Mathematics and Computation

A blog about mathematics for computers

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Posts in the year 2022

Happy birthday, Dana!

Today Dana Scott is celebrating the 90th birthday today. Happy birthday, Dana! I am forever grateful for your kindness and the knowledge that I received from you. I hope to pass at least a part of it onto my students.

On the occasion Steve Awodey assembled selected works by Dana Scott at CMU-HoTT/scott repository. It is an amazing collection of papers that had deep impact on logic, set theory, computation, and programming languages. I hope in the future we can extend it and possibly present it in better format.

As a special treat, I recount here the story the invention of the famous $D_\infty$ model of the untyped $\lambda$-calculus. I heard it first when I was Dana's student. In 2008 I asked Dana to recount it in the form of a short interview.

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One syntax to rule them all

I am at the Syntax and Semantics of Type Theory workshop in Stockholm, a kickoff meeting for WG6 of the EuroProofNet COST network, where I am giving a talk “One syntax to rule them all” based on joint work with Danel Ahman.

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Two new doctors!

Within a month two of my students defended their theses: Dr. Anja Petković Komel just before Christmas, and Dr. Philipp Haselwarter just yesterday. I am very proud of them. Congratulations!

Philipp's thesis An Effective Metatheory for Type Theory has three parts:

  1. A formulation and a study of the notion of finitary type theories and standard type theories. These are closely related to the general type theories that were developed with Peter Lumsdaine, but are tailored for implementation.

  2. A formulation and the study of context-free finitary type theories, which are type theories without explicit contexts. Instead, the variables are annotated with their types. Philipp shows that one can pass between the two versions of type theory.

  3. A novel effectful meta-language Andromeda meta-language (AML) for proof assistants which uses algebraic effects and handlers to allow flexible interaction between a generic proof assistant and the user.

Anja's thesis Meta-analysis of type theories with an application to the design of formal proofs also has three parts:

  1. A formulation and a study of transformations of finitary type theories with an associated category of finitary type theories.

  2. A user-extensible equality checking algorithm for standard type theories which specializes to several existing equality checking algorithms for specific type theories.

  3. A general elaboration theorem in which the transformation of type theories are used to prove that every finitary type theory (not necessarily fully annotated) can be elaborated to a standard type theory (fully annotated one).

In addition, Philipp has done a great amount of work on implementing context-free type theories and the effective meta-language in Andromeda 2, and Anja implemented the generic equality checking algorithm. In the final push to get the theses out the implementation suffered a little bit and is lagging behind. I hope we can bring it up to speed and make it usable. Anja has ideas on how to implement transformations of type theories in a proof assistant.

Of course, I am very happy with the particular results, but I am even happier with the fact that Philipp and Anja made an important step in the development of type theory as a branch of mathematics and computer science: they did not study a particular type theory or a narrow family of them, as has hitherto been the norm, but dependent type theories in general. Their theses contain interesting non-trivial meta-theorems that apply to large classes of type theories, and can no doubt be generalized even further. There is lots of low-hanging fruit out there.

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Jacques Carette asked on Twitter for a refence to the fact that countable choice holds in setoids. I then spent a day formalizing facts about the axiom of choice in setoids in Agda. I noticed something interesting that is worth blogging about.

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