1 research outputs found
An Extension of Parikh's Theorem beyond Idempotence
The commutative ambiguity of a context-free grammar G 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 omega-continuous semirings, we show how to approximate the commutative
ambiguity 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 the commutative ambiguity 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, our results extend the work by Green et al.
on the computation of the provenance of Datalog queries over commutative
omega-continuous semirings