Top Banner
Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this version: Marc Daumas, David W. Matula. Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set. Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology, Springer, 2003, 33 (1-2), pp.7-18. <10.1023/A:1021133616373>. <inria- 00072767> HAL Id: inria-00072767 https://hal.inria.fr/inria-00072767 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.
24

Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

Feb 14, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

Further Reducing the Redundancy of a Notation over a

Minimally Redundant Digit Set

Marc Daumas, David W. Matula

To cite this version:

Marc Daumas, David W. Matula. Further Reducing the Redundancy of a Notation over aMinimally Redundant Digit Set. Journal of VLSI Signal Processing Systems for Signal, Image,and Video Technology, Springer, 2003, 33 (1-2), pp.7-18. <10.1023/A:1021133616373>. <inria-00072767>

HAL Id: inria-00072767

https://hal.inria.fr/inria-00072767

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--38

86--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Further reducing the redundancy of a notationover a minimally redundant digit set

Marc Daumas, CNRS

David W. Matula, SMU

No 3886

Fevrier 2000

THEME 2

Page 3: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this
Page 4: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

��������������� ��������������������� ������������������������� ���!�����"��$#���%�'&������(&���)�)��*�+� ���������"�"�,�������-�./�0�

132547698:2<;+=>2<?A@CBED�F�G8:2IH/JLKNMPO�132<QR;�ST2�@CG�13U

VWYX[Z]\�^"_*`ba[cYde\ fhgjikdhl(de\[f\(mnl(opfql�rsfutwvIZ"x5gjfhdqykrs\zC{|gp}~\(m$�${|a(csojde{|\

�$oj�Y�5gj{�m �Y\ {�\7l�WY\({�l�WY\�c"�k�j�j�k��_*��a(�I{|de\[{n^p�j�k�!_�^j�]�sopik\[t�����[�(���Y�j�j� �$\[��rYc<�YopcAm cYgpm�o�m|degkcst!op{|\�rst�\[��dhZ]�sfedql�dem�fhv9gk{ \����Yfhdhl(d�m|fev�dhc�ZEojcIv9��dhijdem|opfC�Y\[t�deikcst(�VWs\(v:WsoR�j\"x<\[\(c9twm�rs�Yde\7��dhc���\(m|opdhfqt�ojcs��o�ik\(cY\[{|ojfu ¡{�opZE\(¢0gj{|£�dqtb£Acsg�¢ c:m�g�{|\[�Yrsl�\"m�WY\�{�\7��rYc�¤�sopcsl(v�gp ¥o�csgpm|opm�dhgjcE��g�¢ c�m�g!m�Ws\0ZEdhcYdhZEojfefhv"{|\[�YrYcs�YojckmC��dhijdem�t�\�m[��¦�\ �Y{�\7tw\[cAm+Ws\({|\ojc�gk�<\[{|opm�gk{m|g! ¡rY{�m�WY\[{�{|\[��rsl(\m�WY\n{�\7��rYcs�Yojcsl�v�gp ut�rsl�W�o!{�\[�Y{�\7tw\[cAm|o�m|degkcu�/§Tm0��gI\[tCcsgpmC{|\[��rsl(\ m�Ws\$cIrYZ"x5\({Cgj ojfefhg�¢0\[�E��dhijdem|t0twdhcsl�\�{�\[ZEg��AdhcYi�gkcY\n��dhijdem0m�g�o�Z]dhcYdhZ�opfhfev�{|\[��rscs�YopcAm��Ydeikd�mtw\(m0dqt�o"l�gjcI�k\({�twdhgjc]m�go�cYgjc¨{|\[�YrYcs�Yojckm��dhijdem0t�\�m0ojcs�Em|WYdht�dqt�ojc¨\����5\(cst�de�k\ngk�<\[{|opm�dhgjcu�C©�rY{�gk�<\[{|opm�gk{CdecAm|{�g���rsl(\[t0twgkZE\l(gj{|{�\[fhopm�dhgjcEx5\�m~¢0\(\(cEm|WY\!��deikd�m�tªm|g"{|\[��rsl(\$m|WY\ncIrYZ�x<\[{�gp u�5gkt|t�dexYfh\n{|\[�YrYcs�Yojckm�cYgpm�o�m�dhgjc<tª ¡gj{opcIv{|\(�s{�\7tw\[ckm|\[��cIrYZ"x5\({7��VWYdqt${|\[�Yrsl-m|degkc�dqt �IdqtwdhxYfh\ dec�t�Z�opfhfurst�\� ¡rYf�gj�5\({�o�m�gk{|t0fhde£k\!m�WY\"\(fhdeZEdhcso�m|degkcgj ufe\7oj��dhcYi�«(\({|gkt[��¦�\!opfqtwg��Y{|\[t�\(cAm0o"£j\(v�oj�Y�Yfhdhl[o�m�dhgjc�¢ d�m|W�o]¬0­�©!®¨¯0gIgpm�W�{�\7l�g���\[�¨Z"rYfem�dh�Yfhde\[{[�©�rs{nZ"rsf�m|de�Yfhdh\({!dhtbopxsfe\"m�g�okl(l(\(��mbx<gjm�W9o�{�\7��rYcs�YojcAm�gj{bo¨cYgkc�{|\[��rYc<�YopcAm�dhcY�YrYmb¢ dem�W��j\({|v�fedemw¤m|fe\]ZEg���d�°5l(o�m|degkcstnojcs�9ojfeZEgAt~mbcYg��5\(csojf�m~v:dhc�m|deZE\]gj{�t��sokl�\�l(gjZE�soj{�\7��m�g�twm|opm�\(¤Lgj ±¤²m|WY\�¤³op{�mncYgkc{|\[�YrYcs�Yojckm Z�rYf�m|de�sfedh\({�t(�´¶µA·¥¸|¹"º ��»��j� �$\[��rYc<�YopcAm cYgpm�o�m|degkcu¼�½/opcsijrsojij\j¼�­¶rsf�m|de�Yfhdql(o�m|degkcu¼<¬�gjZE�YrYm�\({$oj{�dem�WsZ]\(m�dqlp¼<¬�gkZ�¤�sr�m�\[{$op{�l�WYd�m|\[l�m�rY{|\j¼<¬0­�©!®¨m�\7l�WYcYgkfegkijvj�

¾q¿ÁÀ�ÂTÃ[ÄÀ Å�ÆhÂTÇLÈ�É

Ê/Ë7ÌÎÍ�ÏÁÐ-ÑÓÒ!Ë�Ô�Í+ÕpÖ~Ö~×"ØRÔ�ÑÓÙ²ÌÚÔ�ÛÎÛ Ü ÍLÝ7ØRØpÐ-ÑÓÙ²Ö~Þ�Õ�ÜbÙ²ËRÖßsàÓáuâ ã(ä�åæeѲÐ-ç�Ù²ËRÖ�è�ѲÖ~×RéTË�ê�Ô|Ù²ÌÎÐ-×�Ô�Û<á¥Ö~×(Ù²Ö~Ñ+æhÐ-ÑCâ(éwÌÎÖ~×�ÙLÌ ëRéì/Ö~ÍLÖwÔ�ÑÓéTË]íqáÁê�ì�âRî³ïð Ý7Ù²ËRÐ-ÑÓÍ/ç$Ô~Ü ÕpÖ�ÑÓÖwÔ�éTËRÖ~Þ�ñ[ÌÚÔ�Ö³ò±ç$Ô�ÌÎÛpÔ|ÙÁó Ô�Ѳé-ï ô+Ô�ÝRç$Ô�ÍTõ+ö5ê�â(ò±÷AÜ�Ð-×Aï è�Ñ¥Ô�×RÞbó Ô|Ù²ÝRÛÚÔ�õªâ7ö ð âjï â[ó ø�ï ö<Þ7ÝAïsô+Ôwñ[ÌÎÞ

ó Ô|ÙLÝRÛÚÔ7ù ÍsØpÖ~Ѳç$Ô�×RÖ~×�ÙuÔ�ÞRÞRÑÓÖ~ÍLÍ¥ÌÎÍ<Ô|ÙsÙLËRÖ�â[éTË7Ð[Ð-ÛpÐ�æAö<×Rú-ÌÎ×RÖ~Ö~ÑÓÌÎ×Rú�Ô�×RÞ ð Ø7ØRÛÎÌÎÖ~Þ$â[é~ÌÎÖ~×Ré~ÖwÍ5Ð�ækÙ²ËRÖ�â[Ð-Ý7Ù²ËRÖ~Ѳ×nó!Ö³ÙLË7Ð[ÞRÌÎÍÓÙø�×RÌ ñ�Ö~ÑÓÍLÌ Ù±ÜnÌÎ×!ô+Ô�ÛÎÛÚÔ�Íwû7Ê�Ö³ü7Ô�Í�ïÊ/Ë7ÌÎÍ�Ù²Ö~ü(Ù ÌÎÍbÔ�ÛÎÍLÐ�Ôwñ�Ô�ÌÎÛÚÔ�ÕRÛÎÖ]Ô�Í!Ô¨ÑÓÖ~ÍLÖwÔ�ÑÓéTË�ÑÓÖ~ØpÐ-ÑÓÙ Ð�æ�ÙLË7Ö�÷�Ô�ÕpÐ-ѲÔ|ÙLÐ-ÌÎÑÓÖEÞRÖ�Ûqù à±×7æeÐ-Ѳç$Ô|Ù²ÌÎý[Ý7Ö"Þ7Ý�ßsÔ�ѲÔ�ÛÎÛÎþ~ÛÎÌÎÍLç Ö

ÿ���������������� �������������������������� ï

Unite de recherche INRIA Rhone-Alpes655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN (France)

Telephone : 04 76 61 52 00 - International: +33 4 76 61 52 00Telecopie : 04 76 61 52 52 - International: +33 4 76 61 52 52

Page 5: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

��� �����-�+� � ���0� �+� )������ ���������"���0� ���w����� ���!��� �������.���� ��� � ��.��$&��:)�� ��� �+��������� .,��� ���������"�"�&��(���(&��")

� � ��� � ½u\7t cYgjm|o�m|degkcst {|\[��gkcs�Yojckm|\[t t�gjcAm$r�m|defhdht�a(\7t$��\  Óo���gkc�dhZE�Yfhdhl(d�m|\�gjr:\����Yfhdql�dem�\"�Yojcst$��\csgjZ"xs{�\[r���l�dh{�l�rYdem|t cIrYZEa[{�dqyArY\[t[���Cfhfe\7t gjcAm�a(m�a�a�m|rs��dha(\7t�\(c ��a(m|ojdefC\(m"��\[t�Z]a(m�WYg���\7t�ika(cYa[{|ojfe\7tt�gjcAmªl�gkcYcIrY\[t+�5gjrY{+{|a[��rsde{|\fho�{�\7��gjc<�Yopcsl(\/}~r<t�yAr�� �!o�mwm|\(dhcs��{|\rYcE\(cst�\(Z�xYfe\��\$l�WYd��¥{�\7t�{|\[�Ygjcs�YojcAmZEdhcYdhZEojf²���$gkrst �Y{|a[t�\(cAm�gkcst!dql�d�rsc�gk�<a[{|opm�\[rY{ �<gkrY{�{|a[��rsde{|\�\(csl(gj{|\Efho:{|\[��gkcs�Yojcsl�\���� rYcY\�m|\(fhfe\csgpm|opm�dhgjcu��§³fcY\�{|a[�YrYd�mE�sokt"fh\�cYgkZ"xY{|\��Y\�l�WYd��5{|\[t]oprYm�gj{|dqtwa7t"l(oj{]t�rY�Y�s{�dhZE\({]rYc l�WYd��5{|\���rYcY\csgpm|opm�dhgjc9t�rY{�rYc9\[cstw\[Z"xYfh\��Y\�l�WYd��5{|\[tb��aw}���{|\[��gkcs�Yojckm�Z]dhcYdhZ"rYZ \7t~mbrYcY\El�gkcA�k\({�twdhgjc:�k\({�t$rYcY\csgpm|opm�dhgjc�cYgjc�{�\7��gjcs�sopcAm�\0\�m+rYcY\gj�5a({�o�m|degkc"fhgjcYikrY\j���$gjm�{|\�gk�<a[{|opm�\(rs{�dhcAm�{|gI�YrYd�mC��\7t+l�gj{|{|\(fqo�m�dhgjc<t\[cAm�{|\�fe\7t�l�WYd��5{|\[t!�5gjrY{"{�a7��rYdh{�\�fh\�cYgjZ�xY{|\¨��\[t csgpm|opm�dhgjcst!{|\[�Ygjcs�YojcAm�\[t!�5gkt|t�dexYfh\[t��<gkrY{"l�Wsokykrs\csgjZ"xs{�\�{�\[�Y{�a7tw\[cAm�aj� ¬�\(mwm|\�{|a[��r<l-m�dhgjc>o9rsc \��5\(mE�IdqtwdhxYfh\��5gjrs{E�Y\��5\�m|d�m�tEgj�5a({�o�m�\[rY{�t�m|\(fqt]yArY\f�� a(fhdeZEdhcso�m|degkc���\b«(a[{�gAtCcYgkc�t�deikcYd�°5l(o�m|d� Ót0\[c¨m! (m�\!��� rYc�ZEgjm[�"�$gkrst0�Y{�a7tw\[cAm�gjc<t�ojrst�t�d5rYcY\!op�Y�Yfhdql(o�¤m|degkc�dhZE�<gk{wm�opcAm�\!oR�j\7lnrsc�Z�rYf�m|de�sfedh\(rY{�¬0­�©!®¨xsoktwa t�rY{fe\ l(gI�sopij\!��\ ¯0gIgpm�W/�#�ngpm|{�\!Z"rsf�m|de�Yfhdh\(rY{\7t~mbl(op�<opxYfh\���\!m�{�opdem�\[{$rYcY\"\(cAm�{|a(\ {|\[��gkcs�Yojckm|\ gjr¶cYgkc:{�\7��gjcs�sopcAm�\�oR�k\[l�m�{|X[t �5\(r���\�ZEg���d�°5l(o�¤m|degkcst�\�mªtw\[cst�dexYfh\(ZE\[ckmªopr<l�rYcY\0�5a(csojfedem�a\(c��Id�m|\[t|tw\�\�mª\(c"m|opdhfhfe\l�gkZ]�<op{|a$��rYc"Z�rYf�m|de�sfedh\(rY{Cl[op�sojxYfh\rscYdhyArY\[ZE\(cAmn��\bm�{�opdem�\[{$��\[tcYgkZ"xY{|\[tcsgjc�{�\7��gjcs�sopcAm|t[�% º �7� ¸ �'& � �$gpm�o�m|degkc�{|\[�Ygjcs�YojcAm�\j¼�½�opcYiAopij\k¼�­¶rYfem�dh�Yfedql(opm�dhgjcu¼5�${|d�m|WYZEa�m|dhyArY\���\7tgj{���dhcso�m|\(rY{�t(¼�n{|l�Wsd�m|\[l-m|rY{|\!�Y\[tgj{���dhcso�m|\(rY{�t[¼IV�\[l�WscYgjfhgjikde\�¬0­�©!®¥�

Page 6: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� �

7 8 ���/�+��������������� �����:9���&�&������

�ncIv��5gkt�d�m|de�k\�dhcAm�\(ik\({�W<ojt"o9rYcYdqyArY\¨{�oj��de�<; {�\[�Y{�\7tw\[cAm|o�m|degkc>=�;@?BA ojcs�C;@DFE�G�¢ d�m|W ��dhijdem|tdhc m�WY\¶t�\�mIHKJ�L�M�MNM�LO;QPSR�Tk� ¦ WY\[c>m�WY\¶t�\�m�l�gjcAm�opdhcst]Z]gk{�\�m�W<opcU; \[fe\[Z]\[cAm|t[¼Cm�Ws\�csgpm|opm�dhgjc dht{|\[�YrYcs�Yojckmnojcs�¶t�gjZE\�dhckm|\(ik\({�t WsoR�j\�t�\(�k\({�opfÁ{|\(�Y{|\[t�\(cAm�o�m�dhgjc<t(�0§T ªm�WY\"{�\7��rYcs�YojcAmn{�\[�Y{|\[t�\(cAm|opm�dhgjcgj Co�cIrYZ"x5\({�dht�l�gjZE�YrYm�\[�:xIv:opc�ojr�m�gkZEopm�dql���\[�Adql�\�twrsl�W�okt$o¨l(gjZE�Yr�m|\({7¼s¢0\�ZEdeikWkmbx<\�opxYfh\ m�g{|\�m|{�dh\(�k\bt�gjZE\ l�gk{�{|\(fqo�m|degkcst�x<\(m~¢�\[\(c�m�WY\��Ydeikd�m�t(�©�cY\$t�rsl�W]\��YopZE�Yfh\ dqt+m�WY\ {�oj�Yd���V ¯0gIgpm|W]{|\[l(g���decsibgkc"m|WY\$��dhijdem�tw\(mWHXPYEL�M�MNM�L E�Toktª�Y{|\[t�\(cAm�\7�

dhc[Z]\k¼j^_^Y¢ WY\[{�\ o!��dhijdemC^bl[opc]gjcYfhv�x5\ ¡gjfhfeg�¢0\[�"xAv�o�cs\(ikopm�dh�j\��dhijdem+�5gkt|t�dexYfhv��Y{|\[l(\[��\7�"xIv"o!twm�{|decsigj 0«[\({|gkt[�¨�Ygk{�\��YopZE�Yfh\j¼a`bE�JRKc%d'efE-M�gXhji>JkM�g�l3imRnM�gXoperq�q�dqt cYgjm"o:�Rojfedq� ¯0gAgjm�W {|\[l�g���\7�{|\(�s{�\7tw\[ckm�o�m|degkcu��VWs\b��ojfedq��gjcY\b ¡gj{$�k�]dhts`OR E�J�RKc d e@RWM�gut6iv`OPYE�cwM�g h iQJxMNg l ivRWM�g o �VWY\ l�gkcI�j\({�t�degkc  ¡{|gjZ m|WY\�r<twrsojf�{�oj�Yd��N^ {|\(�Y{|\[t�\(cAm�o�m�dhgjc�m|g m|WY\�{�oj�Yd��>V ¯0gIgpm|WN{|\[l�g���\7�

