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Kleene Algebra with Tests (KAT) provides a framework for algebraic equational reasoning about imperative programs. The recent variant Guarded KAT (GKAT) allows to reason on non-probabilistic properties of probabilistic programs. Here we introduce an extension of this framework called approximate GKAT (aGKAT), which equips GKAT with a partially ordered monoid (real numbers) enabling to express satisfaction of (deterministic) properties except with a probability up to a certain bound. This allows to represent in equational reasoning "à la KAT" proofs of probabilistic programs based on the union bound, a technique from basic probability theory. We show how a propositional variant of approximate Hoare Logic (aHL), a program logic for union bound, can be soundly encoded in our system aGKAT. We then illustrate the use of aGKAT with an example of accuracy analysis from the field of differential privacy. 

Abstract This work is devoted to formal reasoning on relational properties of probabilistic imperative programs. Relational properties are properties which relate the execution of two programs (possibly the same one) on two initial memories. We aim at extending the algebraic approach of Kleene Algebras with Tests (KAT) to relational properties of probabilistic programs. For that we consider the approach of Guarded Kleene Algebras with Tests (GKAT), which can be used for representing probabilistic programs, and define a relational version of it, called Bi-guarded Kleene Algebras with Tests (BiGKAT) together with a semantics. We show that the setting of BiGKAT is expressive enough to encode a finitary version of probabilistic Relational Hoare Logic (pRHL) (without the While rule), a program logic that has been introduced in the literature for the verification of relational properties of probabilistic programs. We also discuss the additional expressivity brought by BiGKAT.