Rule-weighted and terminal-weighted context-free grammars have identical expressivity

TL;DR

This paper demonstrates that rule-weighted and terminal-weighted context-free grammars are equally expressive in terms of the distributions they induce, using a modification of the Greibach Normal Form transformation.

Contribution

It establishes the equivalence in expressivity between rule-weighted and terminal-weighted CFGs through a simple transformation, clarifying their relationship.

Findings

Both formalisms induce identical distributions.

The equivalence is shown via a modified Greibach Normal Form transformation.

The result unifies different approaches to weighted CFGs.

Abstract

Two formalisms, both based on context-free grammars, have recently been proposed as a basis for a non-uniform random generation of combinatorial objects. The former, introduced by Denise et al, associates weights with letters, while the latter, recently explored by Weinberg et al in the context of random generation, associates weights to transitions. In this short note, we use a simple modification of the Greibach Normal Form transformation algorithm, due to Blum and Koch, to show the equivalent expressivities, in term of their induced distributions, of these two formalisms.