Session Types, Concurrent Separation Logic & Algebra

Abstract

This dissertion explores the relation between two formalisms and one algebraic framework for concurrency. Session Types and Concurrent Separation Logic are formalisms that support independent reasoning about concurrent processes, and our motivating question is whether their modularity springs from the same source despite the distance between their models. We first translate a small language we call Baby Session Types (BST), into a ‘basic’ version of Concurrent Separation Logic (BCSL), and we show that the translation is sound. We then describe a model for Separation Logic (SL) based on Actions, which exhibits some of the structure of a Concurrent Kleene Algebra, an algebra where operators for parallel and sequential composition are linked by a version of the exchange law from category theory. The model connects the algebraic notions to locality concepts that underlie Separation Logic. We then move on to provide a more general construction of an algebra model of BCSL, which can be built from (Baby) Session Types. Thus, we end up with a model that brings together concepts from all of Session Types, Separation Logic, and Concurrent Kleene Algebra. Thus, the model links diverse models of concurrency. In addition to this it suggests alterations of the algebraic axioms as well as the foundational models underlying Separation Logic. It is hoped that, apart from these specific results, this dissertation can in some modest way contribute to unification in concurrency theory, a theory (or theories) based presently on diverse models.