Mathematics and Computation

A blog about mathematics for computers

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Posts in the year 2011

HoTT Equivalences

On December 6th 2011 I gave a talk about homotopy equivalences in the context of homotopy type theory at our seminar for foundations of mathematics and theoretical computer science. I discuss the differences and relations between isomorphism (in the sense of type theory), an adjoint equivalence, and a homotopy equivalence. Even though the talk itself was not super-well prepared, I hope the recording will be interesting to some people. I was going fairly slowly, so it should be possible to follow the talk. I apologize for such a long video, but I really did not see how to chop it up into smaller pieces. Also, I need to figure out why I cannot fast forward the video beyond what has been downloaded.

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A talk given at “Mathematics, Algorithms and Proofs 2011” at the Lorentz Center in Leiden, the Netherlands. I explain how to use computational effects to speed up Martin Escardo’s impossible functionals.

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A talk given at “Computation with Infinite Data: Logical and Topological Foundations” Dagstuhl seminar 11411. I describe a realizability model based on infinite-time Turing machines in which it is possible to embed the Baire space (infinite sequences of numbers) into the space of numbers.

Also see the post Constructive gem: an injection from Baire space to natural numbers for written notes on this topic.

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Video lectures as screencasts

Last year I participated in a project whose goal was to record at low cost my lectures on video and put them on-line. Since the most expensive parts of recording are having a camera man and manual post production, we set up a static camera and just uploaded raw video online at videolectures.net. As you can see for yourself, the sound is good (I wore a microphone) but the whiteboard is mostly illegible. In addition, it took about two weeks for the lectures to show up on-line because there were men-in-the-middle. So that got me thinking whether there was a better way.

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In an earlier post I talked about the modulus of continuity functional, where I stated that it cannot be defined without using some form of computational effects. It is a bit hard to find the proof of this fact so I am posting it on my blog in two parts, for Google and everyone else to find more easily. In the first part I show that there is no extensional modulus of continuity. In the second part I will show that every functional that is defined in PCF (simply-typed $\lambda$-calculus with natural numbers and recursion) is extensional. 

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I am not sure whether to call this one a constructive gem or stone. I suppose it is a matter of personal taste. I think it is a gem, albeit a very unusual one: there is a topos in which $\mathbb{N}^\mathbb{N}$ can be embedded into $\mathbb{N}$.

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With Karin Cvetko Vah.

For the last two months or so I got “distracted” by a topic which is not properly my core interest, namely non-commutative algebra. It was very strange at first, but now that I got used to non-commutative lattices (yes, there is such a thing) it’s kind of fun. Anyhow, Karin Cvetko Vah and I worked out Stone duality for skew Boolean algebras with intersections. Classical Stone duality tells us that Boolean algebras are dual to Stone spaces (zero-dimensional compact Hausdorff spaces), and that the generalized Boolean algebras (which are like Boolean algebras without a top element) are dual to Boolean spaces (zero-dimensional locally-compact Hausdorff spaces). Our skew version of duality says that right-handed skew Boolean algebras with intersections are dual to surjective etale maps between Boolean spaces. It is quite a mouthful to say “right-handed skew Boolean algebra with intersections”, let alone get used to it, but in a certain sense this is a very natural non-commutative structure. And we can get rid of the “right-handed” condition to obtain duality for “skew Boolean algebras with intersections”, as well as several other versions. We use the duality to construct a right-handed skew Boolean algebra with intersections which does not have a lattice section. It has been an open question whether such skew lattices exist.

Download: sba.pdf

arXiv: 1106.0425

Abstract: We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.

And now I can get back to homotopy type theory and Coq hacking.

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As a preparation for my part of a joint tutorial Programs from proofs at MFPS 27 at the end of this month with Ulrich Berger, Monika Seisenberger, and Paulo Oliva, I’ve developed in Agda some things we’ve been doing together.

Using

for giving a proof term for classical countable choice, we prove the classical infinite pigeonhole principle in Agda: every infinite boolean sequence has a constant infinite subsequence, where the existential quantification is classical (double negated).

As a corollary, we get the finite pigeonhole principle, using Friedman’s trick to make the existential quantifiers intuitionistic.

This we can run, and it runs fast enough. The point is to illustrate in Agda how we can get witnesses from classical proofs that use countable choice. The finite pigeonhole principle has a simple constructive proof, of course, and hence this is really for illustration only.

The main Agda files are

These are Agda files converted to html so that you can navigate them by clicking at words to go to their definitions. A zip file with all Agda files is available. Not much more information is available here.

The three little modules that implement the Berardi-Bezem-Coquand, Berger-Oliva and Escardo-Oliva functionals disable the termination checker, but no other module does. The type of these functionals in Agda is the J-shift principle, which generalizes the double-negation shift.

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Bob Harper has a blog

Bob Harper of CMU, has recently started a blog, called Existential Type, about programming languages. He is a leading expert in Programming Languages. I remember being deeply inspired the first time I heard him talk. I was an incoming graduate student at CMU and he presented what the programming languages people at CMU did. His posts are fun to read, unreserved and very educational. Highly recommended!

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Next week I am going to a meeting where I am supposed to give a tutorial on the Coq proof assistant. Inspired by the Catsters, I decided to prepare the material in the form of screencasts. You can find the first few tutorials on Youtube in my “Coq tutorials” playlist. So far I have:

You should turn on the high quality HD stream when you watch these. Feedback is welcome (and easy to provide on Youtube). I find it very, very difficult to listen to my own voice. I hope to have many more lectures soon, but I am starting to feel out of my depth, so if anyone wants to help they are welcome!

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Jens Blanck and I presented a paper at Computability and Complexity in Analysis 2009 with a complicated title (I like complicated titles):

Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions

which has been published in Volume 16, Issue 18 of the Journal of Universal Computer Science. I usually just post the abstract, but this time I would like to explain the general idea informally, the way one can do it on a blog. But first, here is the abstract:

Abstract: We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others.

Download paper: effalg.pdf or directly from JUCS

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Alg

Alg is a program for enumeration of finite models of single-sorted first-order theories. These include groups, rings, fields, lattices, posets, graphs, and many more. Alg was written as a class project by Aleš Bizjak, a student of mine whose existence I cannot confirm with a URL. I joined the effort, added bells and whistles, as well as an alternative algorithm that works well for relational structures. Alg is ready for public consumption, although it should be considered of “beta” quality. Instructions for downloading alg are included at the end of this post.

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Alex Simpson, Matija Pretnar and I are organizing a workshop on computational effects. It will take place in Ljubljana on March 17th and 18th 2011. More information is available at the workshop web page.

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I think I am getting addicted to Coq, or more generally to doing mathematics, including the proofs, with computers. I spent last week finalizing a formalization of Gödel’s functional interpretation of logic, also known as the Dialectica interpretation. There does not seem to be one available already, which is a good opportunity for a blog post.

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Postsby categoryby yearall