{|\(�s{�\7tw\[ckm�o�m|degkc�dqt/gkx�m|ojdecs\[�b ¡{|gjZ m�WY\Cxsd�m�m�g��Ydeikd�m�l(gjcI�j\[{|t�dhgjcb�Y{�\7tw\[cAm�\[�!dhc"V�opxYfh\��s¼[�sojij\j\7�Y�/VWsdhtt��5\[l�dqopfÁ{�oj�Yd��[V]csgpm|opm�dhgjc:gjc�m|WY\ {�\7��rYcs�sopcAmn��dhijdemntw\(msHXPYE�LNM�MNM�L E�T dqt cYgjm${�\7��rYcs�YojcAm[�0¬�gkckm|{|oj{�vm|g�ij\[cY\({�opf�x<\[fedh\� Ót[¼ªl�gjcI�k\({�m�dhcYi¶o�¯0gAgjm�W {|\[l(gI�Y\[��cIrYZ"x5\({"m�g�dem|t�l�gjcI�k\(cAm�dhgjcsojfªcYgjc {|\[�YrYcs�Yojckm{|\(�s{�\7tw\[ckm�o�m|degkc"¢0gjrYfq�"csgpmªdecI�kgjfh�j\opcIv"l(oj{�{|v {|de�Y�sfe\$ojcs��l(gjrYfq�]x<\ �5\({� ¡gj{|ZE\[��decE�soj{|ojfefh\(fy=²twWYgkrYfq�ojcIv�rst�\({bx5\"dhckm|\({|\[twm�\7�¶dec���gjdhcYi¨dem�G��!VWsdht��Y{|gj�5\({�m~v�dqtncsgpm��IdqtwdhxYfh\"xIv�fegIgj£IdhcYi�opmnm�Ws\]��dhijdembt�\�mojfegkcY\j�§³c ®I\[l�m�dhgjc�^Y¼5¢0\"dq��\[ckm|d� ¡v�o�yArsopcAm�dem~v�m�Wsopm!�Y\[t|l�{|dex5\[t��sop{�m�gj ªm�WY\El�gk{�{|\(fqo�m|degkc¶gp ªm|WY\]�Ydeikd�m�t(�

VWsdht yArsojckm|d�m~v¶¢ojt�°s{�twm!�s{�\7tw\[ckm|\[�9deczZ �_^�opc<��dem�Wsojtbx5\(\(c rst�\[��dhc{Z Vs¼}|I¼���¼�~I¼u��^L�]§Tm dht l[opfhfe\7�m|WY\� ¡{�ojl�m�dhgjc {|ojcYij\¨okt�t�g�l�dqo�m|\[��m�g9o¶t�\�m�gp  ��opfhdh� {|\(�Y{|\[t�\(cAm�o�m�dhgjc<t'=¡fqopcYikrsopik\KG-��§Tm�dqt m|WY\�t�\�m"gj ojfef$m�WY\¶�5gkt|twdhxYfh\: ¡{|okl-m|degkcst�gp !rYcYdem|t¨dec�m|WY\�fqojtwm¨�<gAtwdem�dhgjc�fhgktwm¨¢ WY\(c�gjcs\:m�{|rYcsl(opm�\7t�opcIv ��opfhdq�{|\(�s{�\7tw\[ckm�o�m|degkc� ¡gj{�opcIv9�5gkt|twdhxYfh\��<gAtwdem�dhgjc dhc�m�WY\¨{|\(�s{�\7tw\[ckm�o�m|degkcu�¨VWYdqt�ZE\[oktwrY{|\E�Y{|g��j\[t�m|g�x<\oj�Y�Y{|gj�Y{|dqo�m�\Edhc {�\(m|ojdecYdhcYi¶ZEgktwm"gj �m|WY\�dhc� ¡gj{|Z�o�m|degkc�¢ Ws\(c l(oj{�{|v�{|\[l(gI�YdecYiAt op{|\¨op�Y�sfedh\[�Á��®Ir<l�Wgk�<\[{|opm�dhgjcst!oj{�\Etw�5\[l(d�°5l���deikd�m�t�\�m"l(gjcI�j\[{|t�degkcst&Z ��^�gj{!{�\[¢ {�dem�dhcYi�{�rsfe\7t0Z]\7��¼�\�\�^Cm�Wsopm�l�gjZE�YrYm�\�dhc�<op{�opfhfe\[f<o'�-R�l(oj{�{|vEopcs�¨ojc�dhc�¤T�Yfhokl�\�{�\7twdq��rsojf5��dhijdem0 ¡gj{0\[ojl�W��<gAtwdem�dhgjcu����deikd�m t�\�m l�gjcI�k\({�twdhgjc�dqt�oik\(cY\[{|ojfedh«[opm�dhgjc¨gj /m|WY\�oj�Y�Yd�m|degkc�gj �cIrYZ"x5\({�tdhc�{|\[��rscs�YopcAm$opc<�/�Rgk{0csgjc�{�\7��rYcs�YojcAmcsgpm|opm�dhgjcst0gkc�<op{�opfhfe\[f�ojcs�9t�\({|dhojfªtwv�twm�\(Z�tkZ�\7^I¼�\7�_^L��VWY\] ¡{|okl-m�dhgjc�{|ojcYij\"m�{�ojl(\[tnm�WY\�l�gk{�{|\(fqo�m|degkc�gp �m�WY\���dhijdem|tdhc o l�gkZ]�sfe\(�>op{|dem�WYZE\�m|dhl9opfhijgj{|dem�WYZ m|Wso�m¨r<tw\7t�gjcYfhv {|\[l(g���decsikt�ojt�m�WYdqt¨dqtEm�Ws\9l(oktw\� ¡gj{�Z]gAt~mdhZE�Yfh\(ZE\(cAm�\7�:op{|d�m|WYZE\�m�dqlbgj�5\({�o�m|gj{�t�gkc�o]{|\[��rscs�YopcAm cYgjm|o�m|degkcu�¦�\!�Y{�\7tw\[cAm0dhc:®�\[l-m|degkc���m�Ws{�\[\boj�Y�Yfhdhl[o�m�dhgjc<t�gj �{|\[��rYc<�Yopcsl(v�{�\7��rsl�m�dhgjc�gj /{�oj��de��m~¢�g]x5gj{|{�g�¢

t|oR�j\�ojcs� l(oj{�{|v9t|oR�j\�cYgjm|o�m|degkcst[¼�gkcY\�gp m�Ws\(Z x<\[decsi9o�¯0gIgpm�W Z�rYf�m|de�sfedh\({�¢ d�m|W gjcY\�{|\[�YrYcs�Yojckmgk�<\[{|ojcs�Á��zC{|\(�Idegkrst�{�\7tw\7op{�l�W]W<ojt0twWYg�¢ c¨m�WY\� ¡\[oktwdhxYdhfedem~vEgp /Z�rYf�m|de�sfedh\({��\[t�dhijcst�\(ZE�Yfhg�vIdecYi"{�\7��rYc�¤�sopcAm�xYdhcsoj{�v]gj�5\({�opcs�st(�+V/g]oR�jgkdh��m�Ws\�ik\(cY\[{|ojfsdhcsl�{|\[oktw\ dhc�Wsoj{|�Y¢0oj{�\$t�de«[\n\[ckm�opdhfe\7��xIv�{|\[�YrYcs�Yojckmxsdecsoj{�v�dhcY�Yr�m7¼5{|\[l(\(cAm�opmwm�\[cAm�dhgjc�Wsoktnx5\(\[c¶ ¡g�l�rst�\[��gjc¶fhdeZEdem�dhcYi�{|\[�YrYcs�Yojckm�decY�sr�m!t�dhZ]�sfev�m�g�m�WY\Z�rYfem�dh�Yfedh\({�{�\7l�g���\({�decY�sr�mYZ�\�VY¼/\K|I¼/\[��¼\�~A¼�\7�_^T��¦ dem�W�ZEdhcYdhZ"rYZ l�dh{|l(rYdem�{|vj¼j¢0\$op{|\ {|\[oj�Yv m�g"�Y\({|de�k\m|WY\0feg�¢¤T�<g�¢0\({/WYgpm�¤LgkcY\n=ÓgjcYfhv gjcY\0twdhijc<opfAdht+t�\�m�opm+ojcIv�m|deZE\_G/t�deikcsopfIl�gkcAm�{|gjfqt��0¤³^Y¼[¤�\j¼p�s¼�\j¼j^x�xZ�\N�_^gk{um|WY\0l(gjZEZEgjc�Z�rYfem�dh�Yfedh\({+l�gkcAm�{|gjfqt���cs\(ikopm�dh�j\X�p��gjrYxsfe\7�! Óojl�m�gj{K��rYcsl�W<opcYik\[�! Óokl-m�gk{���Z ^j��¼�^�\p¼p^j^_^L�¯0r�m�¢�\¨�s{�\( ¡\({ m|g9l�gkZ]�sr�m�\�m�WY\¨\[cYWsojcsl�\7� twdhijc t�\(fh\[l�m"l�gjcAm|{�gkfht0��cY\(iAo�m�dh�j\X�+�<gAtwdem�dh�j\X�C��gjrYxsfe\7� Óokl-m|gj{K�<rYc<l�WsopcYik\[�� Óokl-m�gk{&�#ZÎ^p��^L�b�Ygj{��Y{�ojl-m|dhl[opfu{|\[oktwgkcst[¼Ym�Wsdhtbtw\[\(Z�t$ZEgk{�\"�Y\[t�de{�opxYfh\ m|Wsopc9l(gjc�¤�k\({�m�dhcYi�m|WY\�cIrYZ�x<\[{b ¡{�gkZ%�Ydeikd�m tw\(m�H_JL�MNM�M�L q�TI=¡cYgkc�{�\7��rYcs�sopcAmbdhcY�YrYm�G-¼�H_JLNM�M�M�L���T[=Ól(oj{�{|v¶t�oR�k\dhcY�YrYm�G0gj{nHXPjq�L�MNM�M�L q�Tk=Óx<gk{�{|g�¢ t|oR�j\�decY�sr�m�G�m|g]��dhijdem$tw\(mxHXPYE�LNM�MNM�L E�T!x5\� ¡gj{|\!l�gjcI�k\({�m�dhcYi"\7ojl�W�gj m|WY\��Ydeikd�m�tngkx�m|ojdecY\7��m�g�l�gjcAm|{�gkf/twdhijc<opfqt(� zªojtwm�{|\[l(g���\({�t WsoR�k\"oj�Y�Y\[�¶l�{|dem�dql(opf��so�m|W9��\[fhoRv� ¡gj{�m�WY\

ì�ì ×��N�������

Page 7: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

V � �������k����.'�_5&�������k��,K�*��� ���)��1]�

ZEgk{�\� ¡{|\[yArY\[cAm]l(oktw\�¢ WY\({|\¨cYgkc {�\7��rYcs�sopcAm"xYdhcsop{|v decs�Yr�m�dqt]oR�Rojdefqopxsfe\k��©�rY{"�s{�gk�<gAtw\7� l�dh{|l(rYd�m�YgA\7t0cYgpmfe\[cYipm|WY\(c�m�WY\�m�dhZE\!gp /gkcY\�Z�rYfem�dh�Yfedql(opm�dhgjcu¼Yl(gjZE�soj{�\7�]m�g�m�WY\ twm|o�m|\�¤Tgp ±¤Lm�WY\(¤Toj{wm�\(csl(gI�YdecYide +x<gjm�W:decs�Yr�m|tnop{|\�cYgkc:{�\7��rYcs�YojcAm[�

� �¨�+����������� ��������� ����� �0� �������+� �����������

��� � �����������������VWs\C{|ok��de��;�cYgjm|opm�dhgjc�`����QMNM�M�� o� �"! l MNM�M�$#bc�%!gkc!m�WY\0��dhijdem�t�\�m'&�{|\(�Y{|\[t�\(cAm�t¥m�WY\�{|opm�dhgjcsojfpcIrYZ"x5\({gj �\7ykr<o�m�dhgjc<=%\_G-�CVWY\ t�\�m gp +opfhf¥m�WY\!�5gkt|twdhxYfh\b{|\(�s{�\7tw\[ckm�o�m|degkcstC ¡{|gjZ �5gkt�dem�dhgjcst�(/m�WY{|gjrsijW*) ¢ dem�W�Ydeikd�m�tndhc+&�dqt m|WY\�fqopcsijrsojij\ cYgjm�\[�+& �# =,(.-BJ/- )IG��bVWs\"��opfhrso�m|degkc1020�M3020 %�Z�op�st�& �# decAm�g�m�WY\t�\�m54 gp �m�WY\ {�o�m|degkcsopf¥cIrYZ"x5\({�t[�

020 ��� MNM�M� o6 �"! l MNM�M7�$#7080 %#e�9 :; # �

:;:

¢ d�m|W (<-1=>-1) ? � : ?@& =%\KG

§³c�m�Ws\! ¡gjfhfeg�¢ dhcYis¼�¢0\ op{|\!gjcYfhv¨dhckm|\({|\[twm�\7��dhc:l�gjcAm|deikrYgjrst �Ydeikd�m�t tw\(m|t$l(gjcAm|ojdecYdhcYi]«(\[{�g<¼Im�Wsopm$dht&me HA&CBED F�LNM�M�M�L�&CBEG7H�T¨¢ d�m|W1&CBED FI- JJ-K&CBEGH�fe\7oj��dhcYi¶m|g�o¶{|\[��rYc<�YopcAm�cYgjm|o�m|degkc ojt"t�gIgjc ojt&�BEGH�PL&CBED F+D ;���VWY\���dhijdem�t�\�m"dqt�ZEdhcYdhZEojfefhv�{�\7��rYcs�YojcAm"d� �&�BEGH�PL&CBED F er; opcs� o�l[op{|{�v{|dh�Y�Yfh\"�Y{|g�l�\[t|t$¢ dem�W�x<\7t~m�m�dhZE\]l(gjZE�Yfh\���d�m~vNMk`�O2P$Q/`R) PS(bc�c dqtncs\[l�\7t�t|op{|v�m�g�{|\[�Yrsl�\T&>m�g�gjcYfhvI;\[fe\[ZE\(cAm|t[�©�c opc opr�m|gjZ�o�m|dhl]��\[�Adql�\�t�rsl�W�okt o�l(gjZE�Yr�m|\({�l(de{�l�rYdem[¼�o�cIrYZ�x<\[{ dht!�Y{|g���rsl(\[�9okt o��k\[l�m�gj{

gj 0��dhijdem|t�U eWV ��� MNM�M� o ��! l M�MNM7�$#YXL�]§³c�m�WY\�dhcAm�{|gI�Yrsl-m|degkcu¼Á¢�\EWsoR�k\�t�\(\[c9m|Wso�m!m�Ws\({|\]ZEdhijWAm x<\t�gjZE\El�gk{�{|\(fqo�m|degkcst$x5\�m~¢0\(\[c9m|WY\���deikd�m�tbgp �m|WY\E�j\7l-m�gk{boktbfegkcYi�okt�m�WY\[v¶WsoR�j\]cYgpm!x5\(\(c�ijdh�j\[c9xIvm|WY\ rst�\({$xsr�mnl�gkZ]�sr�m�\7�:xAv�opc:opfhijgk{�dem�WYZ[Z�`R\<c�e]U,m|g��<\[{w ¡gk{�Z twgkZE\!oj{�dem�WsZ]\(m�dql!gj�5\({�o�m|degkcst(�¯0\( ¡gj{|\0£IcYg�¢ dhcYibm�Ws\$ojl-m|rsopfs�Rojfers\ gp 5m�WY\$��dhijdem|t�t~m|gj{|\[��dec]m�Ws\$�j\7l-m�gk{^U¶¼j¢0\ l(ojcE�Y\�°scY\$t�gjZE\ {|\(��¤{|\[t�\(cAm�o�m�dhgjc<tCm|Wso�mnoj{�\!ojl[l�\(�Ym|opxsfe\!opcs���Y{|g��j\�m�Wsopm$twgkZ]\!gjm�WY\[{|tl(ojcYcYgjm0g�l(l(rY{7�+VWsdht ��\(°scY\7tm�WY\fqopcsijrsojij\�Z`_Qe HAZ�`�\ c�Tbac& �# ¢ Wsdhl�W�dqtªm|WY\�t�rYxst�\�m�gj uopfhfYm�Ws\n�5gkt|twdhxYfh\$gjr�m|�Yr�m0{�\[�Y{�\7tw\[cAm|o�m|degkcst[�¯0v¶\(�Am|\(cst�dhgjcu¼¥¢0\�rst�\�m�WY\Et�ojZE\"cYgjm|o�m|degkc� ¡gk{�Z�_ opc<�NZ�� ¦�\���\�°<cY\"m|WY\] ¡{|okl-m|degkc�{|ojcYij\�gp >Zojcs�¶m|WY\� ¡{|okl-m�dhgjc�{|ojcYij\�o�m!�<gAtwdem�dhgjced�xAv¶\7yArso�m|degkcst-=L^�G�opcs�{=Ó�uG-��¦�\�¢ dhfhfCtw\[\"WYg�¢�m|g�r<tw\�m�WY\yArsojcAm�dem~v�opcs��WYg�¢�m|WYdht t�dhcYijfh\]t�\�m�l[op��m|rY{�\7t!opfhgjcY\EtwgkZE\��k\({|v¶{�\[fe\[�RojcAmb�soj{wm!gp �m�WY\�l(gj{|{�\[fhopm�dhgjcx5\�m~¢0\(\[c�m|WY\���dhijdem|t[�

f�gihX`�Z�c�e f�gihX`�Z _ c�ekjl080 J �mhi! l M�M�M��n#�020 % LoV �$� M�MNM7�$#pX}?eZ _5q =²^�G

f�g�`,Znc�erf�g�`�Z _ c�e s#pt"hitl� f�g h `,Z _ c =Ó�XG

�nc¨decs�Yr�m��k\[l�m�gj{�ijdh�j\[c]xIv�m�WY\�rst�\({7¼k¢ WY\[{�\ncYgpm|WYdecsi�dht�£IcYg�¢ c¨opx5gjr�m0opcIv�gj ud�m�t���dhijdem|t[¼AvAdh\(fq�Ytm|WY\ fqop{|ij\[twmY�vu����*"����I ¡{�ojl�m�dhgjcE{|ojcYij\ okt�t�g�l�dqo�m�\7��¢ dem�WEm|WYdht�{|ok��de��opcs��m|WYdht���deikd�m0tw\(m[��§Tm�dhtCx<gkrYcs��\7�

