The Parikh Property for Weighted Context-Free Grammars

TL;DR

This paper investigates the Parikh property in weighted context-free grammars, proving it holds for nonexpansive grammars and providing a decision procedure for rational weights, advancing understanding of weighted CFG equivalences.

Contribution

It proves the Parikh property for weighted nonexpansive CFGs and introduces a decision procedure for rational weights, extending classical results to weighted settings.

Findings

Parikh property holds for weighted nonexpansive CFGs.

A decision procedure is provided for weighted CFGs with rational weights.

The results extend classical Parikh's Theorem to weighted grammars.

Abstract

Parikh's Theorem states that every context-free grammar (CFG) is equivalent to some regular CFG when the ordering of symbols in the words is ignored. The same is not true for the so-called weighted CFGs, which additionally assign a weight to each grammar rule. If the result holds for a given weighted CFG $G$, we say that $G$ satisfies the Parikh property. We prove constructively that the Parikh property holds for every weighted nonexpansive CFG. We also give a decision procedure for the property when the weights are over the rationals.