Introduction CHAPTER Computers

Transcription

1 Iside a Computer

2 CHAPTER Itroductio. Computers A computer is a electroic device that accepts iput, stores data, processes data accordig to a set of istructios (called program), ad produces output i desired form. As show i Figure., it ca be abstracted as a black box that accepts iput ad produces output. The output depeds o the curret program i the block box. Computer iput is the iformatio that is submitted to a computer by a huma, by aother computer, or by its eviromet. Output is the result produced by the computer, such as texts, audio, graphs, ad pictures. Accordig to the Merriam-Webster Dictioary, data is factual iformatio (as measuremets or statistics) used as a basis for reasoig, discussio, or calculatio[2]. I computer sciece, data is such factual iformatio i a form suitable for use with a computer. Sice a computer is a electroic device, the way to store data i the computer is i the form of electrical sigals. These electrical sigals typically have two states represeted by or. Thus, all data from outside a computer are trasformed ito a represetatio that uses or whe stored or processed i the computer. The represetatio is trasformed back to a desired form for output as a result of the computatio. Iput Computer Output Figure.: Computer as a black box. 5

3 6 Itroductio CPU CU ALU RAM Figure.2: The vo Neuma architecture. A bit (a cotractio of biary digit) is the fudametal uit of iformatio i computer sciece that has two possible distict values, or. Iside a computer, data is ecoded as patters of bits (i.e., s ad s). A bit patter is a sequece (or strig) of bits. The meaig of a bit patter depeds o the iterpretatio. Sometimes it is used to represet umeric values; sometimes it represets other symbols such as characters i a alphabet; sometimes a image or audio. A byte is aother uit of iformatio. It cosists of 8 bits. Historically, a byte was used to ecode a sigle character of text. While early computers were built for specific problems ad solved mathematical ad egieerig problems, moder computers are aimed at more geeral problems i a variety of domais. Today, computers are eve embedded i automobiles, airplaes, robots, refrigerators, microwave oves, mobile phoes, MP3 players, etc. A computer system cosists of hardware ad software. Hardware is the physical compoets of the system. Software is the set of programs that istructs the hardware to obtai the output. A computer program is a set of istructios tellig the computer what to do with the iput i order to produce the output. It cotrols the actios of a machie (i.e., computer). The task of writig a program for a computer is called programmig. The perso who writes the program is called a programmer..2 The vo Neuma Architecture Most of moder computers follow the vo Neuma architecture. Theterm is derived from a proposal (First Draft of a Report o the EDVAC[3]) by the early computer scietist Joh vo Neuma ad others i 945. They proposed the idea of a geeral-purpose electroic computer with a stored program. Vo Neuma is ot the first perso who proposed the stored program cocept for

4 .2 The vo Neuma Architecture 7 computers. There are other pioeers i computer sciece, such as Ala Turig, J. Presper Eckert, ad Joh Mauchly, who proposed the same cocept. However, vo Neuma receives the pricipal credit because he istructed the rest of the world about it. A computer that follows the vo Neuma architecture cosists of four basic hardware compoets (Figure.2): iput devices, output devices, mai memory, ad cetral processig uit (CPU). There are may di eret types of iput devices, icludig keyboards, mice, hard disk drives, bar code readers, etc. They trasmit iformatio from the outside world ito mai memory. For example, whe you press a key o a computer keyboard, you sed a character to the mai memory of the computer. The character is ecoded i a sequece of electrical sigals. The, the sigals are trasmitted to ad stored i mai memory. Output devices trasmit iformatio from mai memory to the outside world. They iclude screes, priters, hard disk drives, etc. For example, a character stored i mai memory is trasmitted to a moitor i a sequece of electrical sigals. The moitor decodes the sigals to display the character o its scree. Mai memory stores both the program ad the data beig processed. Data ca be both read ad writte i mai memory as bit patters ad the locatio of data does ot a ect the access speed. This type of memory is ofte referred to as RAM (radom-access memory). I additio, mai memory is typically volatile. That is, whe the power is tured o, the iformatio stored i mai memory is lost. Mai memory does ot distiguish the type of data stored. Programs or iput/output devices are resposible for iterpretig the stored bit patter as a umber, a alphabetical symbol, or some other type. Machie code is the represetatio of a program that is actually read, iterpreted, ad executed by the computer. A program i machie code cosists of a sequece of machie istructios. A machie istructio is represeted as a fiite bit strig. The CPU carries out the istructios of a computer program. Two major compoets of a CPU are the arithmetic logic uit (ALU) ad the cotrol uit (CU). The ALU performs arithmetic ad logical operatios, ad the CU fetches istructios from mai memory, decodes them, ad executes them. The CU uses the ALU to execute the istructios whe ecessary. Mai memory is treated as primary storage. Computers have secodary storage (alteratively referred to as auxiliary storage or exteral memory) that is o-volatile. It does ot lose data stored whe the power is dow. It is typically used as both iput ad output devices, ad for storig programs ad data. The computer usually accesses its secodary storage through a itermediate space i mai memory. I moder computers, hard disk drives or solid-state disk drives are used as secodary storage. The capacity of secodary storage is typically two orders of magitude bigger tha that of mai memory. Some other examples of secodary storage are flash memory (e.g., USB sticks), floppy disks, magetic tape, puched cards, ad paper tape.

