CS 130 : Computer Systems - II. Shankar Balachandran Dept. of Computer Science & Engineering IIT Madras

Transcription

1 CS 3 : Computer Systems - II Shnkr Blchndrn (shnkr@cse.iitm.c.in) Dept. of Computer Science & Engineering IIT Mdrs

2 Recp Differentite Between s nd s Truth Tbles b AND b OR NOT September 4, 27 Introduction 2

3 Recp Logicl AND Logicl OR Logicl NOT Only configurtion which turns bulb ON Only configurtion in which bulb is OFF Humn? Tll? Tll? Algorithm September 4, 27 Introduction 3

4 Logic Functions Truth Tble for n inputs Just like mthemticl functions tht you hve lernt AND, OR nd NOT re the most bsic logic functions Two other importnt functions XOR pronounced ex-or When both re sme, output = XNOR pronounced ex-nor XNOR XOR September 4, 27 Introduction 4

5 Reltionship Between XOR nd XNOR XOR XNOR Whenever XOR is, XNOR is Whenever XOR is, XNOR is XOR(,b) = NOT ( XNOR(,b) ) Also XNOR(,b) = NOT ( XOR(,b) ) September 4, 27 Introduction 5

6 Let s Enumerte Logic Functions One vrible : Wht re the possible functions on? Let s do the truth tbles f f f2 f3 September 4, 27 Introduction 6

7 Four Functions of One Vrible f : Equivlent to f : Equivlent to f2 : Equivlent to NOT() f3 : Equivlent to Nottion : NOT(A) is denoted s Pronounced -br Let s rewrite f s s functions of f = f = f2 = f3 = September 4, 27 Introduction 7

8 Functions of Two Vribles Vribles nd b Let s enumerte ll functions In Truth Tble form b f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 AND XOR OR XNOR September 4, 27 Introduction 8

9 Functions of Two Vribles (Contd.) A few tough functions b f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 b b September 4, 27 Introduction 9

10 Functions of Two Vribles (contd.) A few complex functions b f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 NOT( AND ) = NAND NOT (OR ) = NOR NOT( OR ) NOT( AND ) September 4, 27 Introduction

11 Functions of Two Vribles (contd.) Most complex functions b f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 AND (NOT b) (NOT ) AND b OR (NOT b) (NOT ) OR b September 4, 27 Introduction

12 Summry of functions One vrible 4 functions Two vrible 6 functions N 2 2 Cn you prove this? N vribles - functions Just like mthemticl functions Given function f on N vribles, f(x, x 2,, x N ) is unique x, x 2,, x N ε (,) The functions cn be enumerted Gets very lrge soon though September 4, 27 Introduction 2

13 AND, OR nd NOT Form Universl Set All functions cn be expressed s combintions of AND, OR nd NOT Eg :XOR Eg :XNOR XOR XNOR(,b) = ( AND b) OR ( AND b) XOR(,b) = ( AND b) OR ( AND b) XNOR September 4, 27 Introduction 3

14 Abstrction Using Symbols September 4, 27 Introduction 4

15 Abstrction Using Symbols September 4, 27 Introduction 5

16 Electricl Equivlence NOT Gte (Inverter) Vdd Vdd Bse Output Collector Bse Output Collector Emitter Emitter September 4, 27 Introduction 6

17 Electricl Equivlence NAND Gte Vdd Vdd Vdd Vdd Output b September 4, 27 Introduction 7

18 Electricl Equivlence NOR Gte Vdd Output b September 4, 27 Introduction 8

19 Equivlence x y NAND x y NOR September 4, 27 Introduction 9

20 Introduction 2 September 4, 27 Let s Implement XOR Let s Implement XOR XOR(,b) = ( AND b) OR ( AND b) AND y x b NOT x NOT x AND y x OR y x b AND b AND b XOR

21 XOR Using Gtes b September 4, 27 Introduction 2

22 Electricl Circuits Voltges, Currents Resistors, trnsistors etc. Well known device behviors Equtions for current, voltge etc. Lots of detils!! Work out XOR with trnsistors Evlution of function Simulte nd see Apply rules of the devices September 4, 27 Introduction 22

23 Truth Tbles Logicl function Abstrction Do not hve to worry bout implementtion All nsty trnsistor digrms re gone Mchine understndble You cn look up long the row nd column for vlues Enumertion Cn get clumsy very esily September 4, 27 Introduction 23

24 Gtes Abstrction of Functions Different from truth tbles Gtes nd Truth tbles re equivlent One cn be trnsformed to nother The bstrction using gtes is lso useful Esy to drw schemtics thn write truth tbles Or writing functions directly A nice visul representtion September 4, 27 Introduction 24

25 To Ponder Why need different levels of bstrction? Hints: Humn vs Computer understndbility Sclbility Lrger functions More vribles Automtion Correctness of results Cn you verify? How cn you utomte verifying? We will ddress some of the issues in the next clss September 4, 27 Introduction 25