A Separation Logic for Negative Dependence
Abstract
Formal reasoning about hashingbased 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 nondeterministic, rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Framelike rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hashbased data structures and ballsintobins processes.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.14917
 Bibcode:
 2021arXiv211114917B
 Keywords:

 Computer Science  Programming Languages;
 Computer Science  Logic in Computer Science
 EPrint:
 61 pages, 9 figures, to appear in Proceedings of the ACM on Programming Languages (POPL 2022)