5 8 Itroductio.3 The Stored Program Cocept Early computers were desiged to perform a specific computig task. They had fixed programs. To use them for a di eret task, it was ecessary to rewire, restructure, or redesig the hardware. This requires huma itervetio. For example, ENIAC (Electroic Numerical Itegrator Ad Computer) costructed by the Uiversity of Pesylvaia i 946 was programmed by settig switches ad modifyig wirig to route data ad cotrol sigals. Coceptually, programs ad data are very di eret. The stored program cocept meas that we treat programs as data, ad they ca both be stored i mai memory. With this, the program for a specific task is loaded ito mai memory (e.g., from secodary storage), ad istructios i the program are executed oe after aother without ay huma itervetio. The program is easily replaced by aother program for a di eret task whe ecessary. By storig a program i mai memory, the hours of tedious labor required to reprogram computers ca be elimiated. A moder computer ca solve almost a ifiite variety of problems by just switchig betwee di eret programs..4 Files Accordig to the Merriam-Webster Dictioary, a file is a complete collectio of data treated by a computer as a uit especially for purposes of iput ad output[2]. Computer files ca be cosidered as the couterpart of traditioal files that are kept i o ces. The primary purpose of a computer file is to store data i a more permaet form. Files are typically stored i a secodary storage device. The cotets of a file are ecoded i a sequece of bits. The meaig of the bits totally depeds o the iterpretatio by the program that accesses the file. I geeral, there are two kids of computer files: text files ad biary files. The cotets of a text file are iterpreted as character symbols. Other files tha text files are biary files. A biary file cotais ay type of data ecoded i bits. Text files are cosidered to be di eret from biary files i geeral because biary files cotai more tha just textual data, such as formattig iformatio ecoded i bits. A character is ecoded as a bit strig i a text file. ASCII (America Stadard Code for Iformatio Iterchage) developed by ANSI (America Natioal Stadards Istitute) is the most commo character codig scheme for Eglishlaguage text files. ASCII uses 7 bits for each character symbol. Thus, 28 (2 7 ) di eret character symbols ca be defied with ASCII. However, each byte i a ASCII text file cotais a sigle character. Figure.3 shows the ASCII character table. ASCII was developed a log time ago ad desiged actually for use with teletypes. The first o-pritig characters are rarely used ow. Recetly, the Uicode Cosortium has developed a character codig scheme, called Uicode, to assig a uique value to every character symbol used i every laguage i the world. This allows texts from multiple laguages to appear i

6 .4 Files 9 Dec Bi Char Dec Bi Char Dec Bi Char Dec Bi Char NUL SPC 96 ` SOH 33! 65 A 97 a 2 STX 34 " 66 B 98 b 3 ETX 35 # 67 C 99 c 4 EOT 36 $ 68 D d 5 ENQ 37 % 69 E e 6 ACK 38 & 7 F 2 f 7 BEL 39 ' 7 G 3 g 8 BS 4 ( 72 H 4 h 9 TAB 4 ) 73 I 5 i LF 42 * 74 J 6 j VT K 7 k 2 FF 44, 76 L 8 l 3 CR M 9 m 4 SO N 5 SI 47 / 79 O o 6 DLE 48 8 P 2 p 7 DC 49 8 Q 3 q 8 DC R 4 r 9 DC S 5 s 2 DC T 6 t 2 NAK U 7 u 22 SYN V 8 v 23 ETB W 9 w 24 CAN X 2 x 25 EM Y 2 y 26 SUB 58 : 9 Z 22 z 27 ESC 59 ; 9 [ 23 { 28 FS 6 < 92 \ GS 6 = 93 ] 25 } 3 RS 62 > 94 ^ 26 ~ 3 US 63? 95 _ 27 DEL Figure.3: The ASCII table.

7 Itroductio a sigle text file. A character symbol i Uicode uses 6 bits (2 bytes). Thus, Uicode ca represet up to 65,536 (2 6 ) symbols. Di eret sectios of Uicode are allocated to character symbols from di eret laguages..5 Operatig Systems Applicatio software (also kow as a applicatio) is a set of programs that helps the user to carry out a specific task. For example, a spreadsheet is accoutig software that helps the user to calculate umbers ad orgaize iformatio i a tabular form (e.g., colums ad rows). I cotrast, system software is a set of programs desiged to operate the computer hardware ad to provide a platform for ruig applicatios. Especially, a operatig system (also kow as a OS) is system software that cotrols the operatio of a computer ad directs the processig of programs. It maages computer hardware resources ad provides commo services for applicatio software. A applicatio software ask a service to the operatig system whe ecessary. The operatig system assigs storage spaces i mai memory, cotrols iput ad output fuctios, moitors the uderlyig computer system, etc. It is typical that a user caot ru a applicatio o the computer without a operatig system. Popular operatig systems iclude Microsoft Widows, Mac OS, Liux, Uix, etc. Utility software is system software desiged to help the user maage ad tue the computer hardware ad software. It usually focuses o how the computer hardware ad software operates. It is also referred to as utility, tool, ad service program. Examples of utility software may iclude virus scaers, data compressio utilities, disk partitio utilities, archive utilities, system moitors, text editors, assemblers, etc..6 Programmig Laguages A programmig laguage is a formal laguage i which computer programs are writte. Most programmers write their programs i a high-level programmig laguage, like Java, C, C++, FORTRAN, Scheme, ML, etc. The level of abstractio from the details of the uderlyig computer i a high-level programmig laguage is higher tha a low-level programmig laguage. It is more close to atural laguages ad more uderstadable tha a low-level laguage. Thus, a high-level programmig laguage makes the process of developig programs simpler ad easier. A assembly laguage is a typical example of low-level programmig laguages. It represets machie istructios symbolically. The represetatio is defied by the hardware maufacturer ad specific to a computer architecture. There exists a assembly istructio that correspods to a machie istructio, but ot vice versa. A program called a assembler is used to traslate assembly laguage istructios ito the target computer s machie code istructios.

