A Type Theory for Defining Logics and Proofs

[Philip Thrift]

https://arxiv.org/abs/1905.02617 :

A Type Theory for Defining Logics and Proofs

Brigitte Pientka, David Thibodeau, Andreas Abel, Francisco Ferreira, Rebecca Zucchini

(Submitted on 7 May 2019)

We describe a Martin-Löf-style dependent type theory, called Cocon, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that describes (recursive) computations. We mediate between HOAS representations and computations using contextual modal types. Our type theory also supports an infinite hierarchy of universes and hence supports type-level computation thereby providing metaprogramming and (small-scale) reflection. Our main contribution is the development of a Kripke-style model for Cocon that allows us to prove normalization. From the normalization proof, we derive subject reduction and consistency. Our work lays the foundation to incorporate the methodology of logical frameworks into systems such as Agda and bridges the longstanding gap between these two worlds.

@philipthrift

[Lawrence Crowell]

I don't think I will get around to reading this. It is awfully dense. The subject looks interesting though. I am not sure, but this sort of type theory might be of importance with Laughlin wave functions. The expectation on a 2-dim surface, either a boundary or a graphene sheet <φ(z')φ(z)> ~ -log|z' – z| leads to the Laughlin wave for ψ ~ e^{φ(z')√q}e^{φ(z)√q} and then Z ~ |z' –-z|^q exp(¼|z'|^2), which in general is computed for a product of these in a path integral. These products <φ(z')φ(z)> emerge from the Baker-Campbell-Hausdorff theorem and this has some bearing with the idea of the 2-slit experiment as a logic gate, but where now instead of slits one has fields. However, this paper looks a bit dense to read.

LC