TITLE: Does Homotopy Type Theory have a computational interpretation?
QUESTION [8 upvotes]: In the Robert Harper's course on Homotopy Type Theory, he talks about how it would be desirable to find a computational interpretation of the Univalence Axiom, as it would provide a way of running HoTT programs. At the time (in 2013), this was an open problem. Is it still today an open problem? If it is, do experts believe that a computational interpretation will be found?

While writing this question, I have found this paper by Cohen, Coquand, Huber and Mörtberg (2016); which says

This  paper  presents  a  type  theory  in  which  it  is  possible  to  directly  manipulate n-dimensional cubes (points, lines, squares, cubes, etc.)  based on an interpretation of dependent type theory in a cubical set model.  This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system.
Further, Voevodsky’s univalence axiom is provable in this system.  We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

Maybe in light of this my question is trivial, but my understanding of the topic is very limited. Does this mean that the problem is now closed?

REPLY [3 votes]: This is still a "cutting edge" area, so there's not a complete consensus, but it's reasonable to say that the cubical type theory by Cohen, Coquand, Huber and Mörtberg does provide a computational interpretation of HoTT. If you like this kind of thing they even have a prototype implementation in haskell at https://github.com/mortberg/cubicaltt that you can play around with.
There are still a few questions that were left open by the cubical type theory paper, but I believe the authors have made some more progress since writing it, that is currently unpublished. The paper lacks many higher inductive types that appear in the HoTT book, notably pushouts, and to really prove that the theory is computationally well behaved there should be a proof of normalization. For the latter there is already a partial result in that direction by Huber at https://arxiv.org/abs/1607.04156 .