8 .6 Programmig Laguages #iclude <stdio.h> 2 3 it mai() 4 { 5 pritf("hello, world\ ); 6 } Figure.4: A example C program hello. hello.c source program (text file) Compiler hello executable object program (biary file) Figure.5: The compilatio process. The defiitio of a particular programmig laguage cosists of both sytax (how the various symbols of the laguage are combied) ad sematics (the meaig of the laguage costructs). The sytax ad sematics of a programmig laguage are typically defied i its specificatio. The C programmig laguage was developed from 969 to 973 by Deis Ritchie of Bell Laboratories. It was desiged for a practical purpose to implemet the UNIX operatig system. As UNIX became popular, may software developers were exposed to C. Although it was origially desiged as a systems programmig laguage, it ca be used for writig programs i a variety of differet domais from busiess to egieerig. It is oe of the most widely used programmig laguages i these days. I 978, Bria Kerigha ad Deis Ritchie published the classical book o C, The C Programmig Laguage[]. This book was kow to programmers as K& R ad had served as a iformal specificatio of C util 989. I 989, ANSI itroduced C89 stadard for the C programmig laguage. The same stadard was adopted by ISO (Iteratioal Orgaizatio for Stadardizatio) i 99 ad called C9. The stadard was further revised, leadig to ANSI/ISO C99 stadard that was published i 999. C99 is a iteratioally recogized C laguage specificatio, ad almost all C compilers follow this stadard. The stadard promotes portability, reliability, maitaiability, ad e ciet executio of C programs o a variety of machies. Figure.4 shows a example C program hello. It is itroduced i K&R, ad prits hello, world o the scree whe executed. The programmer creates the program with a text editor ad saves it i a text file hello.c. The text file is stored i secodary storage. A fileame extesio c is used to idicate that the

9 2 Itroductio Algorithm C laguage Assembly laguage Istructio set architecture Computer Sciece ad Egieerig Computer hardware (microarchitecture) Logic gates ad Boolea logic Electroic circuits Figure.6: Abstractios i computer sciece ad egieerig. text file cotais a C program. A fileame extesio is a su x to the fileame (e.g., c that is separated from the base fileame hello by a dot). It idicates the cotet type of the file. A text editor is a utility program to create ad modify text files. Commoly used text editors iclude GNU emacs, UNIX vi, Microsoft word, etc. To ru the hello program o a computer, the C statemets i the source program must be traslated by a compiler ito a sequece of machie istructios. A compiler is a program that automatically traslates aother program from some programmig laguage to machie code. The resultig machie istructios are cotaied i a biary file called a executable object file (or simply executable). After compilig the source file hello.c, the resultig executable hello i Figure.5 is stored i secodary storage. It is ready to be loaded ito the computer s mai memory ad executed..7 Abstractios i Computer Sciece Abstractio plays a key role i computer sciece. Computer sciece is fudametally a sciece of abstractio. The cocept of abstractio is pervasive i may arts ad scieces. Abstractio is the process of cosiderig the exteral properties of a object idepedetly of its iteral details. I other words, by abstractig a object, we are iterested i what the object does without ay iterest i how the object does it. To solve a complex problem, we devise a uderstadable model for it through abstractio. The, we explore appropriate methods to solve it usig the model. For example, the behavior of electroic circuits used to build computers ca be modeled very well by logic gates ad Boolea logic. Although this modelig is ot exact, the model abstracts away may details, such as the behavior of trasistors ad electrical sigals betwee

10 .7 Abstractios i Computer Sciece 3 them. We are able to desig, aalyze, ad maage a complex computer system by applyig abstractios. I this case, abstractios are built layer upo layer. As show i Figure.6, the layer of abstractio starts from electroic circuits. Electroic circuits required to build a computer abstract away the requiremets of the particular solid-state device techology (e.g., CMOS, NMOS, gallium arseide, etc.) used to build the circuit. The, as metioed before, the behavior of the electroic circuits is modeled with logic gates ad Boolea logic. At the ext layer, each compoet of computer hardware (e.g., ALU, CU, ad mai memory) is implemeted with logic gates ad Boolea logic. There are various choices of hardware compoet implemetatios. They di er i performace, power cosumptio, ad cost. However, at the computer hardware level, we do ot worry about such implemetatio details. A istructio set architecture (ISA) is the complete specificatio of the iterface betwee the programmer ad the uderlyig computer hardware. The ISA, ot the computer hardware, is visible to the programmer whe the programmer writes a program. A ISA is implemeted by a microarchitecture that describes how the ISA should be implemeted. Similar to the hardware compoet implemetatios with logic gates, microarchitectures have trade-o s betwee performace, power cosumptio, ad cost. For example, the x86 ISA was origiated by Itel ad has bee implemeted by microprocessors from both Itel ad AMD with radically di eret microarchitectures. O top of the ISA, the assembly laguage provides oe abstractio level. It implemets a symbolic represetatio of machie istructios i the ISA to make the programmig easier. A assembler is the iterface betwee the assembly laguage ad the ISA. The assembly laguage is also abstracted to the C laguage layer i a similar maer. The C compiler is the iterface betwee the C laguage ad the assembly laguage. A C program is traslated ito a assembly laguage program by the C compiler. High-level laguages like C make writig a program simpler ad easier tha a low-level laguage, such as machie ad assembly laguages. Moreover, programs writte i highlevel laguages ca be trasferred from oe computer to aother with little modificatio as log as the target computer has a compiler for the laguage. A algorithm is a sequece of steps for carryig out a task or solvig a problem. Each step is precisely ad uambiguously specified ad ca be carried out by a computer. A algorithm is expressed formally as programs writte i programmig laguages. As show i Figure.6, a algorithm ca be implemeted with a C program.

