# Exact real arithmetic in Haskell

HERA is an implementation of exact real arithmetic in Haskell using the approach by Andrej Bauer and Iztok Kavkler, see these and these slides. It uses the fast multiple precision floating point library MPFR. Download source, and see documentation and examples of usage at my home page.

[Note by Andrej: this is a guest post by AleÅ¡ Bizjak, a first-year student of mathematics at my department. I am very proud of the excellent work he did on his summer project.]

# Efficient computation with Dedekind reals

Two versions of this talk were given at Computability and complexity in analysis 2008 and at Mathematics, Algorithms and Proofs 2008.

Joint work with Paul Taylor.

Abstract: Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of Dedekind reals by constructing them within Abstract Stone Duality (ASD), a computationally meaningful calculus for topology. This provides the theoretical background for a novel way of computing with real numbers in the style of logic programming. Real numbers are defined in terms of (lower and upper) Dedekind cuts, while programs are expressed as statements about real numbers in the language of ASD. By adapting Newton’s method to interval arithmetic we can make the computations as efficient as those based on Cauchy reals.

Slides: slides-map2008.pdf (obsolete version: slides-cca2008.pdf)
Extended abstract: abstract-cca2098.pdf

# Representations of uncomputable and uncountable sets

Occasionally I hear claims that uncountable and uncomputable sets cannot be represented on computers. More generally, there are all sorts of misguided opinions about representations of data on computers, especially infinite data of mathematical nature. Here is a quick tutorial on the matter whose main point is:

It is meaningless to discuss representations of a set by a datatype without also considering operations that we want to perform on the set.

# Seemingly impossible functional programs

Andrej has invited me to write about certain surprising functional
programs.
The first program, due to Ulrich Berger (1990), performs exhaustive
search over the “Cantor space” of infinite sequences of binary
digits. I have included references at the end. A weak form of
exhaustive search amounts to checking whether or not a total predicate
holds for all elements of the Cantor space. Thus, this amounts to
universal quantification over the Cantor space. Can this possibly be
done algorithmically, in finite time?
Continue reading Seemingly impossible functional programs