Edita Pelantová. Katedra matematiky FJFI. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
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1 Redundantní číselné soustavy Edita Pelantová Katedra matematiky FJFI září 2011 Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
2 Problém, jak zapisovat čísla, řeší lidstvo od nepaměti. Poziční zápis, kdy stejný symbol, např. 2, použitý na různých místech vyjadřuje jednou počet dva, jindy počet dvě stě, byl umožněn vynálezem symbolu 0 pro nic. I malému školákovi je jasné, že úloha vynásobit čísla 79 a 37 zapsaná v desítkové soustavě je jednodušší než násobit tato čísla zapsaná římským způsobem LXXIX a XXXVII. Přesto se poziční zápis prosadil v Evropě až ve 13. století. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
3 Používame poziční numerační systém, určený bází β R, β > 1, a konečnou množinou celočíselných cifer nazývanou abeceda A Z β-rozvoj čísla x n x = x k β k, k= kde x k A Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
4 Používame poziční numerační systém, určený bází β R, β > 1, a konečnou množinou celočíselných cifer nazývanou abeceda A Z β-rozvoj čísla x n x = x k β k, k= kde x k A Dekadický numerační systém: báze β = 10, alphabet A = {0, 1, 2,..., 9} Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
5 Používame poziční numerační systém, určený bází β R, β > 1, a konečnou množinou celočíselných cifer nazývanou abeceda A Z β-rozvoj čísla x n x = x k β k, k= kde x k A Dekadický numerační systém: báze β = 10, alphabet A = {0, 1, 2,..., 9} Binární numerační systém: báze β = 2, alphabet A = {0, 1} Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
6 Parallel Addition as a Local Function dita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
7 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = k I x k β k I Z, I finite, x k A} dita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
8 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = k I x k β k I Z, I finite, x k A} We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
9 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = x k β k I Z, I finite, x k A} k I We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., an algorithm which rewrites in time O(1) (x k + y k )β k = } {{ } k I 1 A+A Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
10 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = x k β k I Z, I finite, x k A} k I We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., an algorithm which rewrites in time O(1) (x k + y k )β k = v } {{ } k β }{{} k k I 1 k I 2 A A+A Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
11 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = x k β k I Z, I finite, x k A} k I We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., an algorithm which rewrites in time O(1) (x k + y k )β k = v } {{ } k β }{{} k k I 1 k I 2 A A+A Definition Let A, B be alphabets, and let A Z, B Z be sets of words on these alphabets. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
12 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = x k β k I Z, I finite, x k A} k I We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., an algorithm which rewrites in time O(1) (x k + y k )β k = v } {{ } k β }{{} k k I 1 k I 2 A A+A Definition Let A, B be alphabets, and let A Z, B Z be sets of words on these alphabets. Let r, t be non-negative integers, p = r + t + 1 N, and let φ : B p A. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
13 Parallel Addition as a Local Function Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = x k β k I Z, I finite, x k A} k I We are looking for an algorithm allowing parallel addition in these numeration systems, i.e., an algorithm which rewrites in time O(1) (x k + y k )β k = v } {{ } k β }{{} k k I 1 k I 2 A A+A Definition Let A, B be alphabets, and let A Z, B Z be sets of words on these alphabets. Let r, t be non-negative integers, p = r + t + 1 N, and let φ : B p A. A function ϕ : B Z A Z acting on u B Z by ϕ(u) = v A Z, where v j = φ(u j+t... u j... u j r ), is called a p-local function, with memory r and anticipation t. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
14 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; we perform addition of two elements x, y Fin A (β) dita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
15 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; we perform addition of two elements x, y Fin A (β) x Fin A (β)... x j+t... x j+1 x j x j 1... x j r... x j A y Fin A (β)... y j+t... y j+1 y j y j 1... y j r... y j A dita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