11 CHAPTER 2 Number Systems Iside computers, iformatio is ecoded as patters of bits because it is easy to costruct electroic circuits that exhibit the two alterative states, ad. The meaig of bits depeds o the iterpretatio. Sice the iformatio is processed by the computer essetially i some umerical form, we explore the ways how a computer represet umeric data iterally i this chapter. 2. Positioal Number Systems A umber system is a system of represetig umbers. It is defied by a set of basic symbols called digits or umerals, ad the ways i which the digits ca be combied to represet the umbers i the system. A umber system ca represet itegers, fractios, or mixed umbers. A mixed umber has two parts: a iteger part that tells you the whole ad a fractio part that is less tha oe whole. A radix poit separates the iteger part ad the fractio part. Sice the decimal umber system is the most familiar umber system ad used i our everyday life, our discussio begis with it. The decimal umber system is a positioal umber system. I a positioal umber system, a umber is represeted by a strig of digits. The value of each digit i the strig is determied by the positio it occupies i the umber. That is, each digit positio has a weight associated with it. The rightmost positio i the umber has the lowest weight. The leftmost digit i the umber is called the most sigificat digit (MSD) ad the rightmost digit is called the least sigificat digit (LSD). Other examples of commoly used positioal umber systems iclude biary, octal, ad hexadecimal umber systems. I the decimal umber system, there are digits:,, 2, 3, 4, 5, 6, 7, 8, ad 9. The decimal umber of the form d p d p 2...d d.d d 2...d has the 5

12 6 Number Systems Name Base Digits Biary Octal Decimal Hexadecimal 2, 8,, 2, 3, 4, 5, 6, 7,, 2, 3, 4, 5, 6, 7, 8, 9 6,, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Table 2.: Commoly used umber systems with their bases ad digit sets. value, Xp i= d i i All the weights i the decimal system are powers of the umber. The iteger ad fractio parts are separated by the. symbol, the decimal poit. The fractio part is deoted by a sequece of digits whose weights are egative powers of. The decimal umber system is also kow as the base- umber system because the weight of each positio i the umber is a power of. For example, the decimal umber ca be decomposed as follows: = Whe we replace the base with some other whole umber r, called the base or radix, the we have base-r umber system. This umber system requires r distict digits, ad the weight i positio i is r i. Thus, the base-r umber of the form x p x p 2...x x.x x 2...x has the value, Xp i= x i r i where each x i comes from the set of r distict digits. Traditioally, the first r decimal digits serve as the digits for the base-r systems whe r apple. Whe r>, the first r uppercase letters of the alphabet are used to provide additioal symbols. For example, the three-digit umber deotes five i the biary umber system whereas it deotes oe hudred ad oe i the decimal system. Whe the base of a base-r umber N is uclear from the cotext, we apped a subscript r to it. For example, 2 represet a biary umber. Table 2. shows the bases ad digit sets for some commoly used umber systems. 2.2 Scietific Notatio Scietific otatio is a scheme of represetig decimal umbers that are very small or vary large. I scietific otatio, a umber is represeted i the form: x y

13 2.3 Biary Number System 7 Where x is called the coe ciet (also called the sigificad or matissa) ad is ay real umber, ad y is called the expoet ad must be a iteger. For example, ca be writte as. 7 i scietific otatio. Sice there are may ways to represet a umber i the form of x y, we adopt the covetio of makig the sigificad, x, always i the rage: apple x <. This otatio is ofte referred to as ormalized scietific otatio. Note that we ca ot express zero with the ormalized scietific otatio. I a positioal umber system, the sigificat digits of a umber are those digits whose removal chages the umerical value associated with the umber. Aumber sprecisio or accuracy is the umber of sigificat digits it cotais. Leadig zeroes of a umber are ay cosecutive zeroes that appear i the leftmost positios of the umber s represetatio. O the other had, trailig zeroes are ay cosecutive zeroes i the umber s represetatio after which o other digits follow. Leadig zeroes are isigificat ad do ot a ect the umerical value. Similarly, trailig zeroes that appears to the right of a radix poit do ot a ect the value. They are merely placeholders to idicate the size of the represetatio. For example, the removal of leadig ad trailig zeroes i 3.4 does ot a ect the value of the represetatio. Base-r umbers expressed with a fixed umber of digits ofte have a implicit radix poit at some fixed positio. For example, if the umber is a iteger, the radix poit is immediately to the right of its LSD. If it is a fractio, the radix poit is immediately to the left of its MSD. These umbers are referred to as fixed-poit umbers. I cotrast, floatig-poit umbers have a radix poit that ca be placed aywhere relative to their sigificat digits: the radix poit ca float. The positio of the radix poit is idicated separately ad ecoded i the umber s represetatio. Scietific otatio is closely related to the floatigpoit umbers. We will describe later how the computers express floatig-poit umbers iterally. 2.3 Biary Number System Sice it was easier to build electroic circuits that distiguish betwee two di eret values tha more tha two values, computers use Boolea logic, which is a two-valued logical system, to abstract away the details of electroic circuits. Cosequetly, the biary umber system are directly related to the Boolea logic used i the computer, ad the computer iterally represets umeric data i a biary form. As show i Table 2., the biary umber system uses ad as its digits. The geeral form of a biary umber is b p b p 2...b b.b b 2...b, ad it has the followig iterpretatio, For example, Xp i= b i 2 i 2 = = 9

14 8 Number Systems Biary Octal Hexadecimal Decimal A 3 B 4 C 2 5 D 3 6 E 4 7 F 5 Table 2.2: The correspodece betwee 4-bit biary, octal, hexadecimal, ad decimal umbers.. 2 = =5.25 Similar to the decimal umber system, the leftmost bit b p is called the most sigificat bit (MSB) ad the rightmost bit b is called the least sigificat bit (LSB). The radix poit i the biary umber system is referred to as the biary poit. The octal umber system is useful for represetig multi-bit biary umbers. Sice eight is a power of two (8 = 2 3 ), three bits i a biary umber ca be uiquely represeted with a sigle octal digit. For example, 2 = 2 = =. 2 = Similarly, the hexadecimal umber system is also useful for represetig multi-bit biary umbers. Each group of four bits i a biary umber ca be uiquely represeted by a sigle hexadecimal digit. For example, 2 = 2 =8CE 6. 2 =. 2 =2.B2C 6 Table 2.2 shows the correspodece betwee 4-bit biary, octal, hexadecimal, ad decimal umbers.

15 2.4 Number System Coversio 9 Let N be the decimal umber ad b-b-2 b be the biary umber after the coversio.. Set i to. 2. Divide N by 2 ad obtai a quotiet ad a remaider. 3. Set bi to the remaider ad N to the quotiet. 4. If N is ot zero, set i to i+ ad go to step 2. Figure 2.: A decimal to biary coversio algorithm for a iteger. i N b i 8/2 = 54 (LSB) 54/2 = /2 = 3 3 3/2 =6 4 6/2 =3 5 3/2 = 6 /2 = (MSB) Figure 2.2: Covertig 8 to biary. A biary umber system usig bits ca represet 2 di eret umbers. Whe such a fixed precisio biary umber is used to represet oly positive values, it is called a usiged umber. It is also possible to ecode siged umbers (positive umbers ad egative umbers) i biary. There are several ways to represet the siged umbers i biary. Oe way of represetig a egative umber is settig the MSB to ad usig the remaiig bits to represet the value. I this case, the MSB is referred to as the sig bit. However, to keep the computer hardware implemetatio as simple as possible, almost all today s computers iterally use twos complemet represetatio. 2.4 Number System Coversio Sice humas typically use decimal umbers, but computers use biary umbers iterally, it is importat to kow how to covert decimal umbers to biary umbers ad vice versa. A iteger N ca be coverted from decimal form to biary form by repeatedly dividig N by two ad usig the remaiders. Figure 2. shows a algorithm for such a coversio. For example, a positive iteger 8 is coverted to the biary umber 2. The coversio process that follows the algorithm described i Figure 2. is illustrated i Figure 2.2.

16 2 Number Systems Let N be the decimal fractio ad b-b-2 b- be the biary umber after the coversio.. Set i to. 2. Multiply N by 2 ad set N to the result. 3. If N is less tha, set b-i to. 4. Otherwise, set b-i to ad N to N-. 5. If i is less tha, set i to i+ ad go to step 2. Figure 2.3: A decimal to biary coversio algorithm for fractios. i N b i = = = =.76 Figure 2.4: Covertig.73 to a 4-bit biary fractio. For a mixed umber, we covert its iteger part ad fractio part to biary form separately. The, we combie the results. A algorithm for covertig a decimal fractio to biary form is show i Figure 2.3. To obtai the first bits of a biary fractio usig the algorithm, it is required to perform multiplicatios by two i geeral. For example, the process of covertig a decimal fractio.73 to a 4-bit biary fractio. 2 is illustrated i Figure 2.4. After combiig the results from the iteger coversio ad the fractio coversio, a mixed umber 8.73 i the above examples is coverted to a biary umber. 2. However, the 4-bit biary fractio obtaied i the process of Figure 2.4 is iexact. So far, we assumed that there are as may bits available as required to represet the umber, ad we did ot cosider the size of the represetatio. Whe the umber of bits used is specified to represet a umber, it determies the rage of possible umbers that ca be represeted. For example, 8 bits ca represet 256 = 2 8 distict umeric values. Word is a group of bits that are hadled as a uit by the computer. The umber of bits i a word is a importat characteristic of a computer architecture. The precisio of expressig a umerical quatity strogly depeds o the word size (i.e., the umber of bits i a word). If the word size is oly four bits, the we ca ot express the result of the above example more precisely. A dyadic fractio is a ratioal umber whose deomiator is a power of two;

