The Simplest Proof of Parikh's Theorem via Derivation Trees

TL;DR

This paper presents a concise and straightforward proof of Parikh's theorem, emphasizing simplicity and clarity by leveraging non-constructive techniques to avoid complexities of previous proofs.

Contribution

It offers the simplest proof of Parikh's theorem to date, closely following the original approach while providing clearer exposition and avoiding complex constructions.

Findings

Proof is short and simple compared to previous proofs

Technique avoids many difficulties of earlier methods

Enhances understanding of Parikh's theorem's fundamental nature

Abstract

Parikh's theorem is a fundamental result of the formal language's theory. There had been published many proofs and many papers claimed to provide a simplified proof, but most of them are long and still complicated. We provide the proof that is really short, simple and discloses the nature of this fundamental result. We follow the technique closed to the original Parikh's paper and our proof is similar to the proof by Ryoma Sin'ya 2019, but we provide more detailed exposition and pretend to more simplicity as well. We achieve the simplicity via nonconstructivenes that allows us avoiding many difficulties met by other proofs.