Parikh's theorem for infinite alphabets

TL;DR

This paper extends Parikh's theorem to infinite alphabets by analyzing commutative images of languages recognized by register automata, showing they are always rational but not necessarily semi-linear.

Contribution

It proves that commutative images of one-register automata languages are always rational, extending Parikh's theorem to infinite alphabets and grammars.

Findings

Commutative images of one-register automata are always rational.

They are not necessarily semi-linear.

Results extend to grammars, showing rationality of their commutative images.

Abstract

We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars with arbitrarily many registers.