17 2.4 Number System Coversio 2 i N b i 8/8 = 3 4 (LSB) 3/8 = 5 2 /8 = (MSB) i N b i = = =. 4. 8= Figure 2.5: Covertig 8.73 to a octal umber. i N b i 8/6 = 6 2(C) (LSB) 6/6 = 6 (MSB) i N b i =.76 (B) = 2.6 2(C) = = Figure 2.6: Covertig 8.73 to a hexadecimal umber. for example, /2 or 3/6, but ot /3. Dyadic decimal fractios covert to fiite biary fractios ad are represeted i full precisio if the word size is greater tha or equal to the legth of the biary represetatio. No-dyadic decimal fractios covert to ifiite biary fractios (a repeatig sequece of the same bit patter). Trucatio (also kow as choppig) is the process of discardig ay uwated digits of a umber. Roudig a umber replaces the umber with aother umber that is as close to the origial umber as possible. Decimal to octal ad decimal to hexadecimal coversios are similar to the decimal to biary coversio. A iteger ca be coverted from decimal form to octal (or hexadecimal) form by repeatedly dividig it by eight (or sixtee) ad usig the remaiders. A decimal fractio coverts a octal (or hexadecimal) fractio with digits after performig successive multiplicatios by eight (or sixtee). For example, the coversio process of a mixed umber, 8.73, to octal ( ) ad hexadecimal (6C.BC28 6 ) are illustrated i Figure 2.5 ad Figure 2.6, respectively. Aother way to perform decimal to octal or decimal to hexadecimal coversio for a decimal umber is that covertig it to a biary umber. The, the biary umber ca be coverted to a equivalet octal or hexadecimal umber. For example, 8.73 is coverted to octal ( ) ad hexadecimal (6C.BC28 6 ) i the followig way: 8.73 =. 2 = =. 2 = 6C.BC28 6

18 22 Number Systems 2.5 Usiged Biary Arithmetic Arithmetic operatios performed o usiged biary umbers are much like arithmetic i the decimal umber system that we kow very well. Additio, subtractio, multiplicatio, ad divisio ca be performed o biary umbers. A pair of usiged biary umbers ca be added bit by bit accordig to the same method used i decimal additio. Sice computers use di eret stadard to ecode a mixed umber (typically usig the IEEE 754 stadard), we will restrict our discussio to usiged whole umbers. For example, a pair of usiged 4-bit biary umbers ad ca be added as follows: carry + Whe addig two s i the same colum produces 2, (called carry) will have to be added to the left colum of more sigificat bits. Whe the result of additio exceeds oe, we carry the excess amout to the ext positio usig a carry. The positio has a weight that is higher by a factor equal to two. A overflow occurs whe a computatio produces a value that falls outside the rage of values that ca be represeted. For example, addig two usiged 4-bit biary umbers ad produces a 5-bit value, ad it is a overflow: carry + The carry bit from the MSB tell us that a overflow occurred i the additio. Usiged biary subtractio works i much the same way as decimal subtractio. For example, ca be subtracted from produces as follows: borrow Subtractig a from a i a colum produces by borrowig (called borrow) from the left colum of more sigificat bits. Whe the result of a bit by bit subtractio i a colum is less tha, we borrow from the colum o the left subtractig it from the ext higher positioal value. Usiged biary multiplicatio is also similar to the decimal multiplicatio that adds up partial products to produce the fial result. For example, two

19 2.6 Oes Complemet Represetatio 23 Positive Negative Oes complemet Decimal Oes complemet Decimal Table 2.3: 4-bit oes complemet represetatio. 4-bit usiged biary umbers ad are multiplied as follows: Note that the result ca ot be represeted with 4 bits. I geeral, a -bit biary multiplicatio requires 2 bits to represet the result. Usiged biary divisio is agai similar to the decimal divisio. For example, the process of dividig by produces a quotiet ad a remaider. ) 2.6 Oes Complemet Represetatio Oes complemet represetatio is oe of the methods to represet a egative umber. The oes complemet of a egative biary umber is the bitwise iversio (flippig s for s ad vice-versa) of its positive couterpart. For example, the oes complemet represetatio of 6( 2 ) is 2. The maximum value that ca be represeted by the oes complemet represetatio with bits is 2 ad the miimum is 2. Table 2.3 shows

20 24 Number Systems the umbers that ca be represeted i the oes complemet represetatio with 4 bits. The MSB is the sig bit. Ulike two s complemet represetatio (explaied later), it has two zeroes. For example, 2 ad 2 are the two 4-bit oes complemet represetatios of zero. Addig two biary umbers i the oes complemet represetatio is similar to the usiged biary additio, but there is a coditio called a ed-aroud carry that does ot occur i usiged biary additio. Cosider addig two biary umbers ad i oes complemet represetatio: carry (4) + ( ) (add the ed-aroud carry) (3) For the additio i the oes complemet represetatio, we perform a usiged biary additio. If there is a carry from the MSB (this is called a ed-aroud carry), the add the carry back ito the resultig sum to produce the fial result. Ulike the two s complemet represetatio, the oes complemet represetatio is ot popular i these days because of several issues like egative zero, ed-aroud carry, etc. 2.7 Two s Complemet Represetatio The twos complemet represetatio is the most commo method of represetig siged itegers iterally i computers. It simplifies the complexity of the ALU by allowig usig the same circuit that is used to implemet arithmetic operatios for usiged umbers. The major di erece is the iterpretatio of the result produced by the arithmetic operatio. To covert a umber to two s complemet represetatio, the umber is coverted to its oes complemet first. The oe is added to the result to produce its two s complemet. Aother way of ecodig a egative umber to its twos complemet is flippig all bits but the first least sigificat ad all the trailig s. For example, the twos complemet represetatio of 6 ( 2 ) is 2. The maximum value that ca be represeted by -bit two s complemet represetatio is 2 ad the miimum is 2. Table 2.4 shows the umbers that ca be represeted i the two s complemet represetatio with 4 bits. The MSB is the sig bit. Two s complemet additio is exactly the same as that of usiged biary umbers. For example, addig 4 ( 2 ) ad 5 ( 2 ) i 4-bit two s complemet represetatio gives ( 2 ). Subtractio ca be hadled by covertig it to additio: x y = x +( y)