à¡ê�ìuà ð

Page 8: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� |

xIv¨\7yArso�m|degkc<=*VuG����f�g�` & �# c6e H 080 J � hi! l M�MNM7�$#020 % L ( - d�- ) LoV �$� M�M�M��n#RX}?e& �# Tba R

; P{R M `,& BED F�� & BEG7H c��nHKJT=�VuG

�sgj{u{�oj�Yd���m~¢�g {|\(�Y{|\[t�\(cAm�o�m�dhgjc<t(¼-m|WY\ª ¡{�ojl�m�dhgjc!{|ojcYij\ªgj Io$x<gk{�{|g�¢�t|oR�j\+�j\7l-m|gj{Á¢ WY\[{�\>&+emHXPsR � J � R�Tojcs��o�l(op{|{|v�t|oR�j\��k\[l-m|gj{�¢ WY\({|\ &ze H_J � R � E�T�op{|\"{|\[t��<\7l-m�dh�j\[fev+`%PsRXL�R_c$opc<�SV J�L EXc-�n§³c<��\(\7�¶m�Wsdht Óokl-m¢0okt�{|\[l(gjikcYde«[\[��dhc�§³cAm|\(f�� t��\7t�l({�dh��m�dhgjc�gp um|WY\bz�\[cAm�dhrYZ xYrYi���m�{|rYcsl(opm�dhcYiEo�cIrYZ"x5\({dhc�l[op{|{�vt|oR�j\� ¡gj{|Z�o�m �Y{|g���rsl�\7t opc�\({|{�gk{!¢ d�m|WYdec ^�rYfh�st�l[opfhfe\7��m|WY\�~{|\(ikdegkc9gj �rYcsl(\({�m|ojdecAm~v�[ZÎ^�VY¼/^X|K^T�¨§T m|WY\ cIrYZ"x5\({ dqt twm�gj{|\[��dhc�x5gj{|{|g�¢ t�oR�k\n ¡gk{�Z�opm0m|WY\ \({|{|gj{0dht ¢ dem�Wsdec �-R!rYfh�u�� �5gkt�dem�dh�j\¶l(oj{�{|v {|\[l�g���dhcYi�m�{�opc<t~ ¡gk{�Z�tEo {|ok��de� ;�cYgpm�o�m|degkc `,� � M�MNM7� # c % gjc>m�WY\���deikd�m�t�\�m&CeSHA&CBED F�L�MNM�M�L &CBEG7HuT0m�gbm|WY\{�\[�Y{�\7tw\[cAm|o�m|degkc#` � � _ l � � MNM�M�� # c % ��§Tmªrst�\[tªobyArsopcAm�dem~v�� ? &�l[opfhfe\7�

m|WY\n�sde�kgpmªm�g]��\�°<cY\ m�Ws\n\[yArsopm�dhgjcstn=b|XGªopcs� =²�XGª ¡gj{>('- =^- ) i+R$¢ WY\[{�\�� # er� � _ l e�� � _ h evJY�

� : _ l e�� R de  � : D��J gpm|WY\({|¢ dqtw\ =b|�G

� : _ l ;pi�� : e � : i�� : =Ó�XG¦�\!�j\[{�de ¡v�xIv�decs�Yrsl-m|degkc�gkc*)%m|Wso�m$o�{�\7l�g���dhcYi]�YgA\7t0cYgpm$l�WsojcYij\�m�WY\!��opfhrY\ twm�gj{|\[��ojt0�Y{|\[t�\(cAm�\7�x5\(fhg�¢ � �9 :

; # �:;:e� _ l9 :; # �

:;:

VWs\�{�\7l�g���\7� ��deikd�m�tw\(m�dht H������}`,&CBED F � � Pv;wc�L�MNM�MKL���� �y`,&�BEGH�Pv;Ci R � � c�Tk��VWY\9��dhijdem�t�\�m�dht{|\[�Yrsl�\7��de �&CBED F�iz;r-!��"k&�BEGH��EVWY\Egjr�m|�Yr�m"��dhijdem�tw\(m!dqtbZEdecYdhZ�opfhfev�{�\7��rYcs�sopcAm!okt!twgIgjc ojt& BED F D#�pPC;>opcs�$�0i{;!% & BEGH ¼5m�W<o�m!dht�m�Ws\]l[ojt�\� ¡gj{!\(��ojZE�Yfe\�d� �m�WY\E��dhijdem!t�\�m!dhtbZ�o���dhZ�opfhfev{|\[�YrYcs�Yojckm opc<��peBJY�+§T /m|WY\ gjrYm��Yr�m$��dhijdem$tw\(m dhtZEdhcYdeZ�ojfefhv¨{|\[��rYc<�YopcAm[¼�dem dht�H&�0P ;�LNM�M�MNL �}Tk�+§T �per& BEG7H e & BED F i ;¶m|WY\�gkr�m��sr�m0��dhijdem tw\(m�dqt�ZEdhcYdeZ�ojfefhvE{�\7��rYcs�YojcAm[¼Adh��\[cAm�dql(opf<m�g"m�WY\bdhcY�Yr�m tw\(m[�� cY\(iAo�m|de�k\�l(op{|{|v�l(gjcI�j\[{|t�degkc�dqtbtwdhZEdefqop{nm�g¨m|WY\]�<gAtwdem�dh�j\"l[op{|{�v�l�gkcA�k\({�twdhgjc:\(��l(\(��m�m�Wsopm'� : _ ldqt ��\(°scY\[�� ¡{|gjZ \7yArso�m|degkc<= ~�G-�

� : _ l e � PsR de  � : -��J gjm�WY\[{�¢ dqt�\ =)~�G

VWs\�{|\[l(gI�Y\[�"�Ydeikd�mªtw\(m�dqtaH�������`,& BED F ik;YP�R � ��c�L�MNM�M�L����(��` & BEG7H)� ��i0;�c�TA��VWY\��dhijdemªt�\�m+dqt�{|\[�Yrsl�\7�de ^& BED F "��@-c& BEGH P ;C��§T �m|WY\ gjr�m|�Yr�mn��dhijdemnt�\�m dht$Z]dhcYdhZ�opfhfev�{�\7��rYcs�YojcAm[¼�dem$dhtxH&�}LNM�M�M�L �-iQ;�Tk�§Tm$dqtrYcsl�W<opcYik\[�Á¼�ZEdhcYdeZ�ojfefhv¨{|\[��rYc<�YopcAm[¼sd� *�pe & BED F er& BEGH P ;C�ª¦�\ ¢ dhfef/t�\(\bm|Wso�m$\[�j\(c�¢ WY\[c¢0\��Yg"cYgjm�{|\[��rsl(\$m|WY\b�Ydeikd�mtw\(m�¢ dq�Im�Wu¼Yt�rsl�W�l(gjcI�j\[{|t�dhgjcstª�5\({� ¡gj{|Z twgkZ]\�fe\[ijdem�dhZ�o�m�\�¢�gk{�£]�Idht�dhxYfe\m|WY{|gjrYikW�dem|t\��5\7l-m$gjc�m|WY\! ¡{|okl-m|degkc�{�opcYik\j�+ ÊuËRÖ+Ø7ѲÐ(ÞRÝRé³Ù�Ð�æsÔ�ÍLÖ³Ù-,�Õ�Ü�Ô�Ͳé�Ô�ÛÚÔ�Ñ/.ÌÎÍÁÙ²ËRÖ+ͲÖ~Ù/Ð�æYÔ�×�ÜnÖ~ÛÎÖwç Ö~×�ÙuÐ�æ�,�ÍLéwÔ�ÛÎÖ~ÞnÕ(Ü0.�û .21�,4365&.2187:9/7<;=,?>(ï

ì�ì ×��N�������

Page 9: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

���,� ���`�������o� ������������ � �3���§T <Z3dhto]fhojcYijr<opij\�gp '& �# ¢ d�m|W� ¡{�ojl�m�dhgjc�{�opcsij\�f�g�`,Znc ojcs��� opcIv��<gAtwdem�dh�j\!l(oj{�{|vE{�\7l�g���dhcYi"¢ dem�Wgkr�m��sr�mn��dhijdemntw\(m�H &CBED F�L�MNM�M�L &CBEGH�T!m�Ws\(c��

f�g�`��'`�Znc�c�a R; `,f�g�`�Z�c�i>H &CBED F�L�MNM�MNL�&CBEG7HnPUR�TKc =�XG

� � ºuº�� ½/\�m�U ?+Z ¢ d�m|WNU e V ���QMNM�M��n#pXT¼5¢�\"�Y\�°scY\"m�WY\]l(oj{�{|v��k\[l-m|gj{�� e V �i� _ l � �QM�MNM�� # _ l X ¡{|gjZ \[yArsopm�dhgjc =)|�G�opcs�Em|WY\n{|\[t�rYfem��k\[l-m|gj{��me��'`�Upc�e V �A� _ l �A�CM�MNM�� #YX5 ¡{�gkZ \7yArso�m|degkc =Ó�XG��ª¦�\gkx�m|ojdec�m�Ws\! ¡gjfhfeg�¢ dhcYi]\[yArso�m|degkc� ¡gk{$opcIv¨{�o�m|degkcsopf¥cIrYZ"x5\({�� ?ef�g hX`��'`,Znc�cxIv¨\��Im�{�ojl�m�dhcYi�o�m�\({|Z ¡{|gjZ m�WY\ {�oj��de���5gjfhvIcYgjZEdqopf¥gp ��  ¡{�gkZ \[yArsopm�dhgjc+=%\KG

�peo020 J � hi! l MNM�M&� #080 %�e � hi! l P � hi! l; ir080 J � hi! l � hi! h M�M�M&� #020 %

¦�\ {|\[l�gkijcYdh«(\�m�W<o�mbV � hi! l � hi! h M�MNM�� #YX/dht m�WY\ {|\[l(gI�Y\[�¨�k\[l�m�gj{ gj .V � hi! h MNM�M7�$#pX/ojcs��m�WY\[{�\( ¡gj{|\080 J � hi! l � hi! h M�M�M&� #020 %�eo080 J JT�mhi! h MNM�M�$#7080 %I? R

; f�g h _ l `,Zncva R; f�g�`�Z�c

¦�\�£IcYg�¢  ¡{�gkZ m|WY\���\(°scYdem�dhgjc¶gj �m|WY\"gkr�m��YrYm���dhijdem�tw\(m$m�W<o�m � hi! l P � hi! l e � hi! l P+; ��h " & BEG7Hm|g�opfhfeg�¢ ��hi! l m|g�x<\ dhcsl�gk{��5gj{�o�m|\[��¢ d�m|WYgjr�mnl[op{|{�vE{|de�s�Yfe\k�ª§Tm  ¡gjfhfeg�¢$t0m|Wso�m

f�g�`��'`�Z�cOc�a �"!$#�% ! ls& ; �"!$' (R; `OH*)�T�i1f�g�`�Z�cOc6e R

; `�f�g�`,Znc�i>H &CBED F�LNM�M�MNL�&CBEG7HnPUR�T_c +

¦�\ �Y{|g��j\bdec¶o]�j\[{�v�t�dhZ]dhfqop{¢0oRv¨m�Wsopmde �, dqt oEcY\[iko�m|de�k\bl[op{|{�v�{|\[l(g���decsi"m|WY\(c

f�g�`�,Q`�Z�cOc5a R; ` f�g�`,ZncwizH & BED F ivR�LNM�M�M�L�& BEG7H T�c =*�XG

¬�gkcstwdq��\[{�dhcYi:\[yArsopm�dhgjcst�=²�XG ojcs�z=*�uG!d�m�dht csg�¢ fh\(ikd�m|deZ�o�m|\Em�g��5\({� ¡gj{|Z o¶�Ydeikd�m"t�\�m"{|\[l(g���decsi ¡{|gjZ gjcY\���deikd�m0tw\(mCm�g m|WY\nt|opZE\n��dhijdem�t�\�m�t�dhcsl�\ dem�{|\[��r<l�\[tªm�WY\  ¡{�ojl�m�dhgjc�{�opcYik\gp ¥m�WY\$°scsopf<�j\7l-m�gk{[�¦�\�l[opc {|\[l�g���\¨o:x5gj{|{|g�¢ t|oR�j\��j\7l-m|gj{!m�g9{�\7��rsl�\�d�m�t  ¡{�ojl-m|degkc {|ojcYij\E ¡{|gjZ `�- L/.�c!m�g�0 ! l _21h L43h65¢ dem�W�o �<gAtwdem�dh�j\ {|\[l(g���decsi"opcs�]m|g70 1 h L�l _ 3h85 ¢ dem�W¨o�cY\(iAo�m�dh�j\ {|\[l(g���decsis�+�Ygj{0o�l(oj{�{|v"t|oR�j\ cIrYZ�x<\[{[¼¢0\�l(ojc�{�\7��rsl�\¶dem|t� ¡{|okl-m�dhgjc�{�opcYik\� ¡{�gkZ V -/L9.�c�m�g;: 1 h L l _ 3h85 ¢ dem�WNo �5gkt�d�m|de�k\�{|\[l(g���decsi opcs� m�g: l _21h LNR6i 3h<5 ¢ dem�W¶o]cY\(iAo�m�dh�j\!{|\[l(gI�YdecYi<�¦�\ t~m�op{�m0¢ dem�W:��dhijdem|t0dec�m�WY\ t�\�m�H��kP ;�L�MNM�MNL �kPCR�i+guT ¢ d�m|W R�- g -U;� ¡gk{ o"ik\(cY\[{|ojfedh«[opm�dhgjcgj /m|WYdht$fhokt~m gjx<tw\[{���o�m|degkcu�ª½u\(m=� x5\�o��5gkt�dem�dh�j\!{�\7l�g���dhcYikt0¢ dem�W¶�Ydh�jgjm2�/¼Im�Ws\(c:xIv¨dec<��rsl-m|degkc

f�g�`�� & c.a R; PUR M2> ��P<;�L �'PUR6i g

; &@? � HKJT =O\[�XG

A ÊuËRÖ�ÍLÝRç�Ð�æ�Ù±ÏÁÐ]ͲÖ~Ù²ÍCB�Ô�×RÞ ,NÌÎÍ0ÙLË7Ö!ÍLÖ³Ù Ð�æ�ÙLËRÖ�ÍLÝRç Ð�æ+Ô�×(ÜEٱϥÐ]Ö~ÛÎÖ~ç Öw×�Ù²Í0æhÑÓÐ-çDB�Ô�×RÞ ,0û BFE4, 35 .GE�7 9�H�."I 7KJ*;�BFL=, >(ï

