Abstract
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 \(KAF \subseteq KAR \subseteq KAG\): respectively, these make μ a least fixed point operator, connect it with recursion, and express S. Greibach's method to replace left- by 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.
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© 1992 Springer-Verlag Berlin Heidelberg
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Leiß, H. (1992). Towards Kleene Algebra with recursion. In: Börger, E., Jäger, G., Kleine Büning, H., Richter, M.M. (eds) Computer Science Logic. CSL 1991. Lecture Notes in Computer Science, vol 626. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023771
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DOI: https://doi.org/10.1007/BFb0023771
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