Category Archives: Computation

The Role of the Interval Domain in Modern Exact Real Arithmetic

With Iztok Kavkler.

Abstract: The interval domain was proposed by Dana Scott as a domain-theoretic model for real numbers. It is a successful theoretical idea which also inspired a number of computational models for real numbers. However, current state-of-the-art implementations of real numbers, e.g., Mueller’s iRRAM and Lambov’s RealLib, do not seem to be based on the interval domain. In fact, their authors have observed that domain-theoretic concepts such as monotonicity of functions hinder efficiency of computation.

I will review the data structures and algorithms that are used in modern implementations of exact real arithmetic. They provide important insights, but some questions remain about what theoretical models support them, and how we can show them to be correct. It turns out that the correctness is not always clear, and that the good old interval domain still has a few tricks to offer.

Download slides: domains8-slides.pdf

Implementing real numbers with RZ

With Iztok Kavkler.

Abstract: RZ is a tool which translates axiomatizations of mathematical structures to program specifications 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 specification 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

Are small sentences of Peano arithmetic decidable?

Recently there has been a discussion (here, here, here, and here) on the Foundations of Mathematics mailing list about completeness of Peano arithmetic (PA) with respect to “small” sentences. Harvey Friedman made several conjectures of the following kind: “All true small sentences of PA are provable.” He proposed measures of smallness, such as counting the number of distinct variables or restricting the depth of terms. Here are some statistics concerning such statements.

Continue reading Are small sentences of Peano arithmetic decidable?

Sometimes all functions are continuous

You may have heard at times that there are mathematicians who think that all functions are continuous. One way of explaining this is to show that all computable functions are continuous. The point not appreciated by many (even experts) is that the truth of this claim depends on what programming language we use.
Continue reading Sometimes all functions are continuous