à¡ê�ìuà ð

Page 10: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� ~

¬�gkZ]�<op{|decYiEm|WY\� ¡{�ojl�m�dhgjc¶{�opcYik\�gp Ct�rsl�W�o¨�k\[l�m�gj{n{�\7l�g���\[� )�m|deZE\7t$m|g¨m�Ws\� ¡{�ojl-m|degkc¶{|ojcYij\ gj oEcsgjc:{�\7��rYcs�sopcAm �j\[l�m�gk{ l%�! l MX` �#P ; � ��PURKc-¼<¢�\�t�\(\bm|Wso�m$m|WY\���d��5\[{�\[csl�\�gnMN; ! & dqt$��\7l�{|\[ojt�dhcYiik\(gkZ]\(m�{|dhl[opfhfev:¢ dem�W \[okl�W9cY\[¢ {|\[l(gI�YdecYi<��¦�\�l(ojc�\[�j\(c op�s�Yfev�opcYgjm�WY\[{b{|\[l(g���decsi�m|g:l(\(cAm�\[{!m�WY\ok�Y��\7�"yArsopcAm|d�m~vk��½u\(mG,x5\�m|WY\cY\(iAo�m�dh�j\�{�\7l�g���dhcYin¢ dem�W]�Ydh�jgjm �aP&;C¼Rm|WY\� ¡{�ojl�m�dhgjc�{|ojcYij\�, � & `R\ cdqtx5gjrYcs�Y\[��xIv¨\[yArsopm�dhgjc+=%\�\_G-�CVWY\ l(\(cAm�\[{�dhcYiEdht$x<\7t~m$de w; e E]twdhcsl�\ %�! l% e l% e lh �

f�g�`�, � & c�a R; PUR M > �'P<;pi ;IPUR

; � �'P{R6i ; P{R; i g

; & _ l ? � HKJT =O\�\KG

� � ºuº�� VWs\��Y{|gIgp /gj /\[yArsopm�dhgjc<=%\[�uGCdqt0\[oktwdhfev¨�YgjcY\bxIv�decs�Yrsl-m|degkc�gkc )¥�ª¦�\��s{�g��k\ndem0 ¡gk{C)0emRj��s{�gkZ \[yArso�m|degkc =�V�G-¼I¢�\�£IcYg�¢>m�Wsopmvf�g�`,& �# c.a l%�! l MN` ��P ;�L �'PUR6i g�c��#H_J�Tk�+VWY\�gjr�m|�Yr�m ��deikd�mt�\�m dqt ZEdecsdeZ�opfhfhv¨{�\7��rYcs�sopcAmntwdhcsl(\ �'PUR6iSg "��-iC; ojcs��\7yArso�m|degkc<=²�XG0ijdh�j\7t0rstm�Ws\ decsl(fer<twdhgjc/�

f�g�`��&c a l% `,f�g�`,& �# c}i>H��'P<;�L�MNM�M�L �'PzR�TKcl% � l%�! l ` ��P<;�c}i4�'P ; �@l%�! l ` �'PUR6iSg�cwi6�'PUR�� � HKJ�Tl%"! l � ��P<; � �'P{R6i��% ��� H_J�T

��yArsopm�dhgjcC=%\X\KG�dqt �Y{|g��j\[��dec:o�t�deZEdhfhoj{0¢oRv*�f�g�`�,7� & c a l% 0 f�g�`�� & c�izH&�'P ;pivR�LNM�M�M�L ��T 5

l% � l%�! l ` �'P ;wc}i6�#P ;#ivR �@l%"! l ` ��PUR6i �%�� c}i6� � � HKJTl%"! l � �'P<;pi %"! l% � �'PUR6i %�! l% i �% ��� � � HKJ�T

+¦�\�gjxYm|opdhc:opc:dh�Y\(cAm�dql(ojfu�Y{|gj�5\({�m~v�d� �)�cY\(iAo�m|de�k\b{|\[l(g���decsiktop{|\ op�Y�Yfhdh\[��m�g¨oE�k\[l-m|gj{gp ª��dhijdem|t

dhc m�Ws\�t�\�m�H&��imRsP1g�LNM�M�MNL �#iv;�T�¢ dem�WvR@- gI- ;��¶§³c m�WYdqt�l[ojt�\ , & Wsokt�dem|t  ¡{�ojl�m�dhgjc {�opcYik\x5gjrscs��\[��xIv¶\[yArsopm�dhgjc =O\7^�Gbojcs�7� , & ¢ d�m|W�gkcY\]fhokt~m!�<gAtwdem�dh�j\]{|\[l�g���dhcYi�gp ��sde�kgpm �0iU;>Wsojtbdem|t ¡{�ojl�m�dhgjc�{|ojcYij\bx5gjrYcs�Y\[��xIv¨\[yArsopm�dhgjc+=%\[�uG

f�g�`�, & c.a R; PUR M2> �0i>RWP g

; & L �-iC; ? � H_J�T =O\7^�G

f�g�`�� , & c.a R;IPzR M > �-ivRWP ; P{R

; P g; & _ l � �-iQ; P ; P{R

; ? � H_J�T =O\[�XG

��� � ��� �"�$� �G� �7��@�'�����������+�������������o�$����� �4��� �`��������J�� ��� �� �5���¦�\�¢ dhfef�t�\(\¨®I\7l-m|degkc ��WYg�¢ m�WY\E ¡{�ojl�m�dhgjc�{|ojcYij\]gp  o�xYdhcsop{|v�\(c<l�g���dhcYi¶l(opc x5\�rst�\[�9m�g���\7��rsl�\dhZE�<gk{wm�opcAm��Y{|gj�5\({�m�dh\[t¨gjc�m�Ws\9��dhijdem|t�twm�gk{�\7�>dec o �k\[l�m�gj{7� ¦�\9¢ defhf�}~r<t~m��s{�\7tw\[ckm�WY\({|\�o °s{�t~m�s{|okl-m�dql(ojfÁop�Y�Yfhdql(o�m|degkc�gj �m|WY\! ¡{�ojl-m|degkc�{�opcYik\j�½/\�m U e V ���QMNM�M��n#YXx<\:o¶�k\[l-m|gj{�gp ${�oj��de�<; ��deikd�m�t(��VWY\�¢ dhcs��g�¢ gj  ) ��dhijdem|t]t~m�op{�m�dhcYi9o�m

�5gkt�dem�dhgjc*d�M*):l[opc¶x5\���opfhrY\[�¶ojfegkcY\"ojtnopc¶dhckm|\(ik\({`���h ijdh�j\[c�xIv�\[yArsopm�dhgjcC=%\NVuG-�§T C¢0\"op�s�Yfev�m�Wsdht

ì�ì ×��N�������

Page 11: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

V�opxsfe\0\ �C�$\[��rsl(decsi�m|WY\ {|\[��rYc<�Yopcsl(v�m�gEgkx�m|ojdec¶o]ZEdecYdhZ�opfhfev�{�\7��rYcs�YojcAm �Ydeikd�mnt�\�m${�oj��de� RNJ h �� \7lp� �Y{�ojl-m|degkc�{�opcsij\ ��deikd�m�t {�oj��de� \[�k�

=²�$oj��de� \7�XG ­�o�� ��deikd�mnt�\�m�$gkc R

� Mu`%PsR�L��ji>R_c�e > P R� LNR ? ��� HXPsRXR�L�MNM�M�L ��� T

� R� M > PsR � �ji R

RNJ ? e > P R� � �RNJ ? � JYi R

� HXPsRXR�L�MNM�M�L � J�T

� h R� M > PsR � �ji R

RNJ�J ? e > P R� � � �RNJ�J ? � � i R

� HXPsRXR�LNM�M�M�L�� � T��

�st R� M > PsR � �ji R

RKJ�J�J ? e > P R� � ��� �RNJ�JXJ ? ��� � i R

� HXPsRXR�LNM�M�M�L�� � T��

m|{|ojcstw ¡gj{|ZEopm�dhgjc� ¡gj{nojcAv��p)�)��b-Sd*- ��(��)��!¢0\ gjxYm|opdhc:oEcY\(¢3�j\7l-m�gk{ gp ª��dhijdem|t5� �h {�\[�Y{�\7tw\[cAm�dhcYim|WY\�t|opZE\ cIrYZ"x5\({${�oj�Yd��#; & ojt$t�WYg�¢ c�dec:\[yArsopm�dhgjcC=%\_|�G���¦�\�l�gjZEZEgjcsfev�rstw\!m|WY\�g�l-m�opfuojcs��m�WY\Ws\��Yoj��\7l�dhZ�opfÁcYgjm|o�m|degkcstx5\[l(ojrst�\�m|WYdqt$l�gkcA�k\({�twdhgjc�dht�k\({|v¨\[oktwv¨m�g��5\({� ¡gj{|Z xsojl�£�opcs�� ¡gj{�m�W� ¡{|gjZ{�oj�Yd���^]¢ dem�W )'e>q]gj{�)'evgs�

� �h eo020 ��� h _ l���� & ! l M�MNM7� h � & 080 % e & ! l9 :; o � h � & _

:;:

¢ d�m|W � �h ? ; & P{R; P{R M�V &CBED F � &CBEGH X =O\�VuG

020 ���QM�MNM7�$#0Y0 % e�������� � ��� ��� M�MNM7� � ������ ��� ��� % � =O\K|�G

¦�\�l(ojc�ojfht�g]¢ {�dem�\b���h okt0�Y{�\7tw\[cAm�\[��dhc�\7yArso�m|degkc<=O\[�XG�fh\[ok��decsi�m|g¨opcYgjm�WY\[{decAm�\[{���ojf¥ ¡gj{5���h �� �h e 080 J ��� h _ l���� & ! l MNM�M7�$#080 %wM ; & PT020 J �mh � & M�MNM7�$#7080 % ¢ dem�W � �h ? 0 ; & M f�g�`�U#c�PIf�g�`�Upc 5 =O\[�XG

§T �m�WY\! ¡{�ojl�m�dhgjc�{|ojcYij\ t|o�m|dhtw°s\[tvf�g�`�Upc�aS`�- L/.�c�¼Y\[yArsopm�dhgjc+=%\7�XG�vIde\[fh�Yt0m|Wso�mno�l(gjZ�xYdecs\[����deikd�m���h dqt�x5gjrscs��\[��xIv -�M�; & P . " ���h " .wM�; & P -5�ª¦�\��s{�\7tw\[ckm0V�ojxYfe\7tY\!opc<��^!m~¢0g"\(��ojZE�Yfe\7t�{�oj�Yd��\7�Y¼�¢ d�m|W m�WY\���deikd�m]tw\(m�HXPsRXL�M�MNM�L � T�opcs��rY� m|g9�:�<gAtwdem�dh�j\�l(op{|{|v�{�\7l�g���dhcYikt!¢ dem�W �Yde�kgpm��{e � �VWs\ feg�¢0\({ x5gjrYc<�:gjc�m�WY\�l�gjZ�xYdhcY\[�:��dhijdem�t�\�mndqt ijdh�j\[c:xAv�\[yArsopm�dhgjcC=%\NVuGopc<���YgA\7t$cYgjmnl�Wsopcsij\j�VWs\�ZEop��deZ�rYZ ��ojferY\E ¡gk{�gjcs\¨��deikd�m�dht�l(gjZE�Yr�m|\[�� ¡{|gjZ m�WY\� ¡{�ojl�m�dhgjc {�opcYik\]dhc�m�WY\¨m|opxsfe\k��VWY\°<{|twm {�\7l�g���dhcYiE{�\[Z]g��k\[��op�Y�Y{|gR��deZ�opm�\(fhv¨o]�<gk{wm|degkc l% gp �m�Ws\ �<gAt�t�dexsfe\ ��dhijdem|t$ojcs��m�WY\�t�\[l(gjcs��gjcY\o]�5gj{�m�dhgjc l%� �`b\(mwm�dhcYi�{�dq�:gp CopfhZEgktwm l% gp +m|WY\���dhijdem|t�opfhfeg�¢$t$rst$m|g¨{|\[ojl�W:\7ojt�defhv¨m|WY\"ZEdhcYdeZ�ojfefhv�{|\[�YrYcs�Yojckm

�Ydeikd�mnt�\�m7�ª§Tm$ZEoRv�cYgpmnt�\(\[Z m�g�x5\ dhZ]�5gj{�m|ojcAm¢0gj{|£Idecsi�{�oj��de� \7��xsr�m m�WYdqt$dqt�j\[{�v�{�\[fe\[�RojcAm{�oj�Yd��^]okt¢�\!¢ dhfef/t�\(\!dhc�m�WY\ cY\(�Imntw\7l-m�dhgjc/�! ÊuËRÖªÞRÌÎú-Ì ÙuͲֳÙ�ÌÎÍÁç ÌÎ×RÌÎç$Ô�ÛÎÛ Ü0ѲÖ~ÞRÝ7×RÞ�Ô�×�Ù�ï5Ê/Ë7ÖwÑÓÖªÏ/ÌÎÛÎÛkÕpÖª×RÐ0æeÝRÑ¡ÙLË7ÖwÑ�ÌÎç ØRÑÓÐ�ñ�Ö~ç Ö~×(Ùwï

à¡ê�ìuà ð

Page 12: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� �

V�opxsfe\�^ �C�$\[��rsl(decsi�m|WY\ {|\[��rYc<�Yopcsl(v�m�gEgkx�m|ojdec¶o]ZEdecYdhZ�opfhfev�{�\7��rYcs�YojcAm �Ydeikd�mnt�\�m${�oj��de� RNJXtp�� \7lp� �Y{�ojl-m|degkc�{�opcsij\ ��deikd�m�t {|ok��de� \[�k�j�

=²�$oj��de� \7�XG ­�o�� ��deikd�mnt�\�m�$gkc R

� Mu`%PsR�L��ji>R_c�e > P R� LNR ? �X��� HuPsR�R�RXL�MNM�M�L ����� T

� R� M > PsR � �ji R

RNJ ? e > P R� � �RNJ ? � JXJji R

� HuPsR�R�RXL�MNM�M�L � J�J�T

� h R� M > PsR � �ji R

RNJ�J ? e > P R� � � �RNJ�J ? � � Jji R

� HuPsR�R�RXL�MNM�M�L�� � J�T

�st R� M > PsR � �ji R

RKJ�J�J ? e > P R� � ��� �RNJ�JXJ ? ��� � i R

� HuPsR�RXR�L�MNM�M�L���� � T��

� � �(��� ��� ������)��(���!��������.

��t�¢0\nW<oR�j\�t�\(\[c¨dhc�\[yArsopm�dhgjcstn=%\7�XG-¼y=O\�\KG�¼ =O\7^XGCopc<� =%\7�XG�¼Ao�twWYgk{wmtw\7yArY\(csl(\ngj u{�\7l�g���dhcYikt�l(opc�x<\\[ZE�Yfeg�vk\[�]m|g"{|\[��rsl(\$m|WY\$ ¡{�ojl�m�dhgjc¨{�opcYik\$opcs��op�Y�Y{|gkokl�W]m�WY\b¢ dh��m�W�gj �\$rYfh�u�CVWY\��Y{|okl-m|dhl[opfs��ojferY\gj YxYdhcsop{|v l(op{|{|v�{|\[l(gI�YdecYiAt/dqt/m�Wsopm<}~rstwmªo$ ¡\[¢ {�\7l�g���dhcYikt/�Y{|g��Adq��\��sop{�m�dqopfAl�gjZE�Y{|\[t|t�degkc�twr��¨l(de\[ckm�m�ggkx�m|ojdec�opfhZEgktwmojfef¥m|WY\!x<\[cY\�°Ym�t gp / ¡rYfef�l�gjcI�k\({�twdhgjcEm|gEo]cYgkc�{|\[��rscs�YopcAm cYgjm|o�m|degkc�¢ WYdhfh\!oR�kgjdq��dhcYim|WY\ WYdhijW�l�gAt~mCgj ¥o!l[op{|{�vA¤T{�dh�Y�Yfh\0ok�Y��dem�dhgjcu�+VWY\ ¡gkfefhg�¢ decsibgkxst�\({|�Ropm�dhgjcst+oj{�\ twm�{�opdhijWAmw ¡gk{�¢op{��! ¡{|gjZm|WY\��\�°scsd�m|degkc]ojcs�"t�rY�Y�5gj{�m��<op{�m�dqopfIl�gjZE�Y{|\[t|t�degkc"op�Y�sfedql(opm�dhgjcst�dec�{�gkrYcs��dhcYi<¼�fe\7oj��dhcYibdecst�dhijcYde°<l(ojcAm�Ydeikd�m$��\[fe\(m�dhgjcu¼sojcs��Z"rYfem�dh�Yfhde\[{${�\7l�g���dhcYis�

