# Posts in the category RZ

### Mathematically Structured but not Necessarily Functional Programming

- 28 May 2009
- Computation, Constructive math, Programming, RZ, Talks

These are the slides and the extended abstract from my MSFP 2008 talk. Apparently, I forgot to publish them online. There is a discussion on the Agda mailing list to which the talk is somewhat relevant, so I am publishing now.

**Abstract:** Realizability is an interpretation of intuitionistic logic which subsumes the Curry-Howard interpretation of propositions as types, because it allows the realizers to use computational effects such as non-termination, store and exceptions. Therefore, we can use realizability as a framework for program development and extraction which allows any style of programming, not just the purely functional one that is supported by the Curry-Howard correspondence. In joint work with Christopher A. Stone we developed RZ, a tool which uses realizability to translate specifications written in constructive logic into interface code annotated with logical assertions. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. In our experience, RZ is useful for specification of non-trivial theories. While the use of computational effects does improve efficiency it also makes it difficult to reason about programs and prove their correctness. We demonstrate this fact by considering non-purely functional realizers for a Brouwerian continuity principle.

**Download:** msfp2008-slides.pdf, msfp2008-abstract.pdf

### A constructive theory of domains suitable for implementation

- 31 January 2008
- Constructive math, Publications, RZ

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.

Download: constructive-domains.pdf

→ continue reading### The Role of the Interval Domain in Modern Exact Real Arithmetic

- 17 September 2007
- Computation, Constructive math, RZ, Talks

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

- 11 April 2007
- Computation, Constructive math, Publications, RZ, Talks

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

### RZ: a tool for bringing constructive and computable mathematics closer to programming practice

- 21 January 2007
- Publications, RZ, Talks

With Chris Stone.

**Abstract:**

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 paper:**

- Long version: cie-long.pdf (January 29, 2007)
- Short version: cie-short.pdf (January 29, 2007)

**Download slides:** cie2007-slides.pdf

**Download source code** from RZ web page.

### Specifications via Realizability (CLASE 2005)

- 09 April 2005
- Publications, RZ, Talks