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.

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.