Mathematics and Computation

With Paul Taylor.

Abstract: Abstract Stone Duality (ASD) is an approach to topology that provides an abstract and conceptually satisfying account of topological spaces. The calculus of ASD reveals the computational content of various topological notions and suggests how to compute with them. The distinguishing feature of ASD is a direct axiomatisation in terms of spaces and maps, which does not rely on an underlying set-theoretic or topos-theoretic foundation.

This paper makes the first step in real analysis within ASD, namely the construction of the real line using two-sided Dedekind cuts. Compactness and overtness of the closed interval are proved, and the arithmetic operations are defined. The ASD calculus gives programs for computing the arithmetic operations and the quantifiers that express compactness and overtness.

As the paper aims to be a self-contained introduction to ASD for those interested in constructive and computable topology and analysis, it includes a rapid survey of the ASD calculus. The foundational background to the calculus was covered in detail in earlier work.

Further topics in real analysis within ASD, such as the Intermediate Value Theorem, are presented in a separate paper by Paul Taylor which builds on this one.

To be presented at Computability and Complexity in Analysis 2005, Kyoto, Japan.

Download: an up-to-date version from Paul Taylor's Abstract Stone Duality page.

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So I decided to put all my research papers on the blog. → continue reading

The new address for Math and Computation blog is math.andrej.com → continue reading

In constructive mathematics even very small sets can be quite a bit more interesting than in classical mathematics. Since you will not believe me that sets with at most one element are very interesting, let us look at the set of truth values, which has “two” elements.

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What constructive mathematics actually says about the law of excluded middle. → continue reading (8 comments)

I have found a good way to write math in web pages. ASCIIMathML is a piece of javascript that translates simple-minded Latex-like ASCII math to MathML, but only if the browser supports MathML. Since the input syntax is very simple, the expressions are quite readable in the raw form, as well.

For example, if I type

`forall x in RR exists y in CC. (1-x^2 )/sqrt(1+y^4)=1`

it is seen as `forall x in RR exists y in CC. (1-x^2 )/sqrt(1+y^4)=1`. If you are going to post to the blog, you may be interested in the ASCIIMathML syntax reference page.

To enable MathML on your computer, install mathplayer plugin
if you are using Internet Explorer. For Firefox and Mozilla, you have to install math fonts.

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Abstract: Computability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland's effective topos and Richman's formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Gödel encodings, but rather use familiar concepts from set theory and topology.

Presented at: Mathematical Foundations of Programming Semantics XXI, Birmingham, 2004 (invited talk).

Download paper: synthetic.pdf, synthetic.ps.gz

Download slides: synthetic-slides.pdf

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I recently stumbled upon a blog on machine learning by a good friend of mine, John Langford. The blog has gathered together a community of people who discuss various topics (not limited strictly to machine learning). Naturally I wanted to have a blog, too.

I devote a lot of my time to thinking about the relationship between mathematics and computation. There are two sides of this, which can be expressed by a the slogan “Computable mathematics and mathematics of computation”. Computable mathematics is about how to do mathematics with computers, while mathematics of computation is about mathematics that describes properties of computation in a mathematical, abstract way.

If this is a subject that interests you, I invite you to join me.

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How to build specifications for abstract data types using realizability → continue reading

With Alex Simpson.

Abstract: We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, `X`, in such a way that every sequentially continuous function from Cantor space to `ZZ` extends to a sequentially continuous function from `X` to `RR`. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved.

As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from `X` to `RR` are continuous”, when `X` is an inhabited CSM without isolated points, and when `X` is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non-compact CSM, `X`, generalizes a result previously obtained by Escardo and Streicher in the special case `X = C[0,1]`.

As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space `X`, there exists a Banach-Mazur computable function from `X` to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that `X` is the space of computable real numbers.

Published in: Mathematical Logic Quarterly, 50(4,5):351-369, 2004.

Download: continuity.pdf, continuity.ps.gz

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