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DOI: 10.4230/LIPIcs.LICS.2026.35

Star Complexity of Parikh Images of Languages over Infinite Alphabets

Authors: Yoav Danieli

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

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.

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Yoav Danieli. Star Complexity of Parikh Images of Languages over Infinite Alphabets. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 35:1-35:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{danieli:LIPIcs.LICS.2026.35,
  author =	{Danieli, Yoav},
  title =	{{Star Complexity of Parikh Images of Languages over Infinite Alphabets}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{35:1--35:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.35},
  URN =		{urn:nbn:de:0030-drops-268229},
  doi =		{10.4230/LIPIcs.LICS.2026.35},
  annote =	{Keywords: infinite alphabets, Parikh image, rational sets, star-height}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.29

A Pumping-Like Lemma for Languages over Infinite Alphabets

Authors: Yoav Danieli

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

We prove a kind of a pumping lemma for languages accepted by one-register alternating finite-memory automata. As a corollary, we obtain that the set of lengths of words in such languages is semi-linear.

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Yoav Danieli. A Pumping-Like Lemma for Languages over Infinite Alphabets. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{danieli:LIPIcs.STACS.2026.29,
  author =	{Danieli, Yoav},
  title =	{{A Pumping-Like Lemma for Languages over Infinite Alphabets}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{29:1--29:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.29},
  URN =		{urn:nbn:de:0030-drops-255185},
  doi =		{10.4230/LIPIcs.STACS.2026.29},
  annote =	{Keywords: infinite alphabets, pumping lemma, alternation, semi-linearity}
}

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Documents authored by Danieli, Yoav

Star Complexity of Parikh Images of Languages over Infinite Alphabets

Abstract

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A Pumping-Like Lemma for Languages over Infinite Alphabets

Abstract

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