Infinitary Axiomatization of the Equational Theory of Context-Free Languages

Algebraic reasoning about programming language constructs has been a popular research topic for many years. At the propositional level, the theory of flowchart programs and linear recursion are well handled by such systems as Kleene algebra and iteration theories, systems that characterize the equational theory of the regular sets. To handle more general forms of recursion including procedures with recursive calls, one must extend to the context-free languages, and here the situation is less well understood. One reason for this is that, unlike the equational theory of the regular sets, the equational theory of the contextfree languages is not recursively enumerable. This has led some researchers to declare its complete axiomatization an insurmountable task [13]. Whereas linear recursion can be characterized with the star operator ⋆ of Kleene algebra or the dagger operation † of iteration theories, the theory of context-free languages requires a more general fixpoint operator μ . The characterization of the context-free languages as least solutions of algebraic inequalities involving μ goes back to a 1971 paper of Gruska [7]. 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, 8, 9, 13]. In this paper we consider another class of models satisfying a condition called μ-continuity analogous to the star-continuity condition of Kleene algebra: