7 Search Results for "Xu, Chuangjie"

Artifact
Software
DOI: 10.4230/artifacts.26786
Constructive Notions of Ordinals in Homotopy Type Theory

Authors: Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, Fredrik Nordvall Forsberg, and Chuangjie Xu

Abstract
Cite as

Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, Fredrik Nordvall Forsberg, Chuangjie Xu. Constructive Notions of Ordinals in Homotopy Type Theory (Software, Repository). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@misc{dagstuhl-artifact-26786,
   title = {{Constructive Notions of Ordinals in Homotopy Type Theory}}, 
   author = {de Jong, Tom and Kraus, Nicolai and Mohammadzadeh, Aref and Nordvall Forsberg, Fredrik and Xu, Chuangjie},
   note = {Software, Royal Society URF\R1\191055, RF\ERE\210032, RF\ERE\231052, URF\R\241007, UK National Physical Laboratory Measurement Fellowship project "Dependent types for trustworthy tools", Engineering and Physical Sciences Research Council EP/Y000455/1, EP/Z000602/1, swhId: \href{https://archive.softwareheritage.org/swh:1:snp:af7a3d40f000eeb74aa0f06222f84f76e2046841;origin=https://bitbucket.org/nicolaikraus/constructive-ordinals-in-hott.git}{\texttt{swh:1:snp:af7a3d40f000eeb74aa0f06222f84f76e2046841}} (visited on 2026-07-09)},
   url = {https://bitbucket.org/nicolaikraus/constructive-ordinals-in-hott.git},
   doi = {10.4230/artifacts.26786},
}
Document
DOI: 10.4230/LIPIcs.LICS.2026.59
Generalized Decidability via Brouwer Trees

Authors: Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, and Fredrik Nordvall Forsberg

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

Abstract
In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable". We work in homotopy type theory and use Brouwer tree ordinals to specify the level of decidability of a property. In this framework, we express the property that a proposition is α-decidable, for an ordinal α, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that α-decidable propositions are closed under binary conjunction, and discuss for which α they are closed under binary disjunction. We prove that if each P(i) is semidecidable, then the countable meet ∀ i ∈ ℕ. P(i) is ω²-decidable, and similar results for countable joins and iterated quantifiers. We also discuss the relationship with countable choice. All our results are formalized in Cubical Agda.
Cite as

Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, and Fredrik Nordvall Forsberg. Generalized Decidability via Brouwer Trees. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 59:1-59:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dejong_et_al:LIPIcs.LICS.2026.59,
  author =	{de Jong, Tom and Kraus, Nicolai and Mohammadzadeh, Aref and Nordvall Forsberg, Fredrik},
  title =	{{Generalized Decidability via Brouwer Trees}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{59:1--59:27},
  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.2026.59},
  URN =		{urn:nbn:de:0030-drops-268466},
  doi =		{10.4230/LIPIcs.LICS.2026.59},
  annote =	{Keywords: Decidability in constructive mathematics, homotopy type theory, ordinals, Brouwer trees, countable choice}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSCD.2025.1
Computation First: Rebuilding Constructivism with Effects (Invited Talk)

Authors: Liron Cohen

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
Constructive logic and type theory have traditionally been grounded in pure, effect-free model of computation. This paper argues that such a restriction is not a foundational necessity but a historical artifact, and it advocates for a broader perspective of effectful constructivism, where computational effects, such as state, non-determinism, and exceptions, are directly and internally embedded in the logical and computational foundations. We begin by surveying examples where effects reshape logical principles, and then outline three approaches to effectful constructivism, focusing on realizability models: Monadic Combinatory Algebras, which extend classical partial combinatory algebras with effectful computation; Evidenced Frames, a flexible semantic structure capable of uniformly capturing a wide range of effects; and Effectful Higher-Order Logic (EffHOL), a syntactic approach that directly translates logical propositions into specifications for effectful programs. We further illustrate how concrete type theories can internalize effects, via the family of type theories TT^□_C. Together, these works demonstrate that effectful constructivism is not merely possible but a natural and robust extension of traditional frameworks.
Cite as

Liron Cohen. Computation First: Rebuilding Constructivism with Effects (Invited Talk). In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cohen:LIPIcs.FSCD.2025.1,
  author =	{Cohen, Liron},
  title =	{{Computation First: Rebuilding Constructivism with Effects}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.1},
  URN =		{urn:nbn:de:0030-drops-236167},
  doi =		{10.4230/LIPIcs.FSCD.2025.1},
  annote =	{Keywords: Effectful constructivism, realizability, type theory, monadic combinatory algebras, evidenced frame}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.25
Completeness of the Decreasing Diagrams Method for Proving Confluence of Rewriting Systems of the Least Uncountable Cardinality

