Posets with Interfaces as a Model for Concurrency

TL;DR

This paper introduces posets with interfaces (iposets) and a new gluing composition, providing a categorical framework for modeling concurrency that generalizes existing series-parallel poset models.

Contribution

It develops a new compositional model for concurrency using iposets with a novel gluing operation, extending the algebraic and hierarchical understanding of concurrent systems.

Findings

iposets form a category under gluing composition

They form a 2-category with subsumption order and non-commutative parallel composition

The gluing-parallel hierarchy generalizes series-parallel posets and interval orders

Abstract

We introduce posets with interfaces (iposets) and generalise their standard serial composition to a new gluing composition. In the partial order semantics of concurrency, interfaces and gluing allow modelling events that extend in time and across components. Alternatively, taking a decompositional view, interfaces allow cutting through events, while serial composition may only cut through edges of a poset. We show that iposets under gluing composition form a category, which generalises the monoid of posets under serial composition up to isomorphism. They form a 2-category when a subsumption order and a lax tensor in the form of a non-commutative parallel composition are added, which generalises the interchange monoids used for modelling series-parallel posets. We also study the gluing-parallel hierarchy of iposets, which generalises the standard series-parallel one. The class of gluing-parallel iposets contains that of series-parallel posets and the class of interval orders, which are well studied in concurrency theory, too. We also show that it is strictly contained in the class of all iposets by identifying several forbidden substructures.