A separation logic for negative dependence

Jialu Bao, Marco Gaboardi, Justin Hsu, Joseph Tassarotti

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Article number3498696
JournalProceedings of the ACM on Programming Languages
Volume6
Issue numberPOPL
DOIs
StatePublished - Jan 2022

Keywords

  • negative dependence
  • Probabilistic programs
  • separation logic

ASJC Scopus subject areas

  • Software
  • Safety, Risk, Reliability and Quality

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