21 2.8 Shift Operatios ad Sig Extesio 25 Positive Two s complemet Decimal Negative Two s complemet Decimal Table 2.4: 4-bit two s complemet represetatio.. Add the two operads icludig the sig bits. 2. If the carry ito the MSB is ot equal to the carry out of the MSB, a overflow has occurred. Figure 2.7: Overflow detectio i two s complemet additio. Overflow occurs if + bits are required to cotai the result from a -bit additio or subtractio. If the sig bits were the same for both umbers ad the sig of the result is di eret to them, a overflow has occurred. The process of detectig a overflow after addig two operads is show i Figure 2.7. For example, a overflow occurs i the followig 4-bit two s complemet additio: carry (7) + (7) I this case, the carry ito the MSB () is di eret to the carry () out of the MSB. 2.8 Shift Operatios ad Sig Extesio A shift operatio shifts all of the bits i its operad. Every bit i its operad is moved a give umber of positios to a specified directio. There two di eret types of shift operatios: logical shift ad arithmetic shift. A logical shift operatio moves bits to the left or right. The bits that fall o at the ed of the word are discarded. The vacat positios i the opposite

22 26 Number Systems x>>3 x<<3 Logical shift Arithmetic shift Figure 2.8: The di erece betwee 8-bit logical shift ad arithmetic shift for x =. w bits w+k bits (a) Decimal 4-bit 8-bit 6-6 (b) Figure 2.9: (a) The sig extesio process. (b) The sig-bit repetitio for -6. ed are filled with s. The arithmetic left shift is the same as the logical left shift. I the arithmetic right shift, the leftmost bits are filled with the sig bit of the origial umber. We deote a left shift operatio with << ad right shift operatio with >>. Figure 2.8 shows a example of the di erece betwee logical shift ad arithmetic shift for x =. Usig shift operatios, we ca e cietly perform usiged multiplicatio or divisio by powers of two. A -bit logical left shift operatio o usiged itegers is equivalet to multiplicatio by 2. For example, 2 (3) << 2 = = 2 (52) A -bit logical right shift is equivalet to divisio by 2, ad we obtai the quotiet of the divisio. For example, 2 (53) >> 2 = = 2 (3) Usig arithmetic shifts, we ca e cietly perform multiplicatio or divisio of siged itegers by powers of two. I two s complemet represetatio, arithmetic shift operatios exted the otio of the floor operatio. The floor of a iteger x (deoted by bxc) is defied by the greatest iteger less tha or equal to x. For example, b/2c = 5 ad b 3/2c = 2. A arithmetic shift operatio o a iteger takes the floor of the result of multiplicatio or divisio. For example, 2 ( 4) << = 2 ( 6) 2 ( 4) >> = 2 ( 2) 2 ( 4) >> 2 = 2 ( ) 2 ( 4) << 2 = 2 () Overflow 2 (4) << = 2 ( 8) Overflow

23 2.8 Shift Operatios ad Sig Extesio 27 However, i two s complemet represetatio, a arithmetic right shift is ot equivalet to divisio by a power of two. For example, whe you perform a arithmetic right shift o a 4-bit = 2, you still get = 2.The ANSI C99 stadard does ot specify the defiitio of the C laguage s right shift operator for egative values. The behavior of the right shift operator depeds o the C compiler. Whe covertig a iteger with w bits ito the oe with w + k bits ad the same value, we eed to perform sig extesio i two s complemet represetatio. The MSB (the sig bit) must be repeated i all the k extra bits. Figure 2.9(a) shows this process ad Figure 2.9(b).

24 CHAPTER 3 Logic Circuits The elemetary buildig blocks of a CPU are logic gates. The iput ad output of logic gates assume oly two values, ad. Their behavior is well modeled by laws of Boolea algebra. We start out this chapter with Boolea algebra, ad lear how a arithmetic logic uit ad mai memory are built from logic gates. 3. Boolea Algebra Boolea algebra was developed by George Boole i 854. It is a algebraic represetatio of logic ad the algebra of truth values: true ad false. Edward V. Hutigto gave the formal defiitio of Boolea algebra i 94. It is referred to as Hutigto s postulates. I 937, Claude Shao applied Boolea algebra to electrical switchig circuits to reaso about the behavior of etworks of relay switches i 937. I the rest of this book, the two-elemet Boolea algebra (also kow as switchig algebra) defied by Shao is used ad referred to as Boolea algebra. The two-elemet Boolea algebra is a set of two elemets, B = {, }, together with three operatios _ (OR), ^ (AND), ad (NOT). It satisfies the followig axioms: Axiom (Closure). For all x ad y i B both x _ y ad x ^ y are i B. Axiom 2 (Idetity elemet). There exist distict elemets ad i B such that for all x i B, x _ =x ad x ^ =x. Axiom 3 (Commutativity). For all x ad y i B, x _ y = y _ x ad x ^ y = y ^ x. Axiom 4 (Distributivity). For all x, y, adz i B, x_(y ^z) =(x_y)^(x_z) ad x ^ (y _ z) =(x ^ y) _ (x ^ z). 29

