# Trillium: Higher-Order Concurrent and Distributed Separation Logic for Intensional Refinement

arXiv: https://arxiv.org/abs/2109.07863 · HTML: https://mlsanity.com/paper/2109.07863/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2021.

## Built on by (most cited first, 1 of 1)

- [Harnessing Code Agents for Automatic Software Verification](https://mlsanity.com/paper/2607.06341/) (2026)

## Abstract (arXiv)

Expressive state-of-the-art separation logics rely on step-indexing to model semantically complex features and to support modular reasoning about imperative higher-order concurrent and distributed programs. Step-indexing comes, however, with an inherent cost: it restricts the adequacy theorem of program logics to a fairly simple class of safety properties. In this paper, we explore if and how intensional refinement is a viable methodology for strengthening higher-order concurrent (and distributed) separation logic to prove non-trivial safety and liveness properties. Specifically, we introduce Trillium, a language-agnostic separation logic framework for showing intensional refinement relations between traces of a program and a model. We instantiate Trillium with a concurrent language and develop Fairis, a concurrent separation logic, that we use to show liveness properties of concurrent programs under fair scheduling assumptions through a fair liveness-preserving refinement of a model. We also instantiate Trillium with a distributed language and obtain an extension of Aneris, a distributed separation logic, which we use to show refinement relations between distributed systems and TLA+ models.