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Documents authored by Kovács, András

Document
DOI: 10.4230/LIPIcs.CSL.2022.28
Generalized Universe Hierarchies and First-Class Universe Levels

Authors: András Kovács

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract
In type theories, universe hierarchies are commonly used to increase the expressive power of the theory while avoiding inconsistencies arising from size issues. There are numerous ways to specify universe hierarchies, and theories may differ in details of cumulativity, choice of universe levels, specification of type formers and eliminators, and available internal operations on levels. In the current work, we aim to provide a framework which covers a large part of the design space. First, we develop syntax and semantics for cumulative universe hierarchies, where levels may come from any set equipped with a transitive well-founded ordering. In the semantics, we show that induction-recursion can be used to model transfinite hierarchies, and also support lifting operations on type codes which strictly preserve type formers. Then, we consider a setup where universe levels are first-class types and subject to arbitrary internal reasoning. This generalizes the bounded polymorphism features of Coq and at the same time the internal level computations in Agda.
Cite as

András Kovács. Generalized Universe Hierarchies and First-Class Universe Levels. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kovacs:LIPIcs.CSL.2022.28,
  author =	{Kov\'{a}cs, Andr\'{a}s},
  title =	{{Generalized Universe Hierarchies and First-Class Universe Levels}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.28},
  URN =		{urn:nbn:de:0030-drops-157485},
  doi =		{10.4230/LIPIcs.CSL.2022.28},
  annote =	{Keywords: type theory, universes}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2019.6
For Finitary Induction-Induction, Induction Is Enough

Authors: Ambrus Kaposi, András Kovács, and Ambroise Lafont

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
Inductive-inductive types (IITs) are a generalisation of inductive types in type theory. They allow the mutual definition of types with multiple sorts where later sorts can be indexed by previous ones. An example is the Chapman-style syntax of type theory with conversion relations for each sort where e.g. the sort of types is indexed by contexts. In this paper we show that if a model of extensional type theory (ETT) supports indexed W-types, then it supports finitely branching IITs. We use a small internal type theory called the theory of signatures to specify IITs. We show that if a model of ETT supports the syntax for the theory of signatures, then it supports all IITs. We construct this syntax from indexed W-types using preterms and typing relations and prove its initiality following Streicher. The construction of the syntax and its initiality proof were formalised in Agda.
Cite as

Ambrus Kaposi, András Kovács, and Ambroise Lafont. For Finitary Induction-Induction, Induction Is Enough. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 6:1-6:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaposi_et_al:LIPIcs.TYPES.2019.6,
  author =	{Kaposi, Ambrus and Kov\'{a}cs, Andr\'{a}s and Lafont, Ambroise},
  title =	{{For Finitary Induction-Induction, Induction Is Enough}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{6:1--6:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.6},
  URN =		{urn:nbn:de:0030-drops-130707},
  doi =		{10.4230/LIPIcs.TYPES.2019.6},
  annote =	{Keywords: type theory, inductive types, inductive-inductive types}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2018.20
A Syntax for Higher Inductive-Inductive Types

Authors: Ambrus Kaposi and András Kovács

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract
Higher inductive-inductive types (HIITs) generalise inductive types of dependent type theories in two directions. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalising higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy reals and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a domain-specific type theory. A context in this small type theory encodes a HIIT by listing the type formation rules and constructors. The type of the elimination principle and its beta-rules are computed from the context using a variant of the syntactic logical relation translation. We show that for indexed W-types and various examples of HIITs the computed elimination principles are the expected ones. Showing that the thus specified HIITs exist is left as future work. The type theory specifying HIITs was formalised in Agda together with the syntactic translations. A Haskell implementation converts the types of sorts and constructors into valid Agda code which postulates the elimination principles and computation rules.
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Ambrus Kaposi and András Kovács. A Syntax for Higher Inductive-Inductive Types. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2018.20,
  author =	{Kaposi, Ambrus and Kov\'{a}cs, Andr\'{a}s},
  title =	{{A Syntax for Higher Inductive-Inductive Types}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.20},
  URN =		{urn:nbn:de:0030-drops-91906},
  doi =		{10.4230/LIPIcs.FSCD.2018.20},
  annote =	{Keywords: homotopy type theory, inductive-inductive types, higher inductive types, quotient inductive types, logical relations}
}
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