Authors: Ievgen Ivanov

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
We show that every confluent abstract rewriting system (ARS) of the cardinality that does not exceed the first uncountable cardinal belongs to the class DCR₃, i.e. the class of confluent ARS for which confluence can be proved with the the help of the decreasing diagrams method using the set of labels {0,1,2} ordered in such a way that 0<1<2 (in the general case, the decreasing diagrams method with two labels is not sufficient for proving confluence of such ARS). Under the Continuum Hypothesis this result implies that the decreasing diagrams method is sufficient for establishing confluence of ARS on many structures of interest to applied mathematics and various interdisciplinary fields (confluence of ARS on real numbers, continuous real functions, etc.). We provide a machine-checked formal proof of a formalized version of the main result in Isabelle proof assistant using HOL logic and the HOL-Cardinals theory. An extended version of this formalization is available in the Archive of Formal Proofs.
Cite as

Ievgen Ivanov. Completeness of the Decreasing Diagrams Method for Proving Confluence of Rewriting Systems of the Least Uncountable Cardinality. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ivanov:LIPIcs.FSCD.2025.25,
  author =	{Ivanov, Ievgen},
  title =	{{Completeness of the Decreasing Diagrams Method for Proving Confluence of Rewriting Systems of the Least Uncountable Cardinality}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.25},
  URN =		{urn:nbn:de:0030-drops-236404},
  doi =		{10.4230/LIPIcs.FSCD.2025.25},
  annote =	{Keywords: confluence, decreasing diagrams method, rewriting systems, reduction, formal methods, formal proofs, formal verification, non-discrete models, nondeterministic models, interval models}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.70
Connecting Constructive Notions of Ordinals in Homotopy Type Theory

Authors: Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu

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

Abstract
In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.
Cite as

Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu. Connecting Constructive Notions of Ordinals in Homotopy Type Theory. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kraus_et_al:LIPIcs.MFCS.2021.70,
  author =	{Kraus, Nicolai and Nordvall Forsberg, Fredrik and Xu, Chuangjie},
  title =	{{Connecting Constructive Notions of Ordinals in Homotopy Type Theory}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{70:1--70:16},
  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.2021.70},
  URN =		{urn:nbn:de:0030-drops-145100},
  doi =		{10.4230/LIPIcs.MFCS.2021.70},
  annote =	{Keywords: Constructive ordinals, Cantor normal forms, Brouwer trees}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2020.25
A Gentzen-Style Monadic Translation of Gödel’s System T

Authors: Chuangjie Xu

Published in: LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

Abstract
We introduce a syntactic translation of Gödel’s System 𝖳 parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzen’s negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of 𝖳-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.
Cite as

Chuangjie Xu. A Gentzen-Style Monadic Translation of Gödel’s System T. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{xu:LIPIcs.FSCD.2020.25,
  author =	{Xu, Chuangjie},
  title =	{{A Gentzen-Style Monadic Translation of G\"{o}del’s System T}},
  booktitle =	{5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-155-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{167},
  editor =	{Ariola, Zena M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.25},
  URN =		{urn:nbn:de:0030-drops-123472},
  doi =		{10.4230/LIPIcs.FSCD.2020.25},
  annote =	{Keywords: monadic translation, G\"{o}del’s System T, logical relation, negative translation, majorizability, continuity, bar recursion, Agda}
}
Document
DOI: 10.4230/LIPIcs.TLCA.2015.153
The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation

Authors: Martín Hötzel Escardó and Chuangjie Xu

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract
If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Sigma type via the Curry-Howard interpretation. But with an intuitionistic notion of anonymous existence, defined as the propositional truncation of Sigma, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N -> 2) -> N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn't show Agda code.
Cite as

Martín Hötzel Escardó and Chuangjie Xu. The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 153-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hotzelescardo_et_al:LIPIcs.TLCA.2015.153,
  author =	{H\"{o}tzel Escard\'{o}, Mart{\'\i}n and Xu, Chuangjie},
  title =	{{The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{153--164},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.153},
  URN =		{urn:nbn:de:0030-drops-51618},
  doi =		{10.4230/LIPIcs.TLCA.2015.153},
  annote =	{Keywords: Dependent type, intensional Martin-L\"{o}f type theory, Curry-Howard interpretation, constructive mathematics, Brouwerian continuity axioms, anonymous exi}
}
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