# Quantitative Separation Logic - A Logic for Reasoning about Probabilistic Programs

arXiv: https://arxiv.org/abs/1802.10467 · HTML: https://mlsanity.com/paper/1802.10467/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 2 of 2)

- [Weighted Programming](https://mlsanity.com/paper/2202.07577/) (2022)
- [A separation logic for sequences in pointer programs and its decidability](https://mlsanity.com/paper/2301.06237/) (2023)

## Abstract (arXiv)

We present quantitative separation logic (mathsfQSL). In contrast to classical separation logic, mathsfQSL employs quantities which evaluate to real numbers instead of predicates which evaluate to Boolean values. The connectives of classical separation logic, separating conjunction and separating implication, are lifted from predicates to quantities. This extension is conservative: Both connectives are backward compatible to their classical analogs and obey the same laws, e.g. modus ponens, adjointness, etc. Furthermore, we develop a weakest precondition calculus for quantitative reasoning about probabilistic pointer programs in mathsfQSL. This calculus is a conservative extension of both Reynolds' separation logic for heap-manipulating programs and Kozen's / McIver and Morgan's weakest preexpectations for probabilistic programs. Soundness is proven with respect to an operational semantics based on Markov decision processes. Our calculus preserves O'Hearn's frame rule, which enables local reasoning. We demonstrate that our calculus enables reasoning about quantities such as the probability of terminating with an empty heap, the probability of reaching a certain array permutation, or the expected length of a list.