Star-height of Parikh images is bounded by 2 for one-register automata but the rational conjecture fails for multiple registers, showing Parikh's theorem does not hold over infinite alphabets.
Computing and Combinatorics, 10th Annual International Conference
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Star Complexity of Parikh Images of Languages over Infinite Alphabets
Star-height of Parikh images is bounded by 2 for one-register automata but the rational conjecture fails for multiple registers, showing Parikh's theorem does not hold over infinite alphabets.