Higher-order probabilistic adversarial computations: categorical semantics and program logics

Abstract

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

Higher-Order Weakest Precondition Transformers via a CPS Transformation

“…iszero : N → 1 + 1 in Remark 9) to a λ csignature. Lastly, we would like to look for relations between the CPS transformation and program logics for higher-order languages [36]- [38], aiming at automated verification.…”

“…Several studies provide program logic for higher order programs using higher-order logics [36]- [38] However, these studies do not consider weakest precondition transformers.…”

Concrete categories and higher-order recursion (With applications including probability, differentiability, and full abstraction)

“…These are quite widely used (e.g. [2,29,44,46]). As is well known, the category of quasi-Borel spaces Qbs can be regarded as the category of concrete sheaves on a site (Sbs, 𝐽 Sbs ).…”

“…Notice that in general 𝜔 P is not a 𝐽 -sheaf. The limit of diagram (2) in PSh(C) is the same as in sheaves, i.e. 𝜔.…”

Asynchronous Probabilistic Couplings in Higher-Order Separation Logic

“…In addition, certain important properties of probabilistic programs are naturally expressed in a relational form, such as stability of machine learning algorithms [1], differential privacy [2], and provable security [3]. Consequently, a number of relational program logics and models have been developed for probabilistic programs, e.g., pRHL [4], approximate pRHL [5]- [7], EpRHL [8], HO-RPL [9], Polaris [10], logical relations [11]- [13], and differential logical relations [14].…”