An Extension of Parikh's Theorem beyond Idempotence

TL;DR

This paper extends Parikh's theorem to a broader context, showing how to approximate and represent the commutative ambiguity of context-free grammars using rational formal power series and linear sets, beyond the classical idempotent case.

Contribution

It generalizes Parikh's theorem by demonstrating the rationality and linear set representation of commutative ambiguity under a generalized idempotence condition.

Findings

Commutative ambiguity can be approximated by rational formal power series.

The commutative ambiguity is rational modulo a generalized idempotence identity.

It can be expressed as a weighted sum of linear sets.

Abstract

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.