��� �6� ��� �o� �����'������� �E��� �����1�6 � ���K����� �'�3����§³c�l�gkZ]�sr�m�\[{|t[¼jm|WY\!��deikd�m�j\7l-m|gj{�dqt0cYgpm twm�gj{|\[�Exsr�m \(csl(gI�Y\[��dhcAm�g]xYd�m�t(�+§³c�m�WY\!l(oj{�{|v]t|oR�j\$ ¡gj{|ZEopm[¼\7ojl�W"�Ydeikd�m^� : dqt+t~m|gj{|\[�"¢ d�m|W�m~¢0g�xYdem|t*� : ojcs�� : opcs��m�WY\��dhijdemª�Rojfers\�dqt+��\�°scs\[�]ojt<� : e � : i� : ��¦�\t��5\[l�dqopfhdh«(\bm�WY\"xYd�m�t gp �m�Ws\�{�\7twrsf�m${|\(ikdhtwm�\[{ � : e?� �: i �: t�rsl�W�m�Wsopm � �: _ l twm�gk{�\7t0m�WY\�l[op{|{�v¨xsd�m0� : _ lik\(cY\[{|opm�\7�"xIv"\7yArso�m|degkcI=)|�Gªojcs�� �: t~m|gj{|\[t�m�Ws\${�\7twdq��rsojfYyArsopcAm�dem~v � : P � : e � : P � : _ l ;¶gj Á\[yArso�m|degkc=²�XG���VWYdht�fh\[oj�9rst�m�g�m�WY\]m�{|r�m|W�m|ojxYfh\]ojcs�:m�Ws\]\7yArso�m|degkcst$ ¡gk{nm|WY\]�<gAtwdem�dh�j\El(oj{�{|v�{|\[l(g���decsi�gp �ol[op{|{�v�t�oR�k\ cArsZ"x5\({���dhijrY{|\�\(¤Tos��§T ªm|WY\"cIrYZ�x<\[{|t$oj{�\�t~m|gj{|\[��rst�decYi¨m~¢�g � t$l(gjZE�Yfh\(ZE\(cAm7¼sm|WY\"t�deikc�Ydeikd�m$dqt ZEg��j\7��m|g��<gAtwdem�dhgjce)ri>R�ojcs�� � _ l e� �"�0VWY\ \[yArsopm�dhgjcst gj +��dhijrY{|\'\�¤³oEop{|\bm�WY\�gkcY\[tgj �ojc�W<opfe �oj�s��\({$l(\(fhf�=����nGojt¢0\!¢ defhf/t�\(\ t�gAgkcu�VWY\�cs\(ikopm�dh�j\�l(oj{�{|v�{|\[l(gI�YdecYi�gp  o¶l(oj{�{|v�t|oR�j\EcIrYZ"x5\({�l[opcYcYgjm�x<\¨�Y\�°scY\7� dec t�rsl�W opc�\(fh\�¤

ZE\[ckm�op{|v�Z�opcYcs\({�xYrYm"¢�\�Z�oRv���\�°scs\¨ZEgj{|\�{�\7l�g���dhcYikt!de m�WY\�m�op{|ij\(m!\[csl�g���dhcYi¶dqt�x5gj{|{�g�¢ t�oR�k\dhcstwm�\7oj��gp  l(op{|{|v�t�oR�k\j��� x5gj{|{�g�¢ t|oR�j\E��dhijdem�� : =Ó{�\7tw�/�$� : Gbdqt�t~m|gj{|\[��¢ d�m|W�m~¢�g:xYdem|t�� : opc<��� :=Ó{�\7tw�/� � �: opc<�����: Gopcs��m�Ws\���deikd�m$dqt$��\(°scY\7��okt�� : e?� : P�� : =Ó{�\7tw�/� � : e � �: P����: G-�ª¦�\�tw�5\[l(dhojfedh«(\o¨�<gAtwdem�dh�j\]l(op{|{|v�gj{bo¨cY\[ikopm�dh�j\"l[op{|{�v�{�\7l�g���dhcYi�t�decsl(\"¢0\�l[opc�twm�gj{|\ m|WY\]l[op{|{�v�dec � �: _ l =Ó�<gAtwdem�dh�j\l[op{|{�v�zC¤L{|\[l(gI�YdecYi]gp ª��deikrY{|\k\(¤Lx G�gk{���� : _ l =¡cY\[iko�m|de�k\ l(op{|{|v �$¤L{|\[l(g���decsi]gj ���dhijrY{|\0\�¤³lNG-�

ì�ì ×��N�������

Page 13: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

\7� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

¬® � : � � \ \dec

:� \ � \

��deikd�m`� : � \ \ ^¬® � �: _ l � � � \gjr�m �: � \ \ �

�"�Y�7� · � �: _ l �� : M :

�µ ���²» ��� & �:�

� : � :

=ÓouG0z+gkt�d�m|de�k\!l[op{|{�v�{|\[l(g���decsi�gj +oEl(oj{�{|v¨t|oR�j\bcArsZ"x5\({&=����nG

¯ ® � : � � \ \dhc �

:� \ � \

��dhijdem`� : � ¤�\ \ �¯ ® � �: _ l � � \ �gjrYm ���: � \ \ �

�"�Y�7� · � �: _ l �� : M � :

�µ ���²» ��� & � �:�

� : � � :

=¡x/Gz+gAtwdem�dh�j\ l(oj{�{|v�{|\[l�g���dhcYi]gp ªo]x<gk{�{|g�¢ t|oR�j\bcArsZ"x5\({&=¡zaG

¯ ® � : � � \ \dhc �

:� \ � \

��deikd�m`� : � ¤�\ \ �¯ ® � �: � \ \ �gkr�m � �: _ l � \ � �

�"�Y�7� · ���: _ l �� : M � :

�µ ���²» ��� & � �:�

� : � � :

=ÓlKG �$\[iko�m|de�k\bl[op{|{�v�{|\[l(g���decsi]gj �oEx5gj{|{�g�¢Nt|oR�j\bcIrYZ"x5\({-= �nG

¬® � : � � \ \dec

:� \ � \

��deikd�m`� : � \ \ ^¯ ® � �: _ l � \ \ \gjr�m ���: � \ \ �

�"�Y�7� · � �: _ l �� : i :

�µ ���²» ��� & ���:�

� : � :

=Ó��G �$\(iAo�m|de�k\bl[op{|{�v�{|\[l(gI�YdecYi] ¡{|gjZ,l[op{|{�v¨t|oR�j\bcIrYZ"x5\({m|gEx<gk{�{|g�¢ t|oR�j\b{�\[�Y{�\7tw\[cAm|o�m|degkc =��xG

��deikrY{�\0\��CV/{|r�m�W�m|ojxYfh\�opcs��\[yArsopm�dhgjcstgj �xYdhcsoj{�v�{|\[l(gI�YdecYiAt¢ d�m|W¶o]t��5\[l�dqopfhdh«(\[��l(oj{�{|v�xYdem

à¡ê�ìuà ð

Page 14: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� \X\

�����������

��

�� �����

��deikrY{|\ ^ �"� ¤³l(op{|{|v�{�\7l�g���dhcYiEgp �o]x5gj{|{�g�¢ t�oR�k\�cIrYZ�x<\[{�

��dhijrY{|\ � � �nojf� ªoj�s��\({ l�\[fef6=��n��G=-�i .WevE ��i��

VWY\��<gAtwdem�dh�j\0l(oj{�{|vn{|\[l(gI�YdecYi�gp so$l[op{|{�v!t|oR�j\ªcIrYZ"x5\({�m�g�x<gk{�{|g�¢ t�oR�k\CcYgpm�o�m|degkc dht�cYgpm��<gAt�t�dhxYfe\xsr�m0o"cY\(iAo�m�dh�j\�l(op{|{|v�{|\[l(gI�YdecYi"dht��<gAt�t�dhxYfe\k��§Tmdqt��Y{|\[t�\(cAm|\[����deikrY{|\�\�¤³��ojt �n¤T{�\7l�g���dhcYis����dhijrs{�\�^dhfhferstwm�{�o�m|\[tbWYg�¢ \[ojl�W�gjr�m|�Yr�m���dqopikgjcsojf�gj �m|WY\�E�� `�) P1(yi>E�c�xYdem"op{|{|oRv�dqt ��\�m|\({|Z]dhcY\7�9xIv9ojcdhcY�YrYmnl�gjfhrYZEc:gp �m�WY\�E��+`R) PI(�ivRKc�xYdemnop{|{|oRv] ¡gk{m�WY\ �$¤Tl[op{|{�v�{|\[l(gI�YdecYi<�VWY\]Wsojf� ±¤³oj�Y�Y\({�l(\(fhf�dqtbgjcY\�gp ªm|WY\�x<ojt�dhl�l�\(fhfqtngj ªm�WY\El�gkZ]�sr�m�\[{boj{�dem�WYZE\(m�dql�fhdexY{�op{|dh\[t[���Yrscsl-¤

m|degkcsopfhfhvj¼5d�m!dht�dhZ]�sfe\[Z]\[cAm�\[��¢ d�m|W�opc������ �"!$#&%(')�+* ,�iAo�m�\�opcs��o-�.*�/$%.��01�+0"2)3<=Ót�\(\]��dhijrs{�\"�uG-�§³c l�WYgIgkt�decYi:m|WY\��sfhokl�\�m|g�oj�s� twgkZE\¨cY\(¢4�.* /$%.�"01�5%621'1� ,&7)� ,8#�¼u¢0\��Y\�°scY\�m�WY\�xYdem�fh\(�j\[f�l(\(fhfhtl(gj{|{�\7tw�5gjc<��decsi"m|g�m�WY\"zª¤|¼ � ¤opcs� �n¤�{|\[l(gI�YdecYiAt&=Ót�\(\���dhi�VuG��0VWs\[t�\ decI�j\[{wm|\({�tZ�oRv�gj{nZEoRv�cYgpmvIdh\(fq��ojc�okl-m�r<opf¥�<\[csopfem~v�l(gjZE�sop{|\$m|g�o�twm|ojcs�Yop{��¨W<opfe �oj�s��\({l�\[fefL�C�Ygk{0\(�YopZE�Yfh\j¼Im�Ws\9� ��� �"!�#1%(')�* ,�iko�m|\ Z�oRv�x5\�deZE�Yfh\(ZE\[ckm�rstwdhcYi��sokt�t m�{�opcst�dhtwm�gk{|t okt demndht m�WY\"l[ojt�\ dec�twm|opm�\(¤Lgj ±¤²m|WY\�¤³op{�m$l�dh{|l(rYd�m�Y\[t�deikc[ZÎ^p��^L�+§T utwg<¼jopc�dhcI�j\[{wm|\[�:����� �"!$#&%(')�+* ,"ikopm�\ dqtªgjxYm|opdhcY\7��xIvEtw¢ dem|l�WsdecYi�t�gjZE\ gj <m|WY\$dhcY�Yr�m¢ dh{|\[tgp +oEt~m|{|ojdeikWkm;� ��� �<!$#1%(')��* ,¨iAo�m�\k�¦�\�¢ dhfhf+t�\(\�dec9m�WY\]{|\(Z�opdhcYdhcYi¨gp �m�WY\�m�\(�Ambm�W<o�m!x<\[decYi�opc�WsdeikW9fh\(�j\[f�yArsojcAm�dem~vj¼5m�WY\� ¡{�ojl-m|degkc

{�opcsij\Cl(ojc x<\0\[oktwdhfhv$m�{�ojl�£k\[�bm|WY{�gkrYijW�ojc�opfhijgk{�dem�WYZ opc<�!dht/m�WY\[{�\( ¡gj{|\ª�j\[{�vbrst�\� ¡rYfA¢ WY\(c"Z]gk{�\Cm�WsojcgkcY\${|\[l(gI�YdecYi"dht��<\[{w ¡gk{�ZE\7�Á��©�rY{�{|\[��rscs�YopcAm�¯0gIgpm|W¨{|\[l�g���\[{Y=Ót�\[l�m�dhgjc��s� �uGªrst�\[t�twgkZE\�oj�s��d�m|degkcsopfdhc� ¡gk{�Z�o�m|degkc�gkc�m|WY\ fqojtwm{|\[l(g���decsis�

ì�ì ×��N�������

Page 15: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

\R^ � �������k����.'�_5&�������k��,K�*��� ���)��1]�

������ ���

��

� �� �

� ��� �

� �

��

�� �

����� �� ��������� ����� ��� �

������ �� ��������� ��� � � � � �

� "! �� �������#� ���$� ��� �

��dhijrY{|\xV��C� \7l�g���\({$l(\(fhfht���,� %c������������ �E��� �� ���`�� � ��� ����� �5�2����������§T ª¢0\ m�{|rYcsl[o�m|\!m�WY\�fhg�¢ gj{���\({ �<op{�m$gp Co�¢0gj{��Á¼Y¢0\ ZEoRv:l(ojrst�\�opc:\({|{�gk{ rY�:m�gI�-R rYfe�¶gk{�^ErYfh�st�Y\(�5\(cs��dhcYi�gjc9m�WY\EcYgjm|o�m|degkcu�"©�c�m|WY\Egpm�Ws\({bWsopcs�u¼Ád� 0¢�\�°s{�t~m op�Y�sfev¶o � , & ! l m|{|ojcst~ ¡gk{�Z�opm�dhgjc= )CDrR_G$m|g�m�WY\Ex5gj{|{�g�¢ t|oR�j\�{�\[ijdqt~m|\({!gj{ o � , & ! h'& m�{�opcstw ¡gj{|Z�o�m|degkc =�)QD E�Gnm�g¶o:l(oj{�{|v�t�oR�k\{|\(ikdhtwm�\[{[¼s¢�\ gkx�m|ojdec�oEcIrYZ�x<\[{�twm�gj{|\[��dhc¶x<gk{�{|g�¢ t|oR�j\b ¡gk{�Z�o�m m|Wso�mbWsojtnoE�j\({|v�fhdhZ]dem�\7�: ¡{�ojl-m|degkc{�opcsij\ndhcsl(fers�Y\[��decJV4P lh L lh iQE ! & c��C�nt o]l�gkcstw\7yArY\(csl(\j¼Am�{|rYcsl[o�m|decYi]dem opm$opcIv��<gAtwdem�dhgjc�¢ defhfÁ{|\[t�rYf�mdhc¶opc�\({|{�gk{�fh\[t|tm|Wsopc lh i{E ! & rYfh�u�VWY{|\(\nfedhcY\[t�gp ÁWsojf� /ok�Y��\({��Y{�g��Idq��\[t�o � , h {|gjrYc<��decsi ¢ d�m|W� ¡{|okl-m|degkc]{�opcsij\ 0�P)(*�L lh65 twr��¨l(de\[ckmm|gE{�\7��rsl�\bm|{�rYc<l(o�m|degkc�\[{�{|gj{x5\(fhg�¢+(* rYfh�st[�CVWYdht l[opc�x<\ yArYdem�\ rst�\� ¡rsfÁdec:ZEdhl({�g�l(gI�Y\[�¨{|\[�YrYcs�Yojckm

xsdecsoj{�v ��\[t�deikcst] ¡gj{¨�Yde�Idqtwdhgjcu¼t|ykr<op{|\�{|gIgpm7¼�opc<� m|{|ojcst|l�\(c<��\(cAm|ojfht[� VWY\¶cY\(�Am¨gkxst�\({|�Ropm�dhgjc>Wsojtik{�\7o�m/oj�Y�Yfhdhl[opxYdhfedem~vndhc�\(�Am|{|okl-m|decYi$�Yd �¥\({|\(csl(\[tu ¡{�gkZ3o ¡rscsl-m|degkc m|ojxYfe\C ¡gj{/�5\({� ¡gj{|ZEdecsi decAm�\[{��5gjfqo�m|degkcZÎ^p�Y¼j^X~N^T�ªVWY\jE �k`�)vP (�i RKc�xYdem�op{|{�oRvbZ�oRv�x5\0 ¡gj{|ZE\[�Eojt�m�Ws\$��d �¥\({|\(c<l�\C-�P .�gp sm~¢0g'`R)vPT(�i RKcT¤xsd�m rYcst�dhijcY\7�¨dhckm|\(ik\({�t�ojcs��twm�gk{�\7��dhc�x5gj{|{�g�¢Nt|oR�j\n ¡gj{|Z�o�m¢ d�m|W�cYg]gj�5\({�o�m|degkcu��§³c�m|WYdht l[ojt�\j¼Am~¢0g{|\[l(g���decsiktopfhfeg�¢Nm�g¨��dqt�l[op{��¨m|WY\!fe\7oj��dhcYi���dhijdem|t[�zªop{�m�dqopf+l�gkZE�Y{�\7t�t�dhgjc�opfqt�g¨{�\7opfhde«[\[tn�Adh{�m�rsojfefhv�ojfef�m�Ws\�x5\(cY\(°Ym|tbgj Cfe\7oj��dhcYi��Ydeikd�m ��\(fh\�m|degkcu�!½u\�m\ x<\ o]�5gkt�d�m|de�k\�cIrYZ�x<\[{ {�\[�Y{�\7tw\[cAm�\[��dec�x5gj{|{�g�¢Nt|oR�j\btwrsl�W�m�W<o�m�\ -UE & P Rk�+¦�\!£IcYg�¢ m|Wso�m de , `R\ c�eoV � �� _ l ���� MNM�M���# XT¼Ym|WY\(c@��� & eBJ] ¡gk{$opfhf =aD8)&iBRj��VWAr<tC,Q`R\<c�Z�oRv�x<\!m|{�rYc<l(o�m|\[��m�g¨o

E��+`�)sivRKc�xYdemnop{|{|oRvk��§T +¢�\ l[opc�gjcYfhv¨x5gjrYcs��m�Ws\ ZEojijcYdem�r<��\!gp -,F=%�*(�`0 \ 03-zE & PURKG�¼�¢�\!WsoR�k\m|g�l�gjZE�YrYm�\ � ,Q`�\ c0opcs��¢�\ �s{�g��k\$m�W<o�m`��� �: evJ� ¡gk{$opfhf�=�D )&i{E

