Towards Kleene Algebra with Recursion

We extend Kozen’s theory KA of Kleene Algebra to axiomatize parts of the equational theory of context-free languages, using a least fixed-point operator μ instead of Kleene’s iteration operator ∗. Although the equational theory of context-free languages is not recursively axiomatizable, there are natural axioms for subtheories KAF ⊆ KAR ⊆ KAG: respectively, these make μ a least fixed point operator, connect it with recursion, and express S. Greibach’s method to replace leftby right-recursion and vice versa. Over KAF , there are different candidates to define ∗ in terms of μ, such as tail-recursion and reflexive transitive closure. In KAR, these candidates collapse, whence KAR uniquely defines ∗ and extends Kozen’s theory KA. We show that a model M = (M,+, 0, ·, 1, μ) of KAF is a model of KAG, whenever the partial order ≤ on M induced by + is complete, and + and · are Scott-continuous with respect to ≤. The family of all context-free languages over an alphabet of size n is the free structure for the class of submodels of continuous models of KAF in n generators.