observational type theory

**natural deduction** metalanguage, practical foundations

**type theory** (dependent, intensional, observational type theory, homotopy type theory)

**computational trinitarianism** = **propositions as types** +**programs as proofs** +**relation type theory/category theory**

What is called *observational type theory* (OTT) is a flavor of type theory in between extensional type theory and intensional type theory.

It may be regarded as a first-stage approximation to homotopy type theory (HoTT): the propositions of OTT correspond to the h-propositions of HoTT, and the types in OTT correspond to h-sets in HoTT. The notion of equality on OTT is based on setoids, which is a special case of higher internal groupoids. Since equality is defined type-by-type, OTT enjoys function extensionality, and a form of propositional extensionality at least for a specified universe of propositions (not necessarily including all h-propositions).

There are a few technical differences (e.g. proofs of propositions are definitionally equal in OTT, whereas proofs of hprops are only propositionally equal in HoTT) but on the whole HoTT looks a lot like a higher-dimensional version of OTT.

Observational type theory was introduced in

- Thorsten Altenkirch and Conor McBride,
*Towards observational type theory*(pdf)

A blog post about an Agda implementation, including propositional extensionality (which is not mentioned in the above paper) is at

The above comparison between OTT and HoTT is taken from

Revised on July 29, 2017 19:03:34
by Mike Shulman
(76.167.222.204)