Wreath/cascade products and related decomposition results for the concurrent setting of Mazurkiewicz traces (extended version)

TL;DR

This paper introduces a new algebraic framework for Mazurkiewicz traces that generalizes wreath products to true concurrency, enabling trace language decomposition and automata characterizations.

Contribution

It develops a novel algebraic framework with a local wreath product principle, extending Krohn-Rhodes and Kamp's theorems to concurrent trace languages.

Findings

Decomposition of recognizable trace languages analogous to Krohn-Rhodes theorem

Extension of Kamp's theorem to acyclic architectures

Characterization of aperiodic trace languages using cascade products

Abstract

We develop a new algebraic framework to reason about languages of Mazurkiewicz traces. This framework supports true concurrency and provides a non-trivial generalization of the wreath product operation to the trace setting. A novel local wreath product principle has been established. The new framework is crucially used to propose a decomposition result for recognizable trace languages, which is an analogue of the Krohn-Rhodes theorem. We prove this decomposition result in the special case of acyclic architectures and apply it to extend Kamp's theorem to this setting. We also introduce and analyze distributed automata-theoretic operations called local and global cascade products. Finally, we show that aperiodic trace languages can be characterized using global cascade products of localized and distributed two-state reset automata.