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Thorsten Altenkirch - 1/2 Towards a Syntax for Cubical Type Theory

Institut Henri Poincaré

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  • Recommended by 2 people: 2 in engineering communities.
  • Presented by Thorsten Altenkirch, Professor of Computer Science at the University of Nottingham

Advanced. Deep dive for people who know the field.

11K views12 years ago2 recommendations

Computer scienceProgramming languagesComputation and complexityLanguage design and type systems

Thorsten Altenkirch discusses the challenges of finding computational interpretations for homotopy type theory axioms and proposes a cubical type theory.

One of the key problems of Homotopy Type Theory is that it introduces axioms such as extensionality and univalence for which there is no known computational interpretation. We propose to overcome this by introducing a Type Theory where a heterogenous equality is defined recursively and equality for the universe just is univalence. This cubical type theory is inspired by Bernardy and Moulin's internal parametricity and by Coquand, Bezem and Huber's cubical set model. This is ongoing work with Ambrus Kaposi at Nottingham.

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What people say. Recommended as a deep dive into the theoretical foundations of modern type theory and its relation to programming languages.

u/CodaFiin r/compsci • Recommends

On “Computer Science ∩ Mathematics (Type Theory) - by Computerphile”

Thorstein raised a very interesting point, if only in passing here. For all the fanciness Homotopy Type Theory provides and all the new theorems it's brought into the CS Theory fold we *still* don't know the computational content of some of its core axioms (Extensionality, Univalence). This kind of stuff is what I absolutely love about modern Type Theory as it relates to programming and programming languages. We've got to the point where we have this cart-before-the-horse kind of thing in that we know the benefits and now we have to search for the appropriate semantics. It really feels like a softer science with all these researchers now just throwing "stuff at the wall to see what sticks…
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u/Mentioned_Videosin r/compsci • Recommends

On “Computer Science ∩ Mathematics (Type Theory) - by Computerphile”

Other videos in this thread: Watch Playlist ▶ VIDEO|COMMENT -|- Thorsten Altenkirch - 1/2 Towards a Syntax for Cubical Type Theory|4 - Thorstein raised a very interesting point, if only in passing here. For all the fanciness Homotopy Type Theory provides and all the new theorems it's brought into the CS Theory fold we still don't know the computational content of some of its core a... Type Theory Foundations, Lecture 2|4 - That's how I see it too. It has also been suggested that it is the other way around. (The actual statement comes at 54:39; I've linked to an earlier point for a little context.) CS ∩ Mathematics : LONG VERSION - Computerphile|3 - Slightly extended cut (4 e…
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