A separation logic for negative dependence

Abstract

Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes.

Bunched Fuzz: Sensitivity for Vector Metrics

“…We have already discussed the usefulness for other directions in the deterministic case [14]. One way to approach this problem could be by looking at the family of products recently identified in [1]. These products give a model for a logic to reason about negative dependence between probabilistic variables.…”

“…A recent work by Bao et al [1] introduced a novel bunched logic with indexed products and magic wands with a preorder between the indices. This logic is used as the assertion logic of a separation logic introduced in order to reason about negative dependence between random variables.…”

Asynchronous Probabilistic Couplings in Higher-Order Separation Logic

“…This work focuses on proving quantitative properties of a single program, e.g., bounding the probability that certain events happen. A variety of works have developed separation logics in which the separating conjunction models various forms of probabilistic independence [59]- [61]. For example, the statement P * Q is taken to mean "the distribution of P is independent from the distribution of Q".…”

Tight Bounds for Repeated Balls-into-Bins

“…We will now upper bound E (y ρ i ) 2 . In each round, we will apply the principle of deferred decision to expose Z t j by looking at each of the n trials individually.…”

“…As also pointed out in [8,10], this leads to a non-reversible Markov Chain, which seems to make the computation of the stationary distribution intractable. Furthermore, formal methods have been used to prove guarantees for RBB with m = n [2]. The RBB setting has been also applied to analyze protocols in short packet communications [28].…”