On Polynomial Recursive Sequences

Authors Michaël Cadilhac , Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, Géraud Sénizergues

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Author Details

Michaël Cadilhac
  • DePaul University, Chicago, IL, USA
Filip Mazowiecki
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbücken, Germany
Charles Paperman
  • Université de Lille, Villeneuve d'Ascq, France
Michał Pilipczuk
  • University of Warsaw, Poland
Géraud Sénizergues
  • Université de Bordeaux, France


We thank Maria Donten-Bury for suggesting the proof of Theorem 11 presented here. This proof replaced our previous more elaborate and less transparent argument. We also thank James Worrell, David Purser and Markus Whiteland for helpful comments. The research for this work was carried out in part at the Autobóz Research Camp in 2019 in Firbush, Scotland. Finally, we thank the participants of the automata seminar at the University of Warsaw for an insightful discussion on the class of rational recursive sequences (considered in Section 7).

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Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, and Géraud Sénizergues. On Polynomial Recursive Sequences. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 117:1-117:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is b_n = n!. Our main result is that the sequence u_n = nⁿ is not polynomial recursive.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • recursive sequences
  • expressive power
  • weighted automata
  • higher-order pushdown automata


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