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All Outputs (16)

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.

Shallow Embedding of Type Theory is Morally Correct (2019)
Journal Article
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)
Conference Proceeding
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.

Free Higher Groups in Homotopy Type Theory (2018)
Conference Proceeding
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.

Quotient inductive-inductive types (2018)
Book Chapter
Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., & Nordvall Forsberg, F. (2018). Quotient inductive-inductive types. In C. Baier, & U. Dal Lago (Eds.), FoSSaCS 2018: Foundations of Software Science and Computation Structures (293-310). Cham: Springer Publishing Company. https://doi.org/10.1007/978-3-319-89366-2_16

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of H... Read More about Quotient inductive-inductive types.

Univalent higher categories via complete semi-segal types (2017)
Journal Article
Capriotti, P., & Kraus, N. (2018). Univalent higher categories via complete semi-segal types. Proceedings of the ACM on Programming Languages, 2(POPL), https://doi.org/10.1145/3158132

Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.)solves this by considering only... Read More about Univalent higher categories via complete semi-segal types.

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)
Conference Proceeding
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.

Extending Homotopy Type Theory with Strict Equality (2016)
Conference Proceeding
Altenkirch, T., Capriotti, P., & Kraus, N. (2016). Extending Homotopy Type Theory with Strict Equality. In 25th EACSL Annual Conference on Computer Science Logic (21:1-21:17). https://doi.org/10.4230/LIPIcs.CSL.2016.21

In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult a... Read More about Extending Homotopy Type Theory with Strict Equality.

Constructions with Non-Recursive Higher Inductive Types (2016)
Conference Proceeding
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), 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)
Conference Proceeding
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.