# Category Archives: Publications

Research publications by Andrej Bauer

# Five stages of accepting constructive mathematics

In 2013 I gave a talk about constructive mathematics “Five stages of accepting constructive mathematics” (video) at the Institute for Advanced Study. I turned the talk into a paper, polished it up a bit, added things here and there, and finally it has now been published in the Bulletin of the American Mathematical Society. It is not quite a survey paper, but it is not very technical either. I hope you will enjoy reading it.

Free access to the paper:  Five stages of accepting constructive mathematics (PDF)

# A Brown-Palsberg self-interpreter for Gödel’s System T

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?

# The troublesome reflection rule (TYPES 2015 slides)

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].

# Intuitionistic Mathematics and Realizability in the Physical World

This is a draft version of my contribution to “A Computable Universe: Understanding and Exploring Nature as Computation”, edited by Hector Zenil. Consider it a teaser for the rest of the book, which contains papers by an impressive list of authors.

Abstract: Intuitionistic mathematics perceives subtle variations in meaning where classical mathematics asserts equivalence, and permits geometrically and computationally motivated axioms that classical mathematics prohibits. It is therefore well-suited as a logical foundation on which questions about computability in the real world are studied. The realizability interpretation explains the computational content of intuitionistic mathematics, and relates it to classical models of computation, as well as to more speculative ones that push the laws of physics to their limits. Through the realizability interpretation Brouwerian continuity principles and Markovian computability axioms become statements about the computational nature of the physical world.

Download: real-world-realizability.pdf

# 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 http://homotopytypetheory.org/book/. 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.

# Programming with Algebraic Effects and Handlers

With Matija Pretnar.

Abstract: Eff is a programming language based on the algebraic approach to computational effects, in which effects are viewed as algebraic operations and effect handlers as homomorphisms from free algebras. Eff supports first-class effects and handlers through which we may easily define new computational effects, seamlessly combine existing ones, and handle them in novel ways. We give a denotational semantics of eff and discuss a prototype implementation based on it. Through examples we demonstrate how the standard effects are treated in eff, and how eff supports programming techniques that use various forms of delimited continuations, such as backtracking, breadth-first search, selection functionals, cooperative multi-threading, and others.

Download paper: eff.pdf

ArXiv version: arXiv:1203.1539v1 [cs.PL]

To read more about eff, visit the eff page.

# On the Bourbaki-Witt Principle in Toposes

Abstract: The Bourbaki-Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

Download paper: bw.pdf
ArXiv version: arXiv:1201.0340v1 [math.CT]

This paper is an extension of my previous paper on the Bourbaki-Witt and Knaster-Tarski fixed-point theorems in the effective topos (arXiv:0911.0068v1).

# Stone Duality for Skew Boolean Algebras with Intersections

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.

# Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions

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