Implementing Real Numbers With RZ

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.

A constructive theory of continuous domains suitable for implementation

“…We order approximations according to their quality, which leads to order-theoretic constructions of spaces. We took this approach in our implementation of intervals and real numbers Era [3], which uses the tool RZ [4] to derive specifications (program templates) from axiomatizations of constructive mathematical theories. Therefore, we first looked for a suitable axiomatization of the space of approximations of real numbers.…”

Type classes for efficient exact real arithmetic in Coq

“…To make our implementation of the reals independent of the underlying dense set, we provide an abstract specification of approximate rationals inspired by the notion of approximate fields -a field with approximate operations -which is used in Bauer and Kavler's RZ implementation of the exact reals [BK08]; see also [BT09]. In particular, we provide an implementation of this interface by dyadics based on Coq's machine integers.…”

RZ: a Tool for Bringing Constructive and Computable Mathematics Closer to Programming Practice

“…2 Simple means at most one arrow between any two vertices. 3 We use OCaml notation in which t list classifies finite lists of elements of type t, and t 1 * t 2 classifies pairs containing a value of type t 1 and a value of type t 2 . Figure 1.…”

“…In this section, we look at several small examples which demonstrate various points of RZ. For a serious case study from computable mathematics see the implementation of real numbers with RZ by Bauer and Kavkler [3].…”