Yet another proof of Parikh's Theorem

TL;DR

This paper presents a concise proof of Parikh's Theorem, demonstrating that the Parikh image of any context-free language is semilinear, using Presburger arithmetic and an Eulerian property of derivation trees.

Contribution

It offers a novel, streamlined proof of Parikh's Theorem that avoids Chomsky normal form and leverages an Eulerian property inspired by Hierholzer's algorithm.

Findings

Proof confirms the semilinearity of Parikh images for context-free languages.

Utilizes Presburger arithmetic formulation for a simplified proof.

Avoids traditional normal form transformations in the proof process.

Abstract

Parikh's Theorem says that the Parikh image of a context-free language is semilinear. We give a short proof of Parikh's Theorem using the formulation of Verma, Seidl, and Schwentick in terms of Presburger arithmetic. The proof relies on an Eulerian property of derivation trees of context-free languages and was inspired by Hierholzer's algorithm; it does not use the Chomsky normal form.