A (Co)Algebraic Framework for Ordered Processes

TL;DR

This paper develops a new algebraic and coalgebraic framework for ordered processes, extending previous models to include probabilistic and inequationally specified branching structures, and compares these with coalgebraic similarity.

Contribution

It introduces an alternative framework for ordered process calculi that captures probabilistic extensions and inequational structures, addressing limitations of earlier algebraic models.

Findings

Framework captures probabilistic process calculus.

Probabilistic extension of guarded Kleene algebra is a fragment of the calculus.

Intrinsic order relates to coalgebraic similarity.

Abstract

A recently published paper (Schmid, Rozowski, Silva, and Rot, 2022) offers a (co)algebraic framework for studying processes with algebraic branching structures and recursion operators. The framework captures Milner's algebra of regular behaviours (Milner, 1984) but fails to give an honest account of a closely related calculus of probabilistic processes (Stark and Smolka, 1999). We capture Stark and Smolka's calculus by giving an alternative framework, aimed at studying a family of ordered process calculi with inequationally specified branching structures and recursion operators. We observe that a recent probabilistic extension of guarded Kleene algebra with tests (Rozowski, Kozen, Kappe, Schmid, Silva, 2022) is a fragment of one of our calculi, along with other examples. We also compare the intrinsic order in our process calculi with the notion of similarity in coalgebra.