Category Archives: Constructive math

Topics related to constructive mathematics.

Reductions in computability theory from a constructive point of view

Here are the slides from my Logic Coloquium 2014 talk in Vienna. This is joint work with Kazuto Yoshimura from Japan Advanced Institute for Science and Technology.

Abstract: In constructive mathematics we often consider implications between non-constructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility, which turns out to have a very rich structure. Even better, under Kleene’s function realizability interpretation instance reducibility corresponds to Weihrauch reducibility, while Kleene’s number realizability relates it to truth-table reducibility. We may also ask about a constructive treatment of other reducibilities in computability theory. I shall discuss how one can tackle Turing reducibility constructively via Kleene’s number realizability.

Slides with talk notes:  lc2014-slides-notes.pdf

Seemingly impossible constructive proofs

In the post Seemingly impossible functional programs, I wrote increasingly efficient Haskell programs to realize the mathematical statement

$\forall p : X \to 2. (\exists x:X.p(x)=0) \vee (\forall x:X.p(x)=1)$

for $X=2^\mathbb{N}$, the Cantor set of infinite binary sequences, where $2$ is the set of binary digits. Then in the post A Haskell monad for infinite search in finite time I looked at ways of systematically constructing such sets $X$ with corresponding Haskell realizers of the above omniscience principle.

In this post I give examples of infinite sets $X$ and corresponding constructive proofs of their omniscience that are intended to be valid in Bishop mathematics, and which I have formalized in Martin-Löf type theory in Agda notation. This rules out the example $X=2^\mathbb{N}$, as discussed below, but includes many interesting infinite examples. I also look at ways of constructing new omniscient sets from given ones. Such sets include, in particular, ordinals, for which we can find minimal witnesses if any witness exists.

Agda is a dependently typed functional programming language based on Martin-Löf type theory. By the Curry-Howard correspondence, Agda is also a language for formulating mathematical theorems (types) and writing down their proofs (programs). Agda acts as a thorough referee, only accepting correct theorems and proofs. Moreover, Agda can run your proofs. Here is a graph of the main Agda modules for this post, and here is a full graph with all modules.

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The elements of an inductive type

In the HoTT book issue 460 a question by gluttonousGrandma (where do people get these nicknames?) once more exposed a common misunderstanding that we tried to explain in section 5.8 of the book (many thanks to Bas Spitters for putting the book into Google Books so now we can link to particular pages). Apparently the following belief is widely spread, and I admit to holding it a couple of years ago:

An inductive type contains exactly those elements that we obtain by repeatedly using the constructors.

If you believe the above statement you should keep reading. I am going to convince you that the statement is unfounded, or that at the very least it is preventing you from understanding type theory.

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The HoTT book

The HoTT book is finished!

Since spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. You can get the book freely at Mike Shulman has written about the contents of the book, so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research.

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