On Series-Parallel Pomset Languages: Rationality, Context-Freeness and Automata

TL;DR

This paper establishes a formal connection between bi-Kleene Algebra and pomset automata, characterizing their behaviors and demonstrating the limits of decidability in series-parallel rational languages.

Contribution

It introduces pomset automata as a new automaton model for bi-Kleene Algebra and relates them to series-parallel rational expressions, advancing the understanding of concurrent program semantics.

Findings

PAs can implement BKA semantics of series-parallel rational expressions.

Certain PAs can be translated back into series-parallel rational expressions.

Universality and equivalence for general PAs are undecidable.

Abstract

Concurrent Kleene Algebra (CKA) is a formalism to study concurrent programs. Like previous Kleene Algebra extensions, developing a correspondence between denotational and operational perspectives is important, for both foundations and applications. This paper takes an important step towards such a correspondence, by precisely relating bi-Kleene Algebra (BKA), a fragment of CKA, to a novel type of automata, pomset automata (PAs). We show that PAs can implement the BKA semantics of series-parallel rational expressions, and that a class of PAs can be translated back to these expressions. We also characterise the behavior of general PAs in terms of context-free pomset grammars; consequently, universality, equivalence and series-parallel rationality of general PAs are undecidable.