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DOI: 10.4230/LIPIcs.CSL.2026.39

A Canonical Form for Universe Levels in Impredicative Type Theory

Authors: Yoan Géran

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

The 0-imax-successor algebra, where imax: ℕ × ℕ → ℕ is the function defined by imax(n, 0) = 0 and imax(n, S(m)) = max(n, S(m)), is used to represent universe levels in impredicative type theory, in particular with universe polymorphism which introduces level variables, so it is present in proof systems such as Rocq and Lean. In particular, we need to know when two elements of this algebra are equivalent, and we may also want to decide the inequality. In this article, we introduce a canonical form for the terms of this algebra, and we provide a canonization algorithm. It permits deciding level equivalence by checking the canonical form equality, and also permits easily checking if a level is smaller than another one.

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Yoan Géran. A Canonical Form for Universe Levels in Impredicative Type Theory. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{geran:LIPIcs.CSL.2026.39,
  author =	{G\'{e}ran, Yoan},
  title =	{{A Canonical Form for Universe Levels in Impredicative Type Theory}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{39:1--39:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.39},
  URN =		{urn:nbn:de:0030-drops-254640},
  doi =		{10.4230/LIPIcs.CSL.2026.39},
  annote =	{Keywords: universe levels, canonical form, impredicativity, imax algebra}
}

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DOI: 10.4230/LIPIcs.MFCS.2021.49

Keyboards as a New Model of Computation

Authors: Yoan Géran, Bastien Laboureix, Corto Mascle, and Valentin D. Richard

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract

We introduce a new formalisation of language computation, called keyboards. We consider a set of atomic operations (writing a letter, erasing a letter, going to the right or to the left) and we define a keyboard as a set of finite sequences of such operations, called keys. The generated language is the set of words obtained by applying some non-empty sequence of those keys. Unlike classical models of computation, every key can be applied anytime. We define various classes of languages based on different sets of atomic operations, and compare their expressive powers. We also compare them to rational, context-free and context-sensitive languages. We obtain a strict hierarchy of classes, whose expressiveness is orthogonal to the one of the aforementioned classical models. We also study closure properties of those classes, as well as fundamental complexity problems on keyboards.

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Yoan Géran, Bastien Laboureix, Corto Mascle, and Valentin D. Richard. Keyboards as a New Model of Computation. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 49:1-49:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{geran_et_al:LIPIcs.MFCS.2021.49,
  author =	{G\'{e}ran, Yoan and Laboureix, Bastien and Mascle, Corto and Richard, Valentin D.},
  title =	{{Keyboards as a New Model of Computation}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{49:1--49:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.49},
  URN =		{urn:nbn:de:0030-drops-144896},
  doi =		{10.4230/LIPIcs.MFCS.2021.49},
  annote =	{Keywords: formal languages, models of computation, automata theory}
}

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Documents authored by Géran, Yoan

A Canonical Form for Universe Levels in Impredicative Type Theory

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Keyboards as a New Model of Computation

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