With Davorin Lešnik.
Abstract: We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.
Download paper: csms_in_synthtop.pdf
Abstract: In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarskiâ€™s fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs.
Published in: Theoretical Computer Science Volume 430, 27 April 2012, Pages 43–50. Mathematical Foundations of Programming Semantics (MFPS XXV)
With Iztok Kavkler.
Abstract: We formulate a predicative, constructive theory of continuous domains whose realizability interpretation gives a practical implementation of continuous Ï‰-chain complete posets and continuous maps between them. We apply the theory to implementation of the interval domain and exact real numbers.
With Iztok Kavkler.
Abstract: RZ is a tool which translates axiomatizations of mathematical structures to program speciï¬cations using the realizability interpretation of logic. This helps programmers correctly implement data structures for computable mathematics. RZ does not prescribe a particular method of implementation, but allows programmers to write efficient code by hand, or to extract trusted code from formal proofs, if they so desire. We used this methodology to axiomatize real numbers and implemented the speciï¬cation computed by RZ. The axiomatization is the standard domain-theoretic construction of reals as the maximal elements of the interval domain, while the implementation closely follows current state-of-the-art implementations of exact real arithmetic. Our results shows not only that the theory and practice of computable mathematics can coexist, but also that they work together harmoniously.
Presented at Computability and Complexity in Analysis 2007.
Download paper: rzreals.pdf
Download slides: cca2007-slides.pdf
With Chris Stone.
Realizability theory is not only a fundamental tool in logic and computability, but also has direct application to the design and implementation of programs: it can produce interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpretation of constructive mathematics, RZ translates specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools.
Presented at Computablity in Europe 2007.
Download slides: cie2007-slides.pdf
Download source code from RZ web page.
For the benefit of the topology seminar audience at the math department of University of Ljubljana, I have written a self-contained explanation of the Kleene tree, which is an interesting object in computability theory. For the benefit of the rest of the planet, I am publishing it here.
Continue reading König’s Lemma and the Kleene Tree
These are lecture notes for a tutorial seminar which I gave at a satellite seminar of Computability and Complexity in Analysis 2005 in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted at an audience which is familiar with computable mathematics but less so with constructive mathematics, category theory or realizability theory.
Note: I have revised the original version from August 23, 2005 and corrected the horrible error at the beginning of Section 2. I would appreciate reports on other mistakes that you find in these notes.
Download (version of October 16, 2005): c2c.pdf, c2c.ps.gz
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.
I have created a blog category in which I will stick all my research papers. The idea is to allow people to comment on the papers and have an opportunity for discussion. There are some obvious advantages to this, such as: bug reports, opinions, references to related and relevant topics, a paper and a discussion about it are found in the same place, etc. Right now I do not see any obvious drawbacks, so I hope it will turn out to be a good idea.
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