Concurrent separation logic is distinguished by transfer of state ownership upon parallel composition and framing. The algebraic structure that underpins ownership transfer is that of partial commutative monoids (PCMs). Extant research considers ownership transfer primarily from the logical perspective while comparatively less attention is drawn to the algebraic considerations. This paper provides an algebraic formalization of ownership transfer in concurrent separation logic by means of structure-preserving partial functions (i.e., morphisms) between PCMs, and an associated notion of separating relations. Morphisms of structures are a standard concept in algebra and category theory, but haven't seen ubiquitous use in separation logic before. Separating relations. are binary relations that generalize disjointness and characterize the inputs on which morphisms preserve structure. The two abstractions facilitate verification by enabling concise ways of writing specs, by providing abstract views of threads' states that are preserved under ownership transfer, and by enabling user-level construction of new PCMs out of existing ones.

In addition to pre- and postconditions, program specifications in recent separation logics for concurrency have employed an algebraic structure of resources —a form of state transition systems—to describe the state-based program invariants that must be preserved, and to record the permissible atomic changes to program state. In this paper we introduce a novel notion of resource morphism , i.e. structure-preserving function on resources, and show how to effectively integrate it into separation logic, using an associated notion of morphism-specific simulation . We apply morphisms and simulations to programs verified under one resource, to compositionally adapt them to operate under another resource, thus facilitating proof reuse.

Arguments about correctness of a concurrent data structure are typically carried out by using the notion of linearizability and specifying the linearization points of the data structure's procedures. Such arguments are often cumbersome as the linearization points' position in time can be dynamic (depend on the interference, run-time values and events from the past, or even future), non-local (appear in procedures other than the one considered), and whose position in the execution trace may only be determined after the considered procedure has already terminated. In this paper we propose a new method, based on a separation-style logic, for reasoning about concurrent objects with such linearization points. We embrace the dynamic nature of linearization points, and encode it as part of the data structure's auxiliary state, so that it can be dynamically modified in place by auxiliary code, as needed when some appropriate run-time event occurs. We name the idea linking-in-time, because it reduces temporal reasoning to spatial reasoning. For example, modifying a temporal position of a linearization point can be modeled similarly to a pointer update in separation logic. Furthermore, the auxiliary state provides a convenient way to concisely express the properties essential for reasoning about clients of such concurrent objects. We illustrate the method by verifying (mechanically in Coq) an intricate optimal snapshot algorithm due to Jayanti, as well as some clients.

Designing efficient concurrent objects often requires abandoning the standard specification technique of linearizability in favor of more relaxed correctness conditions. However, the variety of alternatives makes it difficult to choose which condition to employ, and how to compose them when using objects specified by different conditions.

Continuations are programming abstractions that allow for manipulating the "future" of a computation. Amongst their many applications, they enable implementing unstructured program flow through higher-order control operators such as callcc. In this paper we develop a Hoare-style logic for the verification of programs with higher-order control, in the presence of dynamic state. This is done by designing a dependent type theory with first class callcc and abort operators, where pre- and postconditions of programs are tracked through types. Our operators are algebraic in the sense of Plotkin and Power, and Jaskelioff, to reduce the annotation burden and enable verification by symbolic evaluation. We illustrate working with the logic by verifying a number of characteristic examples.