An Extension of Parikh's Theorem beyond Idempotence

The commutative ambiguity cambG,X of a context-free grammar G with start symbol X assigns to each Parikh vector v the number of distinct leftmost derivations yielding a word with Parikh vector v. Based on the results on the generalization of Newton’s method to ω-continuous semirings [EKL07b, EKL07a, EKL10], we show how to approximate cambG,X by means of rational formal power series, and give a lower bound on the convergence speed of these approximations. From the latter result we deduce that cambG,X itself is rational modulo the generalized idempotence identity k = k + 1 (for k some positive integer), and, subsequently, that it can be represented as a weighted sum of linear sets. This extends Parikh’s well-known result that the commutative image of context-free languages is semilinear (k = 1). Based on the well-known relationship between context-free grammars and algebraic systems over semirings [CS63, SS78, BR82, Kui97, Boz99], our results extend the work by Green et al. [GKT07] on the computation of the provenance of Datalog queries over commutative ω-continuous semirings.