��� � .J�v�E�/0%c� �C�=�5� �� {�oj��de�>V>¯0gIgpm|WNZ"rYfem�dh�Yfhde\[{¶l�gjZE�YrYm�\[t�dhcNm|WY{�\[\�twm�\[�st�m�WY\�{|\(�Y{|\[t�\(cAm|opm�dhgjc gj  U e Z �21¢ Ws\({|\"m|WY\Em~¢�g�cIrYZ"x5\({�t�Z opcs�31%op{|\�{|\(�Y{|\[t�\(cAm|\[��{�oj�Yd���^�xIvQ`�- � MNM�M/- # c h opcs�U`�. �54 MNM�M9. #64 c h �VWsdhtEgk{�iAopcYdh«[opm�dhgjc dqtE�Adqt�dexYfh\:��dhijrs{�\:� opcs� decSZÎ^p��¼�^\p¼�^p�_^T� VWs\�°<{|twm¨t~m|\(��l�gjcI�k\({�m|t�1 m|g�o{�oj�Yd��I¤ VbZEdhcYdhZEojfefhv]{�\7��rYcs�YojcAm�{�\7l�g���dhcYi gjc]m|WY\���dhijdem�t�\�mjHuPYE�LNM�M�M�L�E�T�=��w¯0gIgpm|W�\(csl(gI�YdecYi(%G���VWsdhtl(gjcI�j\[{|t�dhgjc:dht��5\({� ¡gj{|Z]\7�: ¡gj{ �R( � �E��T-cd -07R) � �E�8�xIv:fegIgj£IdhcYi�opm�xYdem|t . h h _ l ¼ . h h]ojcs�7. h hi! l m�g

à¡ê�ìuà ð

Page 16: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� \7�

V�opxYfh\ � �ª�$ok��d��[V�¯�gIgjm�W�{�\7l�g���dhcYi

­�rYf�m|de�sfedh\({${|\(�s{�\7tw\[ckm�o�m|degkc ¯0gIgpm|W���dhijdem ®Idhijc¶t�\(fh\[l�m0 ¡{|gjZ Z ^j�_^. h h _ l . h h . h hi! l . �h e@PYE . h h _ l i . h h�i . h hi! l \ h �5\ h ���Eh �+h� � � � � \ � �� � \ \ \ � \ �� \ � \ \ � \ �� \ \ ^ � \ \ �\ � � ¤³^ � \ � \\ � \ ¤ \ \ � � \\ \ � ¤ \ \ � � \\ \ \ � � \ � �

l(gjZE�Yr�m|\]�Ydeikd�m . �h ojtb�Y{|\[t�\(cAm|\[�¶dhc�V�opxYfh\E�Y��©�cY\]l(opc�l�WY\[l�£:xIv:dec<��rsl-m|degkc9m|Wso�mbm�Wsdhtbgj�5\({�o�m|degkc�YgA\7t"csgpm]l�W<opcYik\�m�WY\�{|\(�Y{|\[t�\(cAm�\7� ��opfhrY\�okt"¢ {|d�m�m�\[c dec m�WY\�°s{�t~m]�sop{�m�gj n\[yArsopm�dhgjcv=%\�~�G���VWY\t�\[l(gjcs��twm�\(�>=��wV�{�\[\E{�\7��rsl-m|degkc)OG�okl(l(rYZ"rYfqo�m|\[t�m�Ws\��sop{�m�dqopfª�Y{|g���rsl�m|t . �h MmZ M�g h dhc o�{|\[�YrYcs�Yojckm ¡gk{�Z�opm�m�g]l�gkZ]�sr�m�\�m�WY\b�Y{|g���rsl-mojt0twrsijij\7t~m|\[�ExIv]m�WY\!t�\[l�gkcs���soj{wm0gp /\7yArso�m|degkc =%\_~�G-�ªVWY\�m�WYdh{��twm�\[� l�gkcA�k\({�m|t�m�Ws\¨{�\7��rYcs�sopcAm"{|\[t�rYf�m m|g:m�WY\�rst�rsopfCcsgjc {�\7��rYcs�sopcAm�xYdhcsop{|v9{|\(�Y{|\[t�\(cAm�o�m�dhgjc opc<��5gkt|t�dexYfhv¨{|gjrYc<��demnojl[l�gj{���dhcYi�m�g]m�WY\�okl-m|de�k\b{|gjrscs��dhcYi�ZEg���\�=���^�¤ \!l�gkZ]�s{�\7t�t�degkc)OG-�

�9 :; # .

:E:e

� ��� h��9h ; � #� h � . �h gh opc<� Z � 1 e

�9 :; # .

:M Z>MKE

:e

� ��� h�9h ; � #� h � . �h M ZzM�g h =O\_~�G

¬�gkZ]�<op{|\[� m|g�m�WY\:rst�rsopf Z�rYfem�dh�Yfedql(opm�dhgjcu¼¯0gIgpm�W {�\7l�g���dhcYi ��de�Idq��\[tExIv m~¢0g�m�WY\:cIrYZ"x5\({�gj �<op{�m�dqopfj�s{�g���rsl�m|t/ij\(cs\({�o�m�\7� opcs�"ojl(l(rYZ"rsfhopm�\[�!xsr�m/m|WY\[t�\��Y{|g���rsl�m|t�op{|\CZEgj{|\���d ��l(rYfem�m|g$ij\[cY\({�o�m|\t�dhcsl�\!m|WY\(v�l(opcscYgpm$x5\ gjx�m�opdhcY\[��xIv�o �.* /$%.�"01�+0�2)3�ikopm�\ ojt m�WYdqt dqtm�WY\�l(ojt�\!¢ WY\(c¶¢0\ Z"rsf�m|de�YfhvgkcYfhv¨xIv . : ?<HKJ � R�TA��� csojde�k\!deZE�Yfh\(ZE\(cAm�o�m�dhgjc:l(ojc�{|rYdhc�ojfef¥m|WY\�oj�Y�RojcAm|opik\[t0gp +¯0gAgjm�W�{�\7l�g���dhcYis�®Im|o�m|\ gp ¥m�Ws\$op{�mCdeZE�Yfh\(ZE\(cAm�o�m�dhgjc<tªrst�rsopfhfev��Y{|\[t�\(cAmªm~¢�g�l(\(fhfht��um|WY\$\[csl�g���\[{ªm�WsopmCdhtC{�\7tw�5gjcst�dhxYfe\ gj ik\(cY\[{|opm�dhcYi$o$t�\�m�gp Yl�gkckm|{�gkfkt�deikcsopfqt¥ ¡{|gjZ3m�Ws\�dhcY�Yr�m�xYd�m�t . h h _ l ¼�. h h opcs� . h hi! l opc<�bm�WY\�Z"rYfem�dh�Yfh\���\({m|Wso�m�l�gjZE�YrYm�\[t�o{�\[�Y{|\[t�\(cAm|opm�dhgjcbgp �. �h M Z xsoktw\7�bgjc�`�- � M�M�M - # c h ojcs�bm�WY\�l�gjcAm|{�gkfjtwdhijc<opfqt¥ij\(cs\({�o�m�\7�xIv m|WY\$\[csl�g���\[{[���ntªcYgjm�\7��dhcIZ ^j�_^T¼kopcE§ � � �ª¤%~�|�V�twm|ojcs�Yop{��pZ ^j�_^<��gkrYxYfh\ �Y{�\7l�dqtwdhgjc�Z"rYfem�dh�Yfhde\[{Crst�\[t^u~�\[csl�g���\[{|t!ojcs� \_|j^u~EZ"rsf�m|de�Yfh\���\[{|tn_ rY��m�g �j���gp �m�WY\�op{|\[o�gp �m�WY\�l(de{�l�rYdem dec t�gjZE\�\7op{|fedh\({�srYxYfhdhl[o�m�dhgjc<ZÎ^�\�^T�¦�\!�Y{�\7tw\[cAmdhc:V�ojxYfe\!��m�WY\ t�\�m gp �l�gjcAm|{�gkf¥t�dhijcsojfht0l�gkZ]�<o�m�dhxYfh\!¢ d�m|W�m|WY\!Z"rYfem�dh�Yfh\���\({ rst�\[��dhc

ZÎ^p��^L��VWY\]cIrYZ"x5\({!gkx�m|ojdecY\7��dqt�m�WY\]gjcY\�� t!l(gjZE�Yfh\(ZE\(cAm!{�\[�Y{�\7tw\[cAm|o�m|degkc¶gp >Z �7. �h ��½/d�m�m�fh\]\(�Im�{�ol(de{�l�rsd�m|{�v¨dqt oj�s��\[��m�gEm�op£j\ l[op{|\�gj ª^ � t l(gjZE�Yfh\(ZE\(cAm fegkijdqlkZ ^K^T�VWY\�cY\��ImE{�\7twrsf�mEtwrY�s�<gk{wm�t m�Ws\� Óokl-m�gk{�dhcYi�gj $Z"rYfem�dh�Yfhde\[{�{|\[l(g���decsi�decAm|g�o¶m~¢0g9twm�\[� �Y{|gIl(\[t|t(�

zªoj{wm|dhojf5l(gjZE�Y{|\[t|twdhgjc�dqt�°s{|twm0oj�Y�Yfhde\7��m|g�{|\[l(gI�Y\�o�{|\[�YrYcs�Yojckm� ¡gj{|Z�o�m twg�dec�o"t�\[l�gkcs�¨twm�\[��demZ�oRvx5\ �sokt�t�\[��m�Ws{�gkrYijW:o�¯0gAgjm�W:{|\[l�g���\[{ ¢ d�m|W¶�j\[{�v� ¡\[¢ �<\[csopfem~vj�0VWY\ fe\7oj��dhcYi�cY\[ikopm�dh�j\!¢�\[deikWAmxsd�mgj �opcIv E �7)�xsd�m ¢ dhcs��g�¢ gj 0o�x<gk{�{|g�¢ t�oR�k\"cIrYZ�x<\[{�� , & `R\ c�dhcs��dql(o�m|\[t�m�Ws\Et�dhijc9gj �m|WY\���deikd�m

ì�ì ×��N�������

Page 17: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

\NV � �������k����.'�_5&�������k��,K�*��� ���)��1]�

��ojferY\�gp Ám|Wso�m¢ dec<��g�¢ ¢ WY\(cs\(�j\[{Cm|Wso�m ��dhijdem�dqt�cYgkc�¤T«(\({|gs�+VWY\bt�ojZ]\�{�\7twrsf�mop�Y�Yfhdh\[t� ¡gj{o ¢ dec<��g�¢gkc�o � , & ! l'& {|\[l(gI�Y\[��cArsZ"x5\({7�� � ºuº�� ½/\�m UFe�, & `R\ cÁ¢ d�m|WbUre V ���QMNM�M��n#pXT¼7¢�\C��\(°scY\ªm|WY\�l[op{|{�v �k\[l-m|gj{2�me V �i� _ l � �QM�MNM�� # _ l X ¡{|gjZ \[yArsopm�dhgjc =b|XGªopc<�]m�WY\${|\[t�rYfem��j\7l-m|gj{��@e �'`,U#c�ekV � � _ l � � MNM�M&� # X< ¡{�gkZ \7yArso�m|degkcI=²�XG+¢ dem�W� : e � : P � : �ª¦�\ gjxYm|opdhc�m�WY\! ¡gkfefhg�¢ decsi]\7yArso�m|degkc� ¡gk{$opcIv¨l�gkZ"xYdhcY\[�:��dhijdem�� �: {|ok��d��IE &

� �: ek020 J ���:_ l � � & ! l M�M�M�� #080 h MKE & PL080 J � : � & MNM�M&� #080 hm

¦�\ \��Im|{|okl-m$o�m�\({|Z  ¡{|gjZ m|WY\ {�oj��de���5gjfhvAcsgjZEdhojfÁgp G�  ¡{|gjZ,\[yArso�m|degkc<=O\KG

� �: e ` ��� : _ l���� & ! l P6��� : _ l���� & ! l cyM�E & ! l ic080 J ��� : _ l���� & ! l ��� : _ l���� & ! h M�M�M&� #020 h M�E & PS080 J � & � � M�M�M�� #080 hn ¦�\"{|\[l(gjikcYde«[\!m�Wsopm V ��� : _ l���� & ! l ��� : _ l � � & ! h MNM�M&� #YX+dqt$m|WY\"{|\[l(gI�Y\[�:�j\7l-m|gj{$gj `V ��� : _ l � � & ! h M�M�M��n#YXªopc<���� : _ l���� & ! l P���� : _ l ��� & ! l emP�� �

:_ l ��� & ! l ojcs�¨m|WY\({|\� ¡gk{�\

� �: ?<P�� �:_ l���� & ! l MKE & ! l i{E & ! l M 0 PYE ! & � R 5 P R

E M 0 PsR3P+E ! & � R 5�sgj{$ojcAv�)Á¼Yd�  � � : _ l���� & ! l emRbm�Ws\(c6� �: - PYE & ! l i{E & ! l i l _ h�� �h opcs� � �: -{J�t�dhcsl�\ � �: ?�� �

+¦�\�l(ojc�cYg�¢ fedqtwm ojfef¥m|WY\ �5gkt|twdhxYfh\bgkr�m��sr�m$gp �m~¢�gE��dhijdem|t�� h h _ l opcs� � h h!gp +o� , h gk{$o � , &{|\[l(g���\({7�ª�nt �Y{|\[t�\(cAm|\[��V�ojxYfe\sV�t�gjZE\!gjr�m|�Yr�m�t$op{|\!cYgpm$��opfhdq��x5\[l[oprst�\bm�WY\[v��Idhgjfqo�m�\bm|WY\  ¡{�ojl-m|degkc

{�opcsij\ gj{$m�WY\�fhokt~mn{|\[t�rYfemC}~rstwm��Y{|\[t�\(cAm|\[�:gj{nx<\7l(opr<tw\ m|WY\(v¶l[opcYcYgjm$x<\�gjx�m�opdhcY\[�� ¡{|gjZm�Ws\"fqojtwm �{|\[l(g���decsis��¦�\EWsoR�j\Eopfqtwg�fhdhtwm�\7��V�opxYfh\kV:ojfef�m|WY\E�Rojfedq��l(ojt�\[tbde �¢0\�t~m|gj{|\. h h _ l dhc�� h h _ l ¼ . h h dhc� h h _ l opc<� � h h ojcs�:°scsojfefhvF. h hi! l dhc�� h h �nVWY\���opfhrY\�gp ªm|WY\�twm�gk{�\7�:cIrYZ"x5\({�dhtnrYcsl�WsojcYij\7�¶twdhcsl�\¢0\!�Ide{�m�rsojfefhv�l(gjZE�Yr�m|\sE 1mP 1 okt�Y{|\[t�\(cAm�\7��dhc¶��\�m�opdhfhtxIv¨m|WY\�oprYm�WYgk{|t0dhc<Z �_^T�¦�\]�Y\[t�deikc:m�WY\�cY\(¢�\[csl�g���\[{��Y{|\[t�\(cAm|\[�¶��deikrY{|\-|R¤Tx¶m�g��Y{�g���r<l�\�m|WY\�l(gj{|{�\7l-mn{|\[t�rYfem$ ¡gj{b\[okl�W

okl(l(\(��m�opxYfh\�dhcY�YrYm¨gp  � h h _ l opc<� � h hA�>§Tm¨t�WYg�¢$t]WYg�¢o�x5gj{|{�g�¢ t|oR�j\�{|\(ijdqtwm�\({�l[opc>x<\:rst�\[��ojtm|WY\�dhcY�YrYm�m|g�m�Ws\�¯0gIgpm|W \(csl(gI�YdecYi�fegkijdqlEm�g�t~m|gj{|\�\[d�m|WY\({Eo � , & �s{�\7l�g���\[� l[op{|{�v t|oR�j\�{|\[t�rYf�mgk{"o¶cYgkc {�\7��rYcs�sopcAm"cIrYZ"x5\({"¢ WYdhl�W W<ojt�x5\(\[c l�gkcI�j\({�m�\7��m|g�x5gj{|{�g�¢ t�oR�k\ErstwdhcYi¶m|WY\�l(\(cAm�\[{�\7�l(gjcI�j\[{|t�dhgjcu��VWs\ Z]g���de°s\7�¶¯�gIgjm�W¶Z"rsf�m|de�Yfhdh\({$gj ª��dhijrY{|\���ojl[l�\[��m|t$o¨{�\7��rYcs�YojcAm$cArsZ"x5\({nokt gjcY\gj /dem|t0gj�5\({�opcs�st(¼IxIvEm�WY\brst�\�gj �o��Y{�\7l�g���\[{�¢ WYdql�W��YgA\7t0cYgpmij\[cY\({�o�m|\�opcIv¨ok�Y��dem�dhgjcsojf5��\[fhoRvE ¡gk{0ocsgjc�¤T{�\7��rYcs�sopcAm0gk�<\[{|ojcs�Á�

