All Outputs (18)

Set-Theoretic and Type-Theoretic Ordinals Coincide (2023)
Presentation / Conference Contribution
KRAUS, N., de Jong, T., Nordvall Forsberg, F., & Xu, C. (2023, June). Set-Theoretic and Type-Theoretic Ordinals Coincide. Paper presented at Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Boston, USA

In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (... Read More about Set-Theoretic and Type-Theoretic Ordinals Coincide.

Two-level type theory and applications (2023)
Journal Article
Annenkov, D., Capriotti, P., Kraus, N., & Sattler, C. (2023). Two-level type theory and applications. Mathematical Structures in Computer Science, 33(8), 688-743. https://doi.org/10.1017/s0960129523000130

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy typ... Read More about Two-level type theory and applications.

Type-theoretic approaches to ordinals (2023)
Journal Article
Kraus, N., Nordvall Forsberg, F., & Xu, C. (2023). Type-theoretic approaches to ordinals. Theoretical Computer Science, 957, Article 113843. https://doi.org/10.1016/j.tcs.2023.113843

In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensi... Read More about Type-theoretic approaches to ordinals.

A rewriting coherence theorem with applications in homotopy type theory (2022)
Journal Article
Kraus, N., & von Raumer, J. (2022). A rewriting coherence theorem with applications in homotopy type theory. Mathematical Structures in Computer Science, 32(7), 982-1014. https://doi.org/10.1017/s0960129523000026

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a... Read More about A rewriting coherence theorem with applications in homotopy type theory.

Connecting Constructive Notions of Ordinals in Homotopy Type Theory (2021)
Presentation / Conference Contribution
Kraus, N., Nordvall Forsberg, F., & Xu, C. (2021). Connecting Constructive Notions of Ordinals in Homotopy Type Theory.

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 ordinal... Read More about Connecting Constructive Notions of Ordinals in Homotopy Type Theory.

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT (2021)
Presentation / Conference Contribution
Kraus, N. (2021). Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT. In 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (1-14). https://doi.org/10.1109/LICS52264.2021.9470667

Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer'... Read More about Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT.

Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory (2020)
Presentation / Conference Contribution
Kraus, N., & Von Raumer, J. (2020). Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory. In LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (662-675). https://doi.org/10.1145/3373718.3394800

Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of inducti... Read More about Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory.

Shallow Embedding of Type Theory is Morally Correct (2019)
Presentation / Conference Contribution
Kaposi, A., Kovács, A., & Kraus, N. (2019). Shallow Embedding of Type Theory is Morally Correct. Lecture Notes in Artificial Intelligence, 11825 LNCS, 329-365. https://doi.org/10.1007/978-3-030-33636-3_12

© 2019, Springer Nature Switzerland AG. There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for exampl... Read More about Shallow Embedding of Type Theory is Morally Correct.

Path Spaces of Higher Inductive Types in Homotopy Type Theory (2019)
Presentation / Conference Contribution
Kraus, N., & von Raumer, J. (2019). Path Spaces of Higher Inductive Types in Homotopy Type Theory. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). https://doi.org/10.1109/LICS.2019.8785661

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pus... Read More about Path Spaces of Higher Inductive Types in Homotopy Type Theory.

From Cubes to Twisted Cubes via Graph Morphisms in Type Theory (2019)
Presentation / Conference Contribution
Pinyo, G., & Kraus, N. (2019, June). From Cubes to Twisted Cubes via Graph Morphisms in Type Theory. Paper presented at TYPES 2019, Oslo, Norway

Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher groupoids... Read More about From Cubes to Twisted Cubes via Graph Morphisms in Type Theory.

Free Higher Groups in Homotopy Type Theory (2018)
Presentation / Conference Contribution
Kraus, N., & Altenkirch, T. (2018). Free Higher Groups in Homotopy Type Theory. In LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (599-608). https://doi.org/10.1145/3209108.3209183

© 2018 ACM. Given a type A in homotopy type theory (HoTT), we can define the free∞-group onA as the loop space of the suspension ofA+1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit : F(A),... Read More about Free Higher Groups in Homotopy Type Theory.

Notions of anonymous existence in Martin-Löf type theory (2017)
Journal Article
Kraus, N., Escardo, M., Coquand, T., & Altenkirch, T. (in press). Notions of anonymous existence in Martin-Löf type theory. Logical Methods in Computer Science, 13(1),

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have uniqu... Read More about Notions of anonymous existence in Martin-Löf type theory.

Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type (2017)
Presentation / Conference Contribution
Altenkirch, T., Danielson, N. A., & Kraus, N. (2017). Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type. In Foundations of Software Science and Computation Structures: FoSSaCS 2017 (534-549). https://doi.org/10.1007/978-3-662-54458-7_31

Capretta's delay monad can be used to model partial computations, but it has the ``wrong'' notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the ``right''notion of equality, weak bisimilarity. However,... Read More about Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type.

Constructions with Non-Recursive Higher Inductive Types (2016)
Presentation / Conference Contribution
Kraus, N. (2016). Constructions with Non-Recursive Higher Inductive Types. In LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (595-604). https://doi.org/10.1145/2933575.2933586

© 2016 ACM. Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say tha... Read More about Constructions with Non-Recursive Higher Inductive Types.

The General Universal Property of the Propositional Truncation (2015)
Journal Article
Kraus, N. (2015). The General Universal Property of the Propositional Truncation. LIPIcs, 39, 111-145. https://doi.org/10.4230/LIPIcs.TYPES.2014.111

In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence c... Read More about The General Universal Property of the Propositional Truncation.

Higher Homotopies in a Hierarchy of Univalent Universes (2015)
Journal Article
Kraus, N., & Sattler, C. (2015). Higher Homotopies in a Hierarchy of Univalent Universes. ACM Transactions on Computational Logic, 16(2), Article 18. https://doi.org/10.1145/2729979

For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher... Read More about Higher Homotopies in a Hierarchy of Univalent Universes.

Generalizations of Hedberg’s Theorem (2013)
Presentation / Conference Contribution
Kraus, N., Escardó, M., Coquand, T., & Altenkirch, T. (2013). Generalizations of Hedberg’s Theorem. In Typed Lambda Calculi and Applications: 11th International Conference, TLCA 2013, Eindhoven, The Netherlands, June (173-188). https://doi.org/10.1007/978-3-642-38946-7_14

As the groupoid interpretation by Hofmann and Streicher shows, uniqueness of identity proofs (UIP) is not provable. Generalizing a theorem by Hedberg, we give new characterizations of types that satisfy UIP. It turns out to be natural in this context... Read More about Generalizations of Hedberg’s Theorem.

A Lambda Term Representation Inspired by Linear Ordered Logic (2011)
Journal Article
Abel, A., & Kraus, N. (2011). A Lambda Term Representation Inspired by Linear Ordered Logic. Electronic Proceedings in Theoretical Computer Science, 71, 1-13. https://doi.org/10.4204/EPTCS.71.1

We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where... Read More about A Lambda Term Representation Inspired by Linear Ordered Logic.