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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
first_indexed 2016-11-08T05:18:23Z
last_indexed 2018-02-09T05:13:39Z
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spelling 2017-09-12T16:10:46.7724830 v2 28978 2016-06-21 Logic for Gray-code computation 61199ae25042a5e629c5398c4a40a4f5 Ulrich Berger Ulrich Berger true false 2016-06-21 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 COLLEGE CODE Swansea University 2017-09-12T16:10:46.7724830 2016-06-21T16:08:15.4018719 Ulrich Berger 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 2016-07-31T03:54:36Z
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score 11.038833

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