# Completeness Theorems for Pomset Languages and Concurrent Kleene Algebras

## TL;DR

This paper establishes completeness theorems for pomset languages using bi-Kleene algebra operations, connecting their equational theory to regular and commutative-regular languages, and explores properties of rational pomset languages and ideals.

## Contribution

It proves that all valid equations over pomset languages are derivable from known algebraic theories and characterizes the structure of rational pomset languages and their ideals.

## Key findings

- Valid universal equalities follow from regular and commutative-regular language theories.
- Rational pomset languages are closed under Boolean operations.
- Equivalence of ideal languages is provable from Kleene and exchange axioms.

## Abstract

Pomsets constitute one of the most basic models of concurrency. A pomset is a generalisation of a word over an alphabet in that letters may be partially ordered. A term $t$ using the bi-Kleene operations $0,1, +, \cdot\, ,^*, \parallel, ^{(*)}$ defines a language $ \mathopen{[\![ } t \mathclose{]\!] } $ of pomsets in a natural way. We prove that every valid universal equality over pomset languages using these operations is a consequence of the equational theory of regular languages (in which parallel multiplication and iteration are undefined) plus that of the commutative-regular languages (in which sequential multiplication and iteration are undefined). We also show that the class of $\textit{rational}$ pomset languages (that is, those languages generated from singleton pomsets using the bi-Kleene operations) is closed under all Boolean operations. An $ \textit{ideal}$ of a pomset $p$ is a pomset using the letters of $p$, but having an ordering at least as strict as $p$. A bi-Kleene term $t$ thus defines the set $ \textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) $ of ideals of pomsets in $ \mathopen{[\![ } t \mathclose{]\!] } $. We prove that if $t$ does not contain commutative iteration $^{(*)}$ (in our terminology, $t$ is bw-rational) then $\textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) \cap \textbf{Pom}_{sp}$, where $ \textbf{Pom}_{sp}$ is the set of pomsets generated from singleton pomsets using sequential and parallel multiplication ($ \cdot$ and $ \parallel$) is defined by a bw-rational term, and if two such terms $t,t'$ define the same ideal language, then $t'=t$ is provable from the Kleene axioms for $0,1, +, \cdot\, ,^*$ plus the commutative idempotent semiring axioms for $0,1, +, \parallel$ plus the exchange law $ (u \parallel v)\cdot ( x \parallel y) \le v \cdot y \parallel u \cdot x $.