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. Stepindexing 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.

Conflict-free Replicated Datatypes (CRDTs) are a class of distributed data structures that are highly-available and weakly consistent. The CRDT taxonomy is further divided into two subclasses: state-based and operation-based (op-based). Recent prior work showed how to use separation logic to verify convergence and functional correctness of op-based CRDTs while (a) verifying implementations (as opposed to high-level protocols), (b) giving high level specifications that abstract from low-level implementation details, and (c) providing specifications that are modular (i.e. allow client code to use the CRDT like an abstract data type). We extend this separation logic approach to verification of CRDTs to handle state-based CRDTs, while respecting the desiderata (a)-(c). The key idea is to track the state of a CRDT as a function of the set of operations that produced that state. Using the observation that state-based CRDTs are automatically causally-consistent, we obtain CRDT specifications that are agnostic to whether a CRDT is state- or op-based. When taken together with prior work, our technique thus provides a unified approach to specification and verification of op- and state-based CRDTs. We have tested our approach by verifying StateLib, a library for building state-based CRDTs. Using StateLib, we have further verified convergence and functional correctness of multiple example CRDTs from the literature. Our proofs are written in the Aneris distributed separation logic and are mechanized in Coq.