Towards Kleene Algebra with recursion

Infinitary Axiomatization of the Equational Theory of Context-Free Languages

“…Our main result is that the µ-continuity condition, along with the axioms of idempotent semirings, completely axiomatize the equational theory of the context-free languages. This is the first completeness result for the equational theory of the contextfree languages, answering a question of Leiß [3].…”

“…Leiß [3] investigates three classes of idempotent semirings with a syntactic least fixpoint operator µ. The three classes are called KAF, KAR, and KAG in increasing order of specificity.…”

“…The characterization of the context-free languages as least solutions of algebraic inequalities involving µ goes back to a 1971 paper of Gruska [4]. More recently, several researchers have given equational axioms for semirings with µ and have developed fragments of the equational theory of context-free languages [3,5,6,7,8,9].…”

Context-Free Languages, Coalgebraically

“…In [9,2], Kleene algebras have been extended with a least fixed point operator to axiomatize fragments of the theory of context-free languages. We take a similar approach, but coalgebraic in nature and substituting the Kleene star with a unique fixed point operator.…”

“…We take a similar approach, but coalgebraic in nature and substituting the Kleene star with a unique fixed point operator. Whereas [9,2] are interested in providing solutions to systems of equations of the form x = t using least fixed points, we look at systems of behavioural differential equations and give a semantic solution in terms of context-free languages (in Sect. 4) and syntactic solutions in terms of regular expressions with unique fixed points (in Sect.…”