� � ������)���./�(���

¦�\!WsoR�j\b�Y{|\[t�\(cAm�\7��o]ij\[cY\({�opf< ¡gj{|Z�opfhdht�Z  ¡gj{m|WY\ t~m|rs��v¨gj ��soj{wm|dhojfÁl�gjZE�Y{|\[t|t�degkc�opcs��{|gjrYcs�YdecYiAt(�VWsdhtbop�Y�s{�gAojl�W:�Y{|g��j\[t  ¡{�rYdemw ¡rsf�dhc�{|\[��r<l�dhcYi�m�Ws\"{|\[��rYc<�Yopcsl(v:gp �o�x5gj{|{�g�¢ t|oR�j\�gk{�o¨l[op{|{�v:t�oR�k\xsd�m$op{|{�oRv�m|g�opfhfeg�¢N{�oj�Yd��pE & ¯0gIgpm�W�{|\[l�g���dhcYi�¢ d�m|W�ZEdhcYdhZEojful�dh{|l(rYdem�{|vj�ª� \[�Yrsl�dhcYiE{�\7��rYcs�sopcsl(v�dhtr<tw\( ¡rYf/dhc�opcIv�oj�Y�Yfhdhl[o�m|degkcstm�W<o�m���g�cYgpm�ojfefhg�¢ ojcIv� ¡rYfhf/{�opcsij\ {|\[��rYc<�YopcAm$cIrYZ"x5\({bojt dhcY�YrYmnxYr�m�Yg]csgpm${|\[yArYdh{�\bcYgjc�{|\[��rYc<�YopcAm$dhcY�Yr�m�t(�

à¡ê�ìuà ð

Page 18: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� \_|

V�ojxYfe\xV��C�$oj��de�#V � , h gk{ � , & {|\[l(gI�Y\[��x<gk{�{|g�¢ t|oR�j\bcArsZ"x5\({§³cY�Yr�m$xsd�m�t �$oj��de�#V � l[l�\(�Ym|opxsfe\ ��dhijdem��� h h _ l � h h _ l � h h � h h �Ydeikd�m �$\[l�g���\7� �$gjc:{|\[��rscs�YopcAm

� � � � � �C\[t �C\[t� � � \ ¤�\ �$g �ng� � \ � \ �C\[t �C\[t� � \ \ � �C\[t �ng� \ � � ¤~^ �C\[t �C\[t� \ � \ ¤³� �$g �ng� \ \ � ¤�\ �C\[t �C\[t� \ \ \ ¤~^ �C\[t �ng\ � � � ^ �$g �ng\ � � \ \ �C\[t �C\[t\ � \ � � �$g �ng\ � \ \ ^ �C\[t �C\[t\ \ � � � �$g �ng\ \ � \ ¤�\ �C\[t �C\[t\ \ \ � \ �$g �ng\ \ \ \ � �C\[t �C\[t

ì�ì ×��N�������

Page 19: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

\7� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

����������� �� ����������� �� �

�� � ����� � � �� ���� �� � E ���� � E �� �� ��� � �� �� � �����

� ��� � E �� � E �

���������� � ��� ��� � � � � � ������ �!� � � � ��"$#&%�')(

��dhijrY{|\&| ��*nt�rsojfucYgjc�{|\[��rYc<�YopcAm$opcs��{|\[��rYc<�YopcAmnoR¢op{|\n\(csWsopcsl(\[��twdhijc:t�\(fh\[l-mn¯0gIgpm|W�\[csl�g���\[{|tT

ree

redu

ctio

n

Boo

th E

ncod

ing

2-1

com

pres

sion

Con

ditio

nnal

Sum

Add

er

Com

pres

sed

resu

lt

+ � � � � � � � � �

, ����� � � .- � �/� �

� � !��� �� � � � �

+ � � � � � � � � � �

��dhijrY{|\� ��`b\[cY\({�opfI�YrY{|�5gkt�\0¯0gIgpm|W"Z"rsf�m|de�Yfhdh\({+¢ d�m|W" Óokt~m+ ¡\[\[��xsokl�£�l(oj�sojl(d�m|de\7tum|WY{�gkrYijW]on�s{�\7l�g���\({

à¡ê�ìuà ð

Page 20: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� \�~

�� � E � �� �

� � E � � �

� � E �� � � �

���� �

� �

�� �

��� H ��� 9 � J��� H � 9 � J��� H � � 9 � J��� H ���� 9 J

� � E �� � E �� � E �

� �

�����

�� �� �

� ��� �

�� ��� 9 J��� � � 9 � J��� � � � 9 � J��� � ���� 9 J

� � E � � �

� �

���

����� � � � � � � � ��� � � � �� � � � ���

�" � � �

� � E �� �

��� H ��� 9 � J��� � � � 9 � J��� � � 9 � J

� ��� 9 J

� ��� � � �� � � � ������� ��� � � �� � � � ���

��dhijrY{|\�~)� *nt�\� ¡rsfÁm�{�opcstw ¡gj{|Z�o�m|degkcst¦�\�oj{�\�opxYfh\¨m|g�xsrYdefq� m|WY{|\(\�rstw\( ¡rYf{�\7l�g���\[{�t�\(ijZE\[ckm�t"¢ dem�W gjrY{Eopc<opfhvIt�dqt#=Ót�\(\:��dei<� ~�G-� V/g

gkx�m|ojdec�m|WY\"l�gk{�{|\[t��5gjcs��dhcYi]{�\7l�g���\[{[¼�m|WY\"��\7twdh{|\[��tw\[ijZE\(cAm$dqt${|\(�Yfhdhl[o�m|\[� )�m�dhZ]\7t(¼<¢ WY\({|\ )�dht$m�WY\�Y\[t�de{|\[��decs�Yr�mnfh\(cYijm�Wu�C­�gI�Yd�°s\7���j\({�t�degkcst0gp �m�WY\�t�\(ikZ]\[cAm oj{�\!rst�\[�:opm0m|WY\�\[cs�Yt[¼Im�g¨oR�jgkdh��rst�decYifhgjikdhl]t�deZE�Yfhv¶m�g�ij\[cY\({�o�m|\]l(gjcstwm|ojcAm|t[��VWY\7tw\]�Y{|\[l�g���\[{|t!ojfefhg�¢�m�{�opc<t~ ¡gk{�Z�o�m|degkc�gp l(oj{�{|v¶t�oR�k\�m�gt�dhijc9��dhijdem'=Ót�\(\�°sis�}~7¤³o�ojcs�QZ �_^ª ¡gk{bo�ZEgj{|\���\(m|opdhfh\[�9�Ydht|l�rst|t�degkc�gjc�twdhijcY\7����dhijdembcsgpm|opm�dhgjc/G�opc<�gj �x5gj{|{|g�¢ t�oR�k\!gj{$l(oj{�{|v¨t|oR�j\bm�g�{�oj��de�#V�¯0gIgpm�W:{|\[l(gI�YdecYi =²��dhis��~7¤Tx:ojcs� ~7¤³lNG���VWY\! ¡gj{|Z�o�m$opc<�m|WY\! ¡{�ojl-m|degkc�{�opcsij\!op{|\�dhcs��dql(opm�\[�:op ±m�\({ \7ojl�W�{�\7l�g���dhcYis�¦ dem�W9m�WY\�m�\7l�WYcYgjfhgjikv�r<tw\7�¶dec�m�WY\E��\[t�dhijc��Y{|\[t�\(cAm|\[�:dhc ZÎ^p�_^T¼5m�Wsopmbdqt��s� ^X|��ÁZ ¬0­�©!®�m|\[l�W�¤

csgjfhgjijv ¢ dem�WN^Y� |����5g�¢�\[{¨twrY�s�Yfevk�NVWY\�oprYm�WYgk{|tEgkx�m|ojdec�o���g ����g�Z�rYfem�dh�Yfedh\({�¢ d�m|W�o l(feg�l�£l(v�l�fh\�gj �V<�]\:cst[�N�Y{|gjZ tw�sdhl(\:t�dhZ"rYfqo�m|degkcstE�Y{|\[t�\(cAm�\7�BZÎ^p�_^�¢�\�l[opc>�Y{�\7��dql-m�m�W<o�m[¼0\[okl�W {|g�¢,gj ZEg���de°s\[��Wsojf� ±¤³oj�s��\({�tb¢ defhf���\[fedh�j\[{|t!dem|t gkr�m��sr�m�¢ d�m|W o:��\(fqoRv�x5\�m~¢0\(\(cz\7�j���st�opc<� ^j�j���st�=²��\�¤�5\(c<��decsi¶gjc \(fh\[l�m�{|dhl[opfª�Y{|gj�5\({�m�dh\[t�G-�¨��t�o�{�\7twrYfem[¼/m�WY\¨l(de{�l�rsd�m��Y{|\[t�\(cAm|\[����dhijrs{�\¨�:l(ojc �<\[{w ¡gk{�Zo�cYgkc {|\[�YrYcs�Yojckm�Z�rYf�m|de�sfedql(opm�dhgjc>decUVs��\:cstEgj{E�Y{|\(�soj{�\:o�{�\7��rYcs�sopcAm�{�\7twrYfemEm�g x5\¶{�\[rstw\7� xIvojcYgpm|WY\({0Z"rsf�m|de�Yfhdql(o�m|degkc�dhc:�Y� V"cst[�C�Ygk{0\(�YopZE�Yfh\bl(gjZE�Yr�m|decYiEo��Y{�g���r<l-m gp �\7�"cIrYZ�x<\[{|t0¢ dhfef¥m�op£k\g/Rncst0¢ dem�W�m�WY\bgj{|deikdec<opf5Z"rYfem�dh�Yfhde\[{ojcs�#q�� cst0¢ dem�W�gjrY{0ZEgI�Yd�°s\7�¨gkcY\j� �$dhijWs\({ t�oR�IdhcYikt�l�gkrYfh��x<\okt�t�\[t|t�\[�¨xIv�ojc�\[fe\7l-m|{�dql(ojfutwdhZ"rYfqo�m|degkc�gj �m|WY\ � , & {�\7l�g���dhcYis�©nm|WY\({ op�s�Yfedql(opm�dhgjcstoj{�dqtw\�dec�m�WY\b ¡gk{�¢op{���dhcYi"opc<�¨ ¡\[\[��xsokl�£Egp �cIrYZ�x<\[{decAm�\[{�c<opf5m�gEo��<gko�m|decsi

�5gjdhcAm�rYcYdem m�g¶{|\[��r<l�\�{|\[��rYc<�Yopcsl(v9ojt!�s{�\7tw\[ckm|\[��deczZ V�^T�¨zC{|\[l(gI�Y\({�t op{|\]r<tw\7�9m|g� ¡gj{|¢0oj{|��twgkZE\�<op{�m|t0gp um|WY\!{�\7��rYcs�sopcAm0{|\[t�rYfem$o� ±m�\[{{�\7��rYcs�Yojcsl�v�dqt t�r��¨l�dh\(cAm�fhv�{�\7��rsl(\[�Á�ªVWYdqt opfhfeg�¢$t0opc�§ � � �twm|ojcs�Yoj{|� x5\(W<oR�Adhgj{�gp $m|WY\:{�gkrYcs��dhcYi�rYcYdemEojfex5\(dem�m|WY\:cIrYZ"x5\({�t�oj{�\�cYgpm�l�gkZE�Y{�\7t�t�\[� m�g o�cYgkc{|\[�YrYcs�Yojckm� ¡gj{|Z�o�m[�ªVWY\!l�gkcstwm�{|rsl-m|degkc¨gp �o]twm|opc<�Yop{��¨oj�s��\({0rst�decYi�m�Wsdht�m�\7l�WYcYdqyArY\�dqt��\(�sdhl�m�\[��dhcZ |K^T�

ì�ì ×��N�������

Page 21: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

\7� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

©nm|WY\({op�Y�sfedql(opm�dhgjcst�¢ defhf5oj�Y�5\[op{0dhc�l�gkcA�k\({�m�dhcYi!m�Ws\nm~¢0g�x5gjrYcs�st�gj /opc¨dhcAm�\[{���opfsm�g"m�WY\!t~m|\(�¨gj o�fhdecY\7op{!dhckm|\({|�<gkfhopm�dhgjc�ojtbd�m dqt!�<\[{w ¡gk{�ZE\7�¶ ¡gj{b Óokt~m {|\[l�dh�Y{|g�l(opf+opc<��t|yArsop{|\"{|gIgpm op�Y�Y{|gR��deZ�opm�dhgjcZÎ^p��^L�ªVWYdqt�¢ dhfhf<opfhfhg�¢ m|g� ¡gk{�¢op{����Yde{|\[l�m�fhv�m|WY\�l(gjrY�sfe\$gj u{�\7l�g���\7�Ex<gkrYcs�Yt�dec�x<gk{�{|g�¢ t|oR�j\  ¡gk{�Z�opmm|g�o�ZEg���de°s\[�:¯0gIgpm|W�Z�rYf�m|de�sfedh\({7�

� � ���E����� )(� ���"&�� ���

¦�\¨¢ dqtwW�m|g¶m�WsojcY£�m|WY\�t|l�dh\(cAm�de°<l¨l(gjZEZ"rscYd�m~v�gp m�Ws\¨§ � � � ®�vAZE�5gkt�dersZ gjc ¬�gkZE�Yr�m�\[{"�${|dem�W�¤ZE\(m�dql! ¡gj{ m|WY\(dh{nl(gjZEZE\(cAm|t gkc�m�WYdqtnt�rYx�}~\7l-m�ojcs�:tw�5\[l(dhojfefhv�z+\�m|\({��ngk{�cY\[{�rs�¨ ¡gj{nWYdhtncYgpm�o�m|degkcst$ojt�s{�\7tw\[ckm|\[��dec+Z �_^�ojcs��Wsdht £Idhcs�:opcs��£IcYg�¢ fh\[��ik\[opxsfe\bWY\(fh��g��j\({0m|WY\!m�dhZE\ gp �m�WYdqt ¢0gj{|£5�

� � ����+� ���0� .

���� ��������������������� �"!#%$'&)(+*'!$',.-"/�0�1�23��!54�2!56),.��!�%$7��&)68��$'!59�1'&%� :�;=<�>�?A@CB8?�D E�F�G.<�?AHI>�DJGLKNMOB"PAQ�>.H'RSP�TU>�HIVW�XYX D RSB�VZM[>.@\Q'B8]^>�@_R_P�TA�a`Y�%2_��b���$'���ac���4'4��'c�d%egf�chbYi��j)kYl�%�

� ch�^mA��no��-�&�$p�rqrG�] X <�@CB8? W ?ARs@\Q�]�B8@_R_P W D tYG�?ARs@\Q�]ZTA�vuj-�&)$���!6)&xwU,.252_�pgk%k�d��� d.�7y[�z�x,�1�0�,��Z,�$a({�Z�z|}�~y�,h��1'25,��~�C�N&�6)��(�&�-����s�.-^4a,h-"��!5,�2=6)��0^4�-�&)�"�"!��$�,.$a(O-���1'$a(�!$'#�� :+�N&��"&g,.-�6A�-�&�4I��-"�NkY�g��i�%�a�J,�*I�.-A,.����!-�&�(�&U2_� mC$��\��-�0�,.��!59�1'&�(�1�uj,h-A,�22�)2!5�"0^&��I��/Y�%$p�'��-A,.$'6)&%��gk�kY���

� b��^�Z�)|}��y�,.��1�25,N,.$a(� ��.y�1�$'�U�N!5&�25�"&�$p�h�Lu~!4I&)2!$'&g(x4a,.68�%&��"�S�s��-"�=,.-A(�!5$�#N�'�Y,.��!$�#N4I��!5$��g�pm8�)�\�%1�$a(',h��!�%$'�,.$a(�,�-���1'$a(�&�-g� :x!$[�=?LGgPABAB"V�R\H�t�T�GLK�@\Q�B^���%@\Q���E�] X G�T�Rs<�]�G.H�qrG�] X <�@CB8? W ?ARs@\Q�]^B8@_RSPZ�S�o�a�J,.$'#��I��� �y[��y�1�225&8-g�',.$a(^�Z���z,���,�#�!S��&g(��)� �8�p��y���$���&�-�&�/����=,�2!3�s�.-�$'!5,Y�8��4'4��a)b%ihf')b����am��v�r�����%0^4'1���&�- ����6)!&���/uj-�&)�"�)�pgk%k%���

� lh�� ���y�1'$��U�o!&)2�"&)$��%���g|¡��y�,.��1'25,����N�.�Z�.�¢/�1���,�$a(�£Z���~`Y&)$p�g��u~!54I&�25!$�&g(�4',�6A�Y&��"�S�s�.-"�¤,h-A(�!$'#¥�a�Y,h��!$'#4I��!$��g��mA�h,�$�,%(�(�&�-g� :r!$��=?LGgP8B�B"V�RsH�t�T¤GLKr@\Q'B ���%@\Q��'E.] X G.TAR\<�]¦G�H�qrG�] X <�@CB8? W ?ARs@\Q�]�B8@_R_Pv�S�o�h�p,�$'#������ �y[��y�1�225&8-g�',.$a(^�Z���z,���,�#�!S��&g(��)� �8�p��y���$���&�-�&�/����=,�2!3�s�.-�$'!5,Y�8��4'4��a)b%§hf'hl%l��am��v�r�����%0^4'1���&�- ����6)!&���/uj-�&)�"�)�pgk%k%���

� e.�� ���y�1'$��^�o!&)2�"&)$p������|}�'y�,.��1�25,��a�N��������/�1p��,.$a(�£Z���~`Y&)$����� ¥$7m��v�r�¨6)�%0^4�25!5,.$��¤�'�Y,h��!5$�#�4I��!5$��,�('(�&�-o���',.�U6��%$��\��-�0^�o� !3�������'&�4'!4I&)2!$'&)(©4',�6A�Y&��o�\��-"�=,.-A(�!$�#�4a,h-A,%(�!5#�0�� :^ª"«r«r«­¬�?">�H�T�>%P�@_RSG�H'T^G�HqrG.] X <�@CB8?ATA�'`Y�%2_��bYk��a$'���'b��'4�4p��d%dgf�bY����c.i%i%i��

