Logic for Gray-code computation

Berger, Ulrich; Miyamoto, Kenji; Schwichtenberg, Helmut; Tsuiki, Hideki

doi:10.1515/9781501502620-005

Book chapter 1031 views

Logic for Gray-code computation

Ulrich Berger

, Kenji Miyamoto, Helmut Schwichtenberg, Hideki Tsuiki

Start page: 69

Swansea University Author: Ulrich Berger

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DOI (Published version): 10.1515/9781501502620-005

Abstract

Gray-code is a well-known binary number system where neighboring values differ in one digit only. Tsuiki (2002) has introduced Gray code to the €field of real number computation. He assigns to each number a unique infinite sequence containing at most one undefined element. In this paper we take a lo...

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ISBN:	9781501502620
Published:	Mouton, Oldenburg, China de Gruyter 2016
Online Access:	http://www.cs.swan.ac.uk/~csulrich/ftp/logic_for_gray_code.pdf
URI:	https://cronfa.swan.ac.uk/Record/cronfa28978
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first_indexed	2016-11-08T05:18:23Z
last_indexed	2018-02-09T05:13:39Z
id	cronfa28978
recordtype	SURis
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spelling	2017-09-12T16:10:46.7724830 v2 28978 2016-06-21 Logic for Gray-code computation 61199ae25042a5e629c5398c4a40a4f5 0000-0002-7677-3582 Ulrich Berger Ulrich Berger true false 2016-06-21 SCS Gray-code is a well-known binary number system where neighboring values differ in one digit only. Tsuiki (2002) has introduced Gray code to the €field of real number computation. He assigns to each number a unique infinite sequence containing at most one undefined element. In this paper we take a logical and constructive approach to study real number computation based on Gray-code. Instead of Tsuiki’s indeterministic multihead Type-2 machine, we use pre-Gray code, which is a representation of Gray-code as a sequence of constructors, to avoid the difculty due to the undefined element which prevents sequential access to a stream.We extract real number algorithms from proofs in an appropriate formal theory involving inductive and coinductive defi€nitions. Examples are algorithms transforming pre-Gray code into signed digit code of real numbers, and conversely, the average for pre-Gray code and a translation of fi€nite segments of pre-Gray code into its normal form. These examples are formalized in the proof assistant Minlog. Book chapter 69 de Gruyter Mouton, Oldenburg, China 9781501502620 Gray-code, Real number computation, Inductive and coinductive definitions, Program extraction. 31 7 2016 2016-07-31 10.1515/9781501502620-005 http://www.cs.swan.ac.uk/~csulrich/ftp/logic_for_gray_code.pdf COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2017-09-12T16:10:46.7724830 2016-06-21T16:08:15.4018719 Ulrich Berger 0000-0002-7677-3582 1 Kenji Miyamoto 2 Helmut Schwichtenberg 3 Hideki Tsuiki 4
title	Logic for Gray-code computation
spellingShingle	Logic for Gray-code computation Ulrich Berger
title_short	Logic for Gray-code computation
title_full	Logic for Gray-code computation
title_fullStr	Logic for Gray-code computation
title_full_unstemmed	Logic for Gray-code computation
title_sort	Logic for Gray-code computation
author_id_str_mv	61199ae25042a5e629c5398c4a40a4f5
author_id_fullname_str_mv	61199ae25042a5e629c5398c4a40a4f5_***_Ulrich Berger
author	Ulrich Berger
author2	Ulrich Berger Kenji Miyamoto Helmut Schwichtenberg Hideki Tsuiki
format	Book chapter
container_start_page	69
publishDate	2016
institution	Swansea University
isbn	9781501502620
doi_str_mv	10.1515/9781501502620-005
publisher	de Gruyter
url	http://www.cs.swan.ac.uk/~csulrich/ftp/logic_for_gray_code.pdf
document_store_str	0
active_str	0
description	Gray-code is a well-known binary number system where neighboring values differ in one digit only. Tsuiki (2002) has introduced Gray code to the €field of real number computation. He assigns to each number a unique infinite sequence containing at most one undefined element. In this paper we take a logical and constructive approach to study real number computation based on Gray-code. Instead of Tsuiki’s indeterministic multihead Type-2 machine, we use pre-Gray code, which is a representation of Gray-code as a sequence of constructors, to avoid the difculty due to the undefined element which prevents sequential access to a stream.We extract real number algorithms from proofs in an appropriate formal theory involving inductive and coinductive defi€nitions. Examples are algorithms transforming pre-Gray code into signed digit code of real numbers, and conversely, the average for pre-Gray code and a translation of fi€nite segments of pre-Gray code into its normal form. These examples are formalized in the proof assistant Minlog.
published_date	2016-07-31T03:35:20Z
_version_	1763751515619590144
score	11.01628

Logic for Gray-code computation

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