Continuous Algebras with Hypotheses

2026-05-18

AI Summary

This work proposes a unified framework for reasoning about the equational theories of Kleene algebra and its variants—including Kleene algebra with tests, commutative Kleene algebra, bi-Kleene algebra, and concurrent Kleene algebra—under additional hypotheses. The framework is grounded in canonical models based on complete lattice-ordered algebras and closed languages, supports least fixed-point semantics, and employs a modular quasi-equational axiomatization that avoids reliance on globally complete axiom systems. Leveraging this approach, the paper establishes novel completeness results for commutative Kleene algebra, bi-Kleene algebra, and regular tree languages under various assumptions, thereby demonstrating the generality and effectiveness of the proposed framework.

Abstract

In the literature on Kleene algebra (KA), a number of variants have been proposed such as Kleene algebra with tests, commutative KA, bi-KA, and concurrent KA. The equational theories of some of these structures have then been studied in the presence of additional assumptions, called hypotheses. We propose a unifying framework encompassing all the previous structures, as well as regular tree languages. This is done by considering algebras ordered by complete lattices, where least fixpoints can be computed. We provide a canonical model consisting of closed languages, which we prove sound and complete with respect to all continuous models. Then we study quasi-equational axiomatisations. It is illusory to hope for a generic axiomatisation which would be sound and complete for all instances. Instead, we provide a generic axiomatisation which we prove sound and we setup tools that make it possible to get complete ones in a modular way, building on previous works from the literature. We showcase these tools by proving new completeness results for commutative KA, bi-KA, and regular tree languages, in each case extended with various hypotheses.

Problem

• Kleene algebra
• hypotheses
• equational theory
• quasi-equational axiomatisation
• regular tree languages

Innovation

• continuous algebras
• Kleene algebra with hypotheses
• complete lattices
• canonical model
• modular completeness