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Professor THORSTEN ALTENKIRCH's Outputs (28)

Type theory in type theory using quotient inductive types (2016)
Presentation / Conference Contribution
Altenkirch, T., & Kaposi, A. (2016, January). Type theory in type theory using quotient inductive types. Presented at POPL '16 The 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, St Petersburg, Florida, USA

We present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids r... Read More about Type theory in type theory using quotient inductive types.

Indexed containers (2015)
Journal Article
Altenkirch, T., Ghani, N., Hancock, P., McBride, C., & Morris, P. (2015). Indexed containers. Journal of Functional Programming, 25, 1-41. https://doi.org/10.1017/s095679681500009x

We show that the syntactically rich notion of strictly positive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. As a result, we show indexed containers provide... Read More about Indexed containers.

Monads need not be endofunctors (2015)
Journal Article
Altenkirch, T., Chapman, J., & Uustalu, T. (2015). Monads need not be endofunctors. Logical Methods in Computer Science, 11(1), 1-40. https://doi.org/10.2168/LMCS-11%281%3A3%292015

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show tha... Read More about Monads need not be endofunctors.

Some constructions on ω-groupoids (2014)
Presentation / Conference Contribution
Altenkirch, T., Li, N., & Ondřej, R. (2014, July). Some constructions on ω-groupoids. Presented at LFMTP '14: Theory and Practice, Vienna, Austria

Weak ω-groupoids are the higher dimensional generalisation of setoids and are an essential ingredient of the constructive semantics of Homotopy Type Theory [13]. Following up on our previous formalisation [3] and Brunerie's notes [6], we present a ne... Read More about Some constructions on ω-groupoids.

Relative monads formalised (2014)
Journal Article
Altenkirch, T., Chapman, J., & Uustalu, T. (2014). Relative monads formalised. Journal of Formalized Reasoning, 7(1), https://doi.org/10.6092/issn.1972-5787/4389

Relative monads are a generalisation of ordinary monads where the underlying functor need not be an endofunctor. In this paper, we describe a formalisation of the basic theory of relative monads in the interactive theorem prover and dependently typed... Read More about Relative monads formalised.

Generalizations of Hedberg’s Theorem (2013)
Presentation / Conference Contribution
Kraus, N., Escardó, M., Coquand, T., & Altenkirch, T. (2013, June). Generalizations of Hedberg’s Theorem. Presented at TLCA: International Conference on Typed Lambda Calculi and Applications, Eindhoven, The Netherlands

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.

When is a function a fold or an unfold? (2001)
Presentation / Conference Contribution
Gibbons, J., Hutton, G., & Altenkirch, T. When is a function a fold or an unfold?. Presented at Workshop on Coalgebraic Methods in Computer Science (4th)

We give a necessary and sufficient condition for when a set-theoretic function can be written using the recursion operator fold, and a dual condition for the recursion operator unfold. The conditions are simple, practically useful, and generic in the... Read More about When is a function a fold or an unfold?.