16 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; we perform addition of two elements x, y Fin A (β) x Fin A (β)... x j+t... x j+1 x j x j 1... x j r... x j A y Fin A (β)... y j+t... y j+1 y j y j 1... y j r... y j A u j = x j + y j... u j+t... u j+1 u j u j 1... u j r } {{ }... u j A + A Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
17 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; we perform addition of two elements x, y Fin A (β) x Fin A (β)... x j+t... x j+1 x j x j 1... x j r... x j A y Fin A (β)... y j+t... y j+1 y j y j 1... y j r... y j A u j = x j + y j... u j+t... u j+1 u j u j 1... u j r } {{ }... u j A + A v j = φ(u j+t... u j r )... v j+t... v j+1 v j v j 1... v j r... v j A Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
18 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; we perform addition of two elements x, y Fin A (β) x Fin A (β)... x j+t... x j+1 x j x j 1... x j r... x j A y Fin A (β)... y j+t... y j+1 y j y j 1... y j r... y j A u j = x j + y j... u j+t... u j+1 u j u j 1... u j r } {{ }... u j A + A v j = φ(u j+t... u j r )... v j+t... v j+1 v j v j 1... v j r... v j A At the end of the process, we obtain the sum x + y = v j β j, with v = (v j ) A Z ; i.e. back again in the original set Fin A (β). Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
19 x Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
20 x Redundancy!!! Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
21 Poziční soustavy s celočíselným základem a rozšířenou množinou cifer, tedy soustavy umožňující paralelní zpracování cifer daného čísla, se dnes běžně využívají ve víceprocesorových aritmetických jednotkách. Ty se uplatňují hlavně při algoritmech pro šifrování, které pracují s velkými čísly, např. metoda RSA. Do povědomí širší odborné veřejnosti totiž vstoupily samy kvůli slavné chybě procesoru Pentium odhalené v roce Pentium (tenkráte ale i dnes) používá pro dělení algoritmus SRT a redundantní soustavu se základem β = 4 a pěti ciframi { 2, 1, 0, 1, 2}, viz Muller J.-M., Elementary Functions, Algorithms and Implementation, 2nd ed., 2006, Birkhäuser Boston. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
22 Algorithm: Base b = 4, alphabet A = { 2 1, 0, 1, 2} Input: two finite sequences of digits (x i ) and (y i ) of { 2 1, 0, 1, 2} Output: a finite sequence of digits (z i ) of { 2 1, 0, 1, 2}, such that xi 4 i + y i 4 i = z i 4 i. for each i in parallel do 0. w i := x i + y { i } wi 3 1. case then q w i = 2 and w i 1 2 i := 1 { } wi 3 case then q w i = 2 and w i 1 2 i := 1 else q i := 0 2. z i := w i q i 4 + q i 1 Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
23 Otázky Jak vypadá p-local function pro předchozí paralelní sčítání, jaké je p? Jak přepsat číslo zapsané ve tvaru x = a k 4 k, kde a k {0, 1, 2, 3} do tvaru x = b k 4 k, kde a k { 2, 1, 0, 1, 2}? Lze to udělat paralelně? Jak porovnávat dvě čísla zapsaná v abeced ve { 2, 1, 0, 1, 2}? Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
24 Algoritmus pro hledání zápisu přirozeného čísla v soustavě s bází β N. Příklad: Při hledání koeficientů a j {0, 1,,..., 6} pro zápis čísla 278 = a k 7 k a a 0 v bázi β = 7, vidíme, že číslo 278 a a 0 musejí mít stejný zbytek po dělení číslem 7. Tedy a 0 = 5. Protože 278 a 0 = 273 = 7 39 = 7 (a k 7 k a 1 ), stačí ted vyjádřit v sedmičkové soustavě číslo 39. Jeho poslední cifru a 1 získáme stejně, tedy a 1 = 4, což je zbytek po dělení čísla 39 sedmičkou. A tak pokračujeme dál. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
25 Hladový algoritmus pro hledání rozvoje v bázi β > 1: Pro zadané x R, x > 0 nalezneme maximální k Z takové, aby β k x a položíme x k = x β k. Z volby k plyne, že 1 x k < β a číslo x lze napsat jako x = x k β k + y, kde 0 y < β k. Je-li y = 0, končíme, jinak stejný postup použijeme na číslo y. Tuto proceduru opakujeme, ale nemusíme se zastavit po konečném počtu kroků. Získáme tak konečný nebo nekonečný řetězec cifer z množiny {m Z 0 m < β} a číslo x zapsané jako x = x k β k + x k 1 β k 1 + x k 2 β k Např. číslo π v sedmičkové soustavě (π) 7 = Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
26 Báze zlatý řez Příklad: Baze β = , nalezněte rozvoj čísla 2 v této bázi. Edita Pelantová (Katedra matematiky FJFI) Redundantní číselné soustavy září / 12
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