� �h��uz�zy[�~��&)!5(�&)2�,�$a({£Z�j�~`Y&)$p�p��wN�g�®0�,.$�/[2��#%!6+2&)`%&)2��(���&)�x�a�Y,h��!$'#���4I��!5$���,%(�(�!3��!�%${-�&g9�1'!3-�&h¯%� :�!$��°Y°%±�ªAH'@CB8?AHI>�@_RSG.HI>%D�qrG.HgKgB8?"B8HIPABZG�H¨qrG�] X <�@CB8?U²ZB8T�R5t�H'�p�S N1��L��!$p���J&�³�,��A�8��4'4��p�b�c)fa)b%k��z)k%k�§��

� §.��uz��y[�'��&)!5(�&)2_�Y�Cwo!#%���"4I&)&)(�-�&g(�1'$'(',�$��r-�&)6)!4�-���6),�2�,�4�4�-"³�!0�,.��!�%$�� : !$©�%?"V�´ B">%D'µU<�]�¶8B8?ATx>.HIV�qrG�]�·X <�@CB8?AT^qrG�H)KgB8?�B8HaP8B8�J�Suj,.-�!�)�I��-A,�$'6�&h�8�'4�4p�'c�)khf�c%c�k��jgk�k%§��

� k.��uz��no�.-�$'&�-�1�4p�I�C�U!#%!3�"���"&��x6)��$�`Y&�-��"!�%$��x#%&)$�&�-A,�2!¸g,h��!5��$'��,�$a(©,�4�4'2!6g,.��!��$'�)� :+ª"«r«v«¹¬'?">�H�T8>YP�@_RSG�H�T7G�HqrG.] X <�@CB8?ATA�'`Y�%2_��bYd��a$'���Il��'4�4p��eYc�cgf�eYc�k��jgk�k�b��

�5)i.���o��y[���=,.-"��&�-=,�$a(������¤���¥�%*I&�-"���"��$p�Y���=�'& �"&��v���'&)�.-"/^���p,h-�!����0^&���!6o(�&)6)��0^4I�%�"!3��!�%$p� :Uª"«r«v«¡¬'?L>�H'TA·>%P�@_RSG�H'TxG�H¨qrG�] X <�@CB8?AT��I`Y�%2_�'d%k��'$����'§��a4�4p��k%k�dhf'gi%i%l��j)k%k%i��

�5���7y[�J�j-�6)&)#%�h`�,�6�,.$a(O�o�z�J,.$'#����Lºo$©-�&)6���(�!$'#�!${,h-�!3���'0^&���!6^,.25#���-�!3���'0^�)� :»F�G�<�?AHI>%D�GLKO¼a½~��ª��'Rt�HI>%D��?"GgPAB8T�T�RsH�t��a`Y�%2_��)b���4'4��Ic.§%dgf�c�k.b��z)k%k�e��

�5gch�� ��I J`�!¾)!&)$�!5�)���"��!#%$'&)(�(�!#%!3�o$�1�0�*I&�-¤-�&�4�-�&)�"&)$��A,h��!5��$'� �s�.-o�_,.�L�U4',.-A,�22&)2j,h-�!3���'0^&���!6%� :�ª"´ «®¬'?">.H'T8>YP)·@_R_G�H'T�G.H�«=D5B"P�@_?"G�H�RSP7qrG.] X <�@CB8?ATA�a`Y�%2_��gi���4'4��ad�§%kgf�bYi�i��z)k%e�%�

à¡ê�ìuà ð

Page 22: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

������������� ������������������������������ ��!#"%$&�'��"��(���)�*"��+"�,���-��.-���/��.0��1�12!��������� �������3���4�����65��� \K�

�5)d.�^�Z�J|}�ry�,h��1'25,�� W�XYX D R�B"V�qrG.] X <�@�>�@_RSG.H}¬pQ'B"G�?AE�� W HI>%D E.T�RsT � ²ZB8T�R5t�H � M[GgV�B�D RsH�t��~68���j�N,%(�!3³{ ¥-�!3�����0^&8��!56��~�U!#%!3�A,�2p N2#%�.-�!3���'0^�¤�s�.-U����0^4'1���&�-¥ ¥-�6A�'!3��&)6���1�-�&%��4'4��ad%�.b.fYb%b%§��¤uj-�&)$���!6)&xwU,.22S��gk%��e���5�b��^���~�z,.�%,.#%!¥,�$'(��I���=,��L!0�,��r�� ­�_,.�L��!��&8-A,.��!`Y&�0Z1'23��!4'2!6g,.��!��$�0^&�������({*�/{-�&)6)��(�!5$�#[!$���&�-�0^&g(�!5,.��&4�-���(�1�6��g� :O!$ ��?"GgPAB�B�V.RsH�t�T[GLK�@\Q'B[���h@\Q µx>�@_RSG�Ha>%D�qrG�H�B8H'@_RSG�H¡GLK�ªAHgK)G.?8]�>�@_RSG�H �IP�RSB8HIPAB8���_nU/Y��������%,�4a,.$I�8��gk�§Y���

�5glh�^�����z,���,�#�!_�p�� ¥-�!3���'0^&8��!56�1'$�!3�x*a,.�"&g([��$O,7�'!#%�[�"4I&)&g(»0�1'23��!4'2!&�-U� !3���O,7-�&g(�1'$a(�,�$��U*�!$a,.-"/©,�('(�!���!��$O�"-�&)&%� :�!$ W V�.>.HIPAB"V©�'Rt�HI>%Dj�=?LGgP8B8T�T�RsH�t W D tYG�?ARs@\Q�]�T � W ?"PAQ�Rs@CB"P�@_<�?"B8T�>.HIV�ªA] X D5B8]^B8H'@�>�@_RSG.H ª"ªA�`%�%2_�pgl�e�e����v��?"GgPAB�B"V�RsH�t�TZGLKU���vª"«��'4�4p�'c.b�b.f�c%l���jgk�k�%�

�5)e.��|¡�U�I�N�v-�!#%#%�[,�$'( ��� |}�Uy�,.��1�25,��x�C�N&)68�A,�$'#�1'25,.-©,h-"-A,)/¡�"!#%$�&g(¡(�!#%!3�©0�1'23��!4'2!&�-g� : �x� uj,.��&)$��l�g§�b�d�g§����x��uj,.��&�$��oº��^6�&%�p)k%k�d��

�5g�h���¥��|¡���Z��|O&)!_�Jw����U1p�J,�$'([wZ�p����&)$p���� 6)��0^4'2&�³»$�1'0Z*I&�-U0Z1'23��!4'2!&�-�1'�"!$�#+-A,%(�!3³[b�(�!#%!3���)� :^!$��?"GgPAB�B"V�RsH�t�T GLK�@\Q'BN���Y@\QZ�'E.] X G�T�Rs<�] G�H�qrG�] X <�@CB8? W ?ARs@\Q�]^B8@_RSP=���¢�hnx$��h� 2&)�z,�$a(x|}�hw���y�6g N22!�L��&�-g�&)(��)� �8�J�S��,.���p�'�~$'#%25,.$a(a�8��4'4���§�b.f�k%i��ImL�r�v� ���%0^4�1���&�-N����6)!&���/�u~-�&��"�)�p)k%kYl��

�5)§.�7�N���Z���¢/�1�,�$'(+����|}�ay�,.��1�2 ,����C�¥&g(�1�$a(',.$��=*'!$',.-"/+���������+-�&)6)��(�!$�#�� :x!$»��?"GgPAB�B"V�RsH�t�TZGLKU@\Q'BZ���Y@\Q��E�] X G�T�Rs<�]¦G�H�qrG�] X <�@CB8? W ?8Rs@\Q�]�B8@_RSP����I�gnx$'�g� 2&)�J,�$a(U|}�)wZ�.y�6g ¥225!�L��&�-g��&g(��)� �8���S�=,h���p�.�~$'#�2 ,.$a('�8�4�4p�'l�igf�l����Im��v�r� ����0^4'1���&8-N����6)!&��C/�uj-�&)�"�)�pgk%k%l��

�5)k.�^���Yy[���%&)�"��,.$'!I,.$a(�y[��u~1��"-�!$����Y�L���%0^4',.-�!�"�%$^�.�¢�"!5$�#%2&N,�$a(^(�1a,.2'4a,.�"�v0Z1'23��!4'23/�,%('(��s1'�"&)(��a�Y,h��!$'#4I��!$�� 1�$'!3���)� :�ª"«r«v« ¬'?">�H�T8>YP�@_RSG�H�T�G.H qrG�] X <�@CB8?AT��I`Y�%2_��b����a$'����k��a4�4p��kYc��gf�k%dY���jgk�k%§��

� c.i.�����jy��.-�!�B8@x>%D���p�� )i�$'�����������%* 4',.-A,�22&)2v�L�"-�1'6���1�-�&g(©�\1'22=,h-"-A,g/[0�1�23��!54�2!5&8-�� !3��� i�� l���� 6)0^�%���&�68��$'�%2��#�/�� :�ª"«r«v« F�G�<�?8Ha>%DzGLKU�aG%D RSV%·S�'@�>�@CB�qjRs?"P�<�Rs@_T��a`%�%2_�Ic.e���$'���'b��a4'4���e%i%igf�e�iYl��Jgk%k�%�

� c���7£Z�¤£o�������o�o�¤��,h�����Ny[�=�o,���,���!0�,��N,�$'( �o� ��1'�Y&)0Z1�-A,��v�� ����� ���%* -�&)#�1'25,.-�23/ �L�"-�1�6���1�-�&)(��"-�&)&0Z1'23��!4'2!&�-g� :�ª"«r«r« F�G�<�?8Ha>%DzG�K��aG%D RSV%·S�'@�>�@CB�qjRs?"P�<�Rs@_T���`Y�%2_�ac%���'$'���ak��'4�4p�¢hc�c�khf'hc.d%e��~)k%kYc��

� c�ch�^w��'y�,���!$'�»B8@=>%D��5���L ¥$"!�# !�$'�$���%�����.��*'!3�¥0�1'23��!4'2!&�-¤� !3������!#%���"4I&)&)(+-�&g(�1�$a(�,�$��¤*'!$',.-"/+,.-�6A�'!3��&)6����1�-�&%� :�ª"«r«r« F�G�<�?8Ha>%DjG�H��aG%D RSV��'@�>.@CB7qjRs?"P�<�Rs@_T��'`Y�%2_�'d�%�'$'���ae��p)k%k%e��

� c.d.�7£Z�~£o����� B8@�>�D��5�z�� ­b��)$��^6)�%0^4',�6��%���&�'���%*�0�1�23��!54�2!5&8-�1���!52!¸)!$'#©�"!#%$��"&)2&)6����v��������&)$'6)��(�&�-��)� :ª"«r«r« F�G�<�?8Ha>%DzG�Kx�aG%D RSV%·S�'@�>.@CB7qjRs?"P�<�Rs@_T��'`Y�%2_�'dYc���$'���p�%�'4'4p�¢)eY��egfa)e%§Yc��~gk%k%���

� chb������ ��y[�Jy�1'22&�-g�¢�� N2#%�.-�!����0^&)�x(�&�(�!`�!�"!�%$©4I�%1�-U0^!568-��%4�-���6�&)�"�"&)1�-����o!2521��L�"-A,.��!�%$)(+2_� ,�!5(�&^(�1»�C*�1'#�:(�1�u~&)$���!1'0�� :7¬¢B"PAQ�H'R+*�<'B�B8@v�IP�R�B8HaPABUªAHgK�G�?8]�>�@_R+*�<'B8T��a`Y��2_�p�b��a$����'§���gk�kYl��

� c�lh�^w��juz�~���a,h-A,�$'#�4a,.$'!=,.$a(¨y[�~�~�~�=,h-"���%$p�j�L���A,h��!�L��!56),�2¤,.$a,�23/��"!���.�¥�'�Y,h��!5$�#»4I�%!$��Z�a,g�®!$¨4I&)$���!51�04�-���6)&)�"�"�.-��)� :o| �'!3��&�uj,�4I&8-��%�'m�$���&)2p����-�4I�.-A,.��!�%$��Jgk�k�b��

� c.e.�^�Z�¤�x,�����,.-�0�,¨,.$a( ���¤|}�¥y�,h��1'25,��N��wo,.-A(��¤,.-�&©-�&)6)!4�-���6),�2x�A,�*�25&[6)�%0^4�-�&)�"�"!�%$-,�(�&�6)�%0^4�-�&)�"�"!�%$��&�68��$'!59�1'&)�)� :�!$ �IP�R�B8H�@_R .�P©qrG�] X <�@_RsH�tO>�HaV ¼�>%D RSVY>.@CB�V�µU<�]^B8?8RSP�T���£Z�j N2&��\&)25(��v ��z��-��%0^0^&8-g�j,�$'(�¥�a�p,�$'#��'&)(��)� �8�z�s|O1'4'4I&8-"�A,�2_�'£o&�-�0�,�$�/'�8��4'4����8f'h���I N��,%(�&�0^!56�/v&8-�2 ,.#��pgk%k%l��

� c��h�7y[�'���I��6A��1�23��&x,�$'(����a� �a����!$'&����� N4�4�-��g³�!0�,.��!$'#�&)2&)0^&)$��A,.-"/^�\1'$�6���!�%$'� � !3�����L/�0^0^&��"-�!6o*�!54',.-"��!3��&�A,.*'2&)�)� :�ª"«r«r« ¬'?">.H'T8>YP�@_RSG�H�TZG�H¨qrG�] X <�@CB8?AT��a`Y��2S��bY§��a$����'§���4'4p��§.b�cgf�§�bY���jgk�k%k��

� c.§.�^�Z�a����&)`Y&)$��"�%$OB8@¤>%D������ ¥$�,.0^&�-�!6g,�$+$a,.��!��$a,�2��L�A,�$'(',h-A(��~mL�r�r���L�A,�$'(',h-A(7�s�.-¥*'!$a,h-"/^�a�Y,h��!$'#�4I��!5$��,h-�!3���'0^&���!6%� : W qJM ��ª�0v�~½ W µ¦µ�G�@_R_P8B8T��'`Y�%2_�ac%c��'$'���Ic��'4�4p��khf�c%l��Jgk%§%���

� c.k.�^w��Iy[���x,h-�25&8/¨B8@¤>�D��5�����z2�Y,h��!$'#^4I�%!$��1,.!$���&)#�&�-o4�-���6)&)�"�"�.-N� !3���»(�!`�!5(�&x,.$a(���9�1',.-�&U-������¥�s1�$'6���!��$'�)� :�U�+uj,h��&)$�� b^§Y�.§�gk%i��-�U�+uj,.��&�$��Nº���6)&%��gk%§�k��

� d�i.��uz�p���',�!oB8@U>%D���¢�� ­hc.i�y��j�zºUuv�O�¤y�ºx���a�%,.��!$'#+4I�%!$��x4�-���6)&��"�"��-g� :^!$¨��?"GgPAB�B"V�RsH�t�T^GLK�@\Q'B7�%°Y°��qj<�T�@�G�] ªAH'@CBSt�?">�@CB"V�qjRs?LP�<�R\@_T�qrG.HgKgB8?"B8HIPAB8������,�$^�o!5&�#%���'�=,.25!3�\��-�$'!5,%�8�'4�4p�'gl��%�Afahl��%� b��Im��v�r�¨���%0��4�1���&�-N����6)!&���/+uj-�&)�"�)�J)k%k�%�

ì�ì ×��N�������

Page 23: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

^j� � �������k����.'�_5&�������k��,K�*��� ���)��1]�

� d���^w��Yn�,.*'1'�+B8@v>%D��5���L ¤6)6�1�-A,.��&¤-���1'$a(�!$'#��"68��&)0^&¤�s�.-=���'& �o&8�¤���%$��o,.4'���"�%$^0^&�������(�1'�"!$'#x-�&g(�1'$a(�,�$��*�!$a,.-"/�-�&)4�-�&)�"&)$��A,.��!��$p� :�ª"«r«v« ¬'?">�H�T8>YP�@_RSG�H�T�G.H�qrG.] X <�@CB8?ATA�'`Y�%2_��bYd��a$'���p��'4�4p��bYdgf�l�%�z)k%k�b��

� d%ch�7�I�'y[���o1'&��¢���~�'wN1p�'���auz�'u~-A,.*'��1p�',�$'(����' ���| ,h-�&%���� ¥4'4a,h-A,.��1��=�s��-o(�&8��&�-�0^!$'!$'#�����������-�&�6)��(�&�-!$�4'1��=6)�%$��"-��%2p�"!#%$a,.2�)� : �x�+uj,h��&)$��ol�c�§�i�bYd�k��-�U�+uj,h��&)$��Uº���6)&%��gk%k.b��

à¡ê�ìuà ð

Page 24: Further Reducing the Redundancy of a Notation over a ...Further Reducing the Redundancy of a Notation over a Minimally Redundant Digit Set Marc Daumas, David W. Matula To cite this

Unit e de recherche INRIA Lorraine, Technopole de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES NANCY

Unit e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnit e de recherche INRIA Rhone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unit e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnit e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

EditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399