the probability that a program returns an erroneous result. By representing error bounds as a resource, we recover the benefits of separation logic, including compositionality, modularity, and dependency between errors and program terms, allowing for more precise specifications. Moreover, we enable novel reasoning principles such as expectation-preserving error composition, amortized error reasoning, and error induction. We illustrate the advantages of our approach by proving amortized error bounds on a range of examples, including collision probabilities in hash functions, which allow us to write more modular specifications for data structures that use them as clients. We also use our logic to prove correctness and almost-sure termination of rejection sampling algorithms. All of our results have been mechanized in the Coq proof assistant using the Iris separation logic framework and the Coquelicot real analysis library.