We present Linearizability Hoare Logic (LHL), the first mechanized, sound, and complete program logic for atomic, set, and interval linearizability. We achieve this by showing soundness and completeness of LHL w.r.t. a more general criterion, compositional linearizability, which subsumes all three criteria. We showcase the expressivity of LHL by verifying an exchanger with a set linearizable specification, the elimination-backoff stack built above the exchanger, a lock with an atomic linearized specification, and a write-snapshot object with an interval linearizable specification. Together with LHL we formalize a modular verification framework for concurrent components based on the theory of compositional linearizability. This allows us to specify components at a high level of abstraction and granularity, and then assemble them into large systems that are correct by construction. As a showcase, we verify the elimination-backoff stack modularly by verifying each of its sub-components against their linearized specifications and then linking them together.

Compositionality is at the core of programming languages research and has become an important goal toward scalable verification of large systems. Despite that, there is no compositional account of linearizability , the gold standard of correctness for concurrent objects. In this article, we develop a compositional semantics for linearizable concurrent objects. We start by showcasing a common issue, which is independent of linearizability, in the construction of compositional models of concurrent computation: interaction with the neutral element for composition can lead to emergent behaviors, a hindrance to compositionality. Category theory provides a solution for the issue in the form of the Karoubi envelope. Surprisingly, and this is the main discovery of our work, this abstract construction is deeply related to linearizability and leads to a novel formulation of it. Notably, this new formulation neither relies on atomicity nor directly upon happens-before ordering and is only possible because of compositionality, revealing that linearizability and compositionality are intrinsically related to each other. We use this new, and compositional, understanding of linearizability to revisit much of the theory of linearizability, providing novel, simple, algebraic proofs of the locality property and of an analogue of the equivalence with observational refinement . We show our techniques can be used in practice by connecting our semantics with a simple program logic that is nonetheless sound concerning this generalized linearizability.

Crash-safety is an important property of real systems, as the main functionality of some systems is resilience to crashes. Toward a compositional verification approach for crash-safety under full-system crashes, one observes that crashes propagate instantaneously to all components across all levels of abstraction, even to unspecified components, hindering compositionality. Furthermore, in the presence of concurrency, a correctness criterion that addresses both crashes and concurrency proves necessary. For this, several adaptations of linearizability have been suggested, each featuring different trade-offs between complexity and expressiveness. The recently proposed compositional linearizability framework shows that to achieve compositionality with linearizability, both a locality and observational refinement property are necessary. Despite that, no linearizability criterion with crashes has been proven to support an observational refinement property. In this paper, we define a compositional model of concurrent computation with full-system crashes. We use this model to develop a compositional theory of linearizability with crashes, which reveals a criterion, crash-aware linearizability , as its inherent notion of linearizability and supports both locality and observational refinement. We then show that strict linearizability and durable linearizability factor through crash-aware linearizability as two different ways of translating between concurrent computation with and without crashes, enabling simple proofs of locality and observational refinement for a generalization of these two criteria. Then, we show how the theory can be connected with a program logic for durable and crash-aware linearizability, which gives the first program logic that verifies a form of linearizability with crashes. We showcase the advantages of compositionality by verifying a library facilitating programming persistent data structures and a fragment of a transactional interface for a file system.

Compositionality is at the core of programming languages research and has become an important goal toward scalable verification of large systems. Despite that, there is no compositional account of linearizability, the gold standard of correctness for concurrent objects. In this paper, we develop a compositional semantics for linearizable concurrent objects. We start by showcasing a common issue, which is independent of linearizability, in the construction of compositional models of concurrent computation: interaction with the neutral element for composition can lead to emergent behaviors, a hindrance to compositionality. Category theory provides a solution for the issue in the form of the Karoubi envelope. Surprisingly, and this is the main discovery of our work, this abstract construction is deeply related to linearizability and leads to a novel formulation of it. Notably, this new formulation neither relies on atomicity nor directly upon happens-before ordering and is only possible because of compositionality, revealing that linearizability and compositionality are intrinsically related to each other. We use this new, and compositional, understanding of linearizability to revisit much of the theory of linearizability, providing novel, simple, algebraic proofs of the locality property and of an analogue of the equivalence with observational refinement. We show our techniques can be used in practice by connecting our semantics with a simple program logic that is nonetheless sound concerning this generalized linearizability.

Large-scale software verification relies critically on the use of compositional languages, semantic models, specifications, and verification techniques. Recent work on certified abstraction layers synthesizes game semantics, the refinement calculus, and algebraic effects to enable the composition of heterogeneous components into larger certified systems. However, in existing models of certified abstraction layers, compositionality is restricted by the lack of encapsulation of state. In this paper, we present a novel game model for certified abstraction layers where the semantics of layer interfaces and implementations are defined solely based on their observable behaviors. Our key idea is to leverage Reddy's pioneer work on modeling the semantics of imperative languages not as functions on global states but as objects with their observable behaviors. We show that a layer interface can be modeled as an object type (i.e., a layer signature) plus an object strategy. A layer implementation is then essentially a regular map, in the sense of Reddy, from an object with the underlay signature to that with the overlay signature. A layer implementation is certified when its composition with the underlay object strategy implements the overlay object strategy. We also describe an extension that allows for non-determinism in layer interfaces. After formulating layer implementations as regular maps between object spaces, we move to concurrency and design a notion of concurrent object space, where sequential traces may be identified modulo permutation of independent operations. We show how to express protected shared object concurrency, and a ticket lock implementation, in a simple model based on regular maps between concurrent object spaces.