Mathematics and Computation

A blog about mathematics for computers

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In a paper accepted at POPL 2016 Matt Brown and Jens Palsberg constructed a self-interpreter for System $F_\omega$, a strongly normalizing typed $\lambda$-calculus. This came as a bit of a surprise as it is “common knowledge” that total programming languages do not have self-interpreters.

Thinking about what they did I realized that their conditions allow a self-interpreter for practically any total language expressive enough to encode numbers and pairs. In the PDF note accompanying this post I give such a self-interpreter for Gödel’s System T, the weakest such calculus. It is clear from the construction that I abused the definition given by Brown and Palsberg. Their self-interpreter has good structural properties which mine obviously lacks. So what we really need is a better definition of self-interpreters, one that captures the desired structural properties. Frank Pfenning and Peter Lee called such properties reflexivity, but only at an informal level. Can someone suggest a good definition?

Note: self-interpreter-for-T.pdf

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Postdoc position in Ljubljana

A postdoc position in the Effmath research project is available at the University of Ljubljana, Faculty of Mathematics and Physics. The precise topic is flexible, but should generally be aligned with the project (see project description). Possible topics include:

The candidate should have a PhD degree in mathematics or computer science, with background knowledge relevant to the project area. The position is available for a period of one year with possibility of extension, preferably starting in early 2016. No knowledge of the Slovene language is required.

The candidates should contact Andrej Bauer by email as soon as possible, but no later than January 8th 2016. Please include a short CV and a statement of interest.

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Agda Writer

My student Marko Koležnik is about to finish his Master’s degree in Mathematics at the University of Ljubljana. He implemented Agda Writer, a graphical user interface  for the Agda proof assistant on the OS X platform. As he puts it, the main advantage of Agda Writer is no Emacs, but the list of cool features is a bit longer:

Agda Writer is open source. Everybody is welcome to help out and participate on the Agda Writer repository.

Who is Agda Writer for? Obviously for students, mathematicians, and other potential users who were not born with Emacs hard-wired into their brains. It is great for teaching Agda as you do not have to spend two weeks explaining Emacs. The only drawback is that it is limited to OS X. Someone should write equivalent Windows and Linux applications. Then perhaps proof assistants will have a chance of being more widely adopted.

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Provably considered harmful

This is officially a rant and should be read as such.

Here is my pet peeve: theoretical computer scientists misuse the word “provably”. Stop it. Stop it!

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I have not written a blog post in a while, so I decided to write up a short observation about truth values in intuitionistic logic which sometimes seems a bit puzzling.

Let $\Omega$ be the set of truth values (in Coq this would be the setoid whose underlying type is $\mathsf{Prop}$ and equality is equivalence $\leftrightarrow$, while in HoTT it is the h-propostions). Call a truth value $p : \Omega$ intermediate if it is neither true nor false, i.e., $p \neq \bot$ and $p \neq \top$. Such a “third” truth value $p$ is proscribed by excluded middle.

The puzzle is to explain how the following two facts fit together:

  1. “There is no intermediate truth value” is an intuitionistic theorem.
  2. There are models of intuitionistic logic with many truth values.
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Here are the slides of my TYPES 2015 talk “The troublesome reflection rule” with fairly detailed presenter notes. The meeting is  taking place in Tallinn, Estonia – a very cool country in many senses (it’s not quite spring yet even though we’re in the second half of May, and it’s the country that gave us Skype).

Download slides: The troublesome reflection rule (TYPES 2015) [PDF].

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Another PhD position in Ljubljana

It is my pleasure to announce a second PhD position in Ljubljana!

A position is available for a PhD student at the University of Ljubljana in the general research area of modelling and reasoning about computational effects. The precise topic is somewhat flexible, and will be decided in discussion with the student. The PhD will be supervised by Alex Simpson who is Professor of Computer Science at the Faculty of Mathematics and Physics.

The position will be funded by the Effmath project (see project description). Full tuition & stipend will be provided.

The candidate should have a master’s (or equivalent) degree in either mathematics or computer science, with background knowledge relevant to the project area. The student will officially enrol in October 2015 at the University of Ljubljana. No knowledge of the Slovene language is required.

The candidates should contact by email as soon as possible. Please include a short CV and a statement of interest.

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I am looking for a PhD student in mathematics. Full tuition & stipend will be provided for a period of three years, which is also the official length of the programme. The topic of research is somewhat flexible and varies from constructive models of homotopy type theory to development of a programming language for a proof assistant based on dependent type theory, see the short summary of the Effmath project for a more detailed description.

The candidate should have as many of the following desiderata as possible, and at the very least a master’s degree (or an equivalent one):

  1. a master’s degree in mathematics, with good knowledge of computer science
  2. a master’s degree in computer science, with good knowledge of mathematics
  3. experience with functional programming
  4. experience with proof assistants
  5. familiarity with homotopy type theory

The student will officially enrol in October 2015 at the University of Ljubljana. No knowledge of Slovene is required. However, it is possible, and even desirable, to start with the actual work (and stipend) earlier, as soon as in the spring of 2015. The candidates should contact me by email as soon as possible. Please include a short CV and a statement of interest.

Update 2015-03-28: the position has been taken.

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TEDx “Zeroes”

I spoke at TEDx University of Ljubljana. The topic was how programming influences various aspects of life. I showed the audence how a bit of simple programming can reveal the beauty of mathematics. Taking John Baez’s The Bauty of Roots as an inspiration, I drew a very large image (20000 by 17500 pixels) of all roots of all polynomials of degree at most 26 whose coefficients are $-1$ or $1$. That’s 268.435.452 polynomials and 6.979.321.752 roots. It is two degrees more than Sam Derbyshire’s image,  so consider the race to be on! Who can give me 30 degrees?

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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

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