Mathematics and Computation

A blog about mathematics for computers

Postsby categoryby yearall

On fixed-point theorems in synthetic computability

I forgot to record the fact that already two years ago I wrote a paper on Lawvere's fixed-point theorem in synthetic computability:

Andrej Bauer: On fixed-point theorems in synthetic computability. Tbilisi Mathematical Journal, Volume 10: Issue 3, pp. 167–181.

It was a special issue in honor of Professors Peter J. Freyd and F. William Lawvere on the occasion of their 80th birthdays.

Lawvere's paper "Diagonal arguments and cartesian closed categories proves a beautifully simple fixed point theorem.

Theorem: (Lawvere) If $e : A \to B^A$ is a surjection then every $f : B \to B$ has a fixed point.

Proof. Because $e$ is a surjection, there is $a \in A$ such that $e(a) = \lambda x : A \,.\, f(e(x)(x))$, but then $e(a)(a) = f(e(a)(a)$. $\Box$

Lawvere's original version is a bit more general, but the one given here makes is very clear that Lawvere's fixed point theorem is the diagonal argument in crystallized form. Indeed, the contrapositive form of the theorem, namely

Corollary: If $f : B \to B$ has no fixed point then there is no surjection $e : A \to B^A$.

immediately implies a number of famous theorems that rely on the diagonal argument. For example, there can be no surjection $A \to \lbrace 0, 1\rbrace^A$ because the map $x \mapsto 1 - x$ has no fixed point in $\lbrace 0, 1\rbrace$ -- and that is Cantors' theorem.

It not easy to find non-trivial instances to which Lawvere's theorem applies. Indeed, if excluded middle holds, then having a surjection $e : A \to B^A$ implies that $B$ is the singleton. We should look for interesting instances in categories other than classical sets. In my paper I do so: I show that countably based $\omega$-cpos in the effective topos are countable and closed under countable products, which gives us a rich supply of objects $B$ such that there is a surjection $\mathbb{N} \to B^\mathbb{N}$.

Enjoy the paper!

Runners in action

It has been almost a decade since Matija Pretnar and I posted the first blog posts about programming with algebraic effects and handlers and the programming language Eff. Since then handlers have become a well-known control mechanism in programming languages.

Handlers and monads excel at simulating effects, either in terms of other effects or as pure computations. For example, the familiar state monad implements mutable state with (pure) state-passing functions, and there are many more examples. But I have always felt that handlers and monads are not very good at explaining how a program interacts with its external environment and how it gets to perform real-world effects.

Danel Ahman and I have worked for a while on attacking the question on how to better model external resources and what programming constructs are appropriate for working with them. The time is right for us to show what we have done so far. The theoretical side of things is explained in our paper Runners in action, Danel implemented a Haskell library Haskell-Coop to go with the paper, and I implemented a programming language Coop.

On complete ordered fields

Joel Hamkins advertised the following theorem on Twitter:

Theorem: All complete ordered fields are isomorphic.

The standard proof posted by Joel has two parts:

1. A complete ordered field is archimedean.
2. Using the fact that the rationals are dense in an archimedean field, we construct an isomorphism between any two complete ordered fields.

The second step is constructive, but the first one is proved using excluded middle, as follows. Suppose $F$ is a complete ordered field. If $b \in F$ is an upper bound for the natural numbers, construed as a subset of $F$, then so $b - 1$, but then no element of $F$ can be the least upper bound of $\mathbb{N}$. By excluded middle, above every $x \in F$ there is $n \in \mathbb{N}$.

So I asked myself and the constructive news mailing list what the constructive status of the theorem is. But something was amiss, as Fred Richman immediately asked me to provide an example of a complete ordered field. Why would he do that, don't we have the MacNeille reals? After agreeing on definitions, Toby Bartels gave the answer, which I am taking the liberty to adapt a bit and present here. I am probably just reinventing the wheel, so if someone knows an original reference, please provide it in the comments.

The theorem holds constructively, but for a bizarre reason: if there exists a complete ordered field, then the law of excluded middle holds, and the standard proof is valid!

What is algebraic about algebraic effects?

Published as arXiv:1807.05923.

Abstract: This note recapitulates and expands the contents of a tutorial on the mathematical theory of algebraic effects and handlers which I gave at the Dagstuhl seminar 18172 "Algebraic effect handlers go mainstream". It is targeted roughly at the level of a doctoral student with some amount of mathematical training, or at anyone already familiar with algebraic effects and handlers as programming concepts who would like to know what they have to do with algebra. We draw an uninterrupted line of thought between algebra and computational effects. We begin on the mathematical side of things, by reviewing the classic notions of universal algebra: signatures, algebraic theories, and their models. We then generalize and adapt the theory so that it applies to computational effects. In the last step we replace traditional mathematical notation with one that is closer to programming languages.

The blog moved from Wordpress to Jekyll

You may have noticed that lately I have had trouble with the blog. It was dying periodically because the backend database kept crashing. It was high time I moved away from Wordpress anyway, so I bit the bullet and ported the blog.

What is an explicit bijection? (FPSAC 2019 slides)

Here are the slides with speaker notes for the talk What is an explicit bijection which I gave at the 31st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2019). It was the "outsider" talk, where they invite someone to tell them something outside of their area.

So how does one sell homotopy type theory to people who are interested in combinatorics? That is a tough sell. I used my MathOverflow question "What is an explicit bijection?" to give a stand-up comedy introduction, after which I plunged into type theory. I am told I plunged a little too hard. For instance, people asked "why are we doing this" because I did not make it clear enough that we are trying to make a distinction between "abstractly exists" and "concretely constructed". Oh well, it’s difficult to explain homotopy type theory in 50 minutes. Anyhow, I hope you can get something useful from the slides.

Video recording of the lecture is now available.

A course on homotopy (type) theory

This semester my colleague Jaka Smrekar and I are teaching a graduate course on homotopy theory and homotopy type theory. The first part was taught by Jaka and was a nice review of classical homotopy theory leading up to Quillen model categories. In the second part I am covering basic homotopy type theory.

The course materials are available at the GitHub repository homotopy-type-theory-course. The homotopy type theory lectures are also recorded on video.

How to implement type theory in an hour

I was purging the disk on my laptop of large files and found a video lecture which I forgot to publish. Here it is with some delay. I lectured on how to implement type theory at the School and Workshop on Univalent Mathematics in December 2017, at the University of Birmingham (UK).

You may watch the video and visit the accompanying GitHub repository spartan-type-theory.

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Algebraic effects and handlers at OPLSS 2018

I have had the honor to lecture at the Oregon Programming Language Summer School 2018 on the topic of algebraic effects and handlers. The notes, materials and the lectures are available online:

I gave four lectures which started with the mathematics of algebraic theories, explained how they can be used to model computational effects, how we make a programming language out of them, and how to program with handlers.