Abstract
We introduce the notion of a connectivity graph —an abstract representation of the topology of concurrently interacting entities, which allows us to encapsulate generic principles of reasoning about deadlock freedom . Connectivity graphs are parametric in their vertices (representing entities like threads and channels) and their edges (representing references between entities) with labels (representing interaction protocols). We prove deadlock and memory leak freedom in the style of progress and preservation and use separation logic as a meta theoretic tool to treat connectivity graph edges and labels substructurally. To prove preservation locally, we distill generic separation logic rules for local graph transformations that preserve acyclicity of the connectivity graph. To prove global progress locally, we introduce a waiting induction principle for acyclic connectivity graphs. We mechanize our results in Coq, and instantiate our method with a higher-order binary session-typed language to obtain the first mechanized proof of deadlock and leak freedom.
Highlights
Binary session types [Honda 1993; Honda et al 1998] are a type discipline for specifying protocols of interactions in message-passing concurrent programs
We show how to use separation logic in a non-standard way as a language for linking our abstract connectivity graphs to a concrete language’s operational semantics and type system
The semantics of most constructs is standard, so we focus on the message passing constructs: 1Due to the session typing discipline, only one of the buffers is expected to be populated at any given time
Summary
Binary session types [Honda 1993; Honda et al 1998] are a type discipline for specifying protocols of interactions in message-passing concurrent programs. As shown by the examples, connectivity graphs describe the types and abstract reference topology of a program’s execution configuration, but not the concrete expressions and values that constitute the threads and channels. All ingredients of our method (the definition of connectivity graph, the separation logic, the graph transformations, and waiting induction) are parametric in the vertices, edges and labels of the connectivity graph This is crucial for mechanization: we can encapsulate our proof method as a library that is independent of the concrete programming language. We show how to use separation logic in a non-standard way as a language for linking our abstract connectivity graphs to a concrete language’s operational semantics and type system. We use our connectivity graph library to obtain the first mechanized proof of deadlock and leak freedom for a binary session-typed λ-calculus with higher-order channels, recursive types, and unrestricted types. An archive of the Coq mechanization can be found at Jacobs et al [2021], and the most recent version at https://github.com/julesjacobs/cgraphs
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