Star Complexity of Parikh Images of Languages over Infinite Alphabets

TL;DR

This paper investigates the complexity of Parikh images of languages over infinite alphabets, showing bounds for one-register automata and disproving conjectures for multiple registers, thus demonstrating limits of Parikh's theorem in this context.

Contribution

It proves that the star-height of Parikh images for one-register automata is bounded, and shows that the conjecture fails for multiple registers and context-free languages over infinite alphabets.

Findings

Star-height of Parikh images for one-register automata is at most two.

Parikh's theorem does not hold for languages over infinite alphabets with multiple registers.

Context-free languages over infinite alphabets can have arbitrarily high star height in their Parikh images.

Abstract

It has been conjectured that the Parikh (commutative) image of every language over an infinite alphabet recognized by an automaton with registers is defined by a rational expression. This conjecture is known to hold for all languages recognized by one-register automata. We refine this result by proving that the star-height of the Parikh image of any language recognized by a one-register automaton is universally bounded by two. Furthermore, we show that one-register context-free languages have rational commutative images of arbitrarily high star height. We then disprove the conjecture for multiple registers, as well as disprove the equivalence of commutative expressive power between context-free grammars and automata over infinite alphabets. In other words, we show that Parikh's theorem fails for infinite alphabets.