25 3 Logic Circuits OR AND NOT x y x y x y x y x x Figure 3.: The three operatios i Boolea algebra: OR, AND, ad NOT. x y z f(x,y,z) = ((x y) z ) Figure 3.2: The truth table of a Boolea fuctio f(x, y, z) =((x _ y) ^ z ). Axiom 5 (Complemet). For each x i B, there exists a elemet i B, deoted x ad called the complemet or egatio of x, such that x_x =ad x^x =. Axiom 6 (Cardiality). There are at least two distict elemets i B. The three operatios, OR (_), AND (^), ad NOT ( ) are defied as follows: x _ y = x ^ y = x = whe either x or y is, or both are otherwise whe both x ad y are otherwise whe x = otherwise Figure 3. defies these operatios with truth tables. The truth table of a Boolea operatio is a list of all combiatios of s ad s that ca be iputs to the operatio ad a list that shows the value of the operatio. There are some other properties (i.e., theorems) of Boolea algebra that ca be derived from the axioms:

26 3. Boolea Algebra 3 f(x, y) xy Commet Costat x y x ad y (AND) x y x but ot y x x x y y but ot x y y (x y ) (x y) x or y but ot both (XOR) x y x or y (OR) (x y) NOR (x y) (x y ) x equals y (XNOR) y NOT y x y If y the x x NOT x x y If x the y (x y) NAND Costat Figure 3.3: The 6 Boolea fuctios of two iput variables, x ad y Property (Uiqueess of ad ). ad are uique. Property 2 (Idempotecy). For all x i B, x _ x = x ad x ^ x = x. Property 3. For all x i B, x _ =ad x ^ =. Property 4 (Absorptio). For all x ad y i B, (x_y)^x = x ad (x^y)_x = x. Property 5 (Associativity). For all x, y, adz i B, (x _ y) _ z = x _ (y _ z) ad (x ^ y) ^ z = x ^ (y ^ z). Property 6 (The uiqueess of complemet). For all x i B, x is uique. Property 7 (Ivolutio). For all x i B, (x ) = x. Property 8 (De Morga s law). For all x ad y i B, (x _ y) = x ^ y ad (x ^ y) = x _ y. A Boolea expressio is a strig of symbols ivolvig costats ad, some variables, ad Boolea operatios _, ^, ad. A Boolea expressio i variables x, x 2,, ad x is defied iductively as follows: Each of the symbols,, x, x 2,, ad x is a Boolea expressio. If e ad e 2 are Boolea expressios, so are e,(e _ e 2 ) ad (e ^ e 2 ).

27 Logic Circuits XOR XNOR NOR NAND x y x y x y x y x y (x y) x y (x y) Figure 3.4: The truth tables of XOR, XNOR, NOR, ad NAND. The followig is some examples of Boolea expressios: ((x _ y) _ (x ^ y)) (((x ^ y) _ (x ^ z )) _ (y _ y) ) For Boolea expressios e ad e 2, e is ofte called as the egatio of e, ad (e _e 2 ) ad (e ^e 2 ) are called as the disjuctio ad cojuctio of e ad e 2,respectively.Ife is a Boolea expressio i variables x, x 2,, ad x, the e defies a Boolea fuctio mappig B ito B whose value at -tuple (a,a 2,,a ) is the elemet of B = {, } obtaied by replacig x by a, x 2 by a 2,, ad x by a i e. A truth table is the simplest way to specify a Boolea fuctio. For example, f(x, y, z) =((x _ y) ^ z ) is a Boolea fuctio defied by Boolea expressio (x _ y) ^ z. This fuctio is also defied by the truth table i Figure 3.2. The umber of di eret Boolea fuctios that ca be defied over biary variables is 2 2. Table 3.3 shows all 6 fuctios of two iput variables, x ad y. A set of Boolea operatios is fuctioally complete if its members ca costruct all other Boolea fuctios over ay give set of iput variables. We assume that these operatios ca be applied as may times as eeded. A well kow complete set of Boolea operatios is {AND, OR, NOT}. Some ew Boolea operatios show i Figure 3.3 ca be composed from these three basic operatios. For two variables x ad y, the Boolea operatios XOR ( ) ad XNOR ( ) are defied by x y = (x ^ y ) _ (x ^ y) = x y = (x ^ y) _ (x ^ y ) = whe x 6= y otherwise whe x = y otherwise XOR ad XNOR are shorthads for exclusive or ad exclusive NOT-OR, respectively. NOR is defied by (x _ y) ad a shorthad for NOT-OR. Similarly, NAND is defied by (x ^ y) ad a shorthad for NOT-AND. {NOR} ad {NAND} are also fuctioally complete. The truth tables of XOR, XNOR, NOR, ad NAND are show i Figure 3.4.

28 3.2 Logic Gates 33 Name Symbol Fuctio NOT x f f = x' OR x y f f = x y AND x y f f = x y NOR x y f f = (x y)' NAND x y f f = (x y)' XOR x y f f = x y XNOR x y f f = x y Figure 3.5: The graphic symbols of some elemetary logic gates. 3.2 Logic Gates A logic gate is a coceptual or physical device that performs oe or more Boolea operatios. A Boolea fuctio ca be implemeted with a logic gate. Logic gates are typically made from trasistors. However, we focus o the abstract otio of logic gates; we do ot worry about their implemetatio details or how electroic circuits implemet them. A logic gate ca be viewed as a black box device. If a Boolea fuctio f maps B ito B m (i.e., it has iput variables ad m outputs), the logic gate that implemets f has iput pis ad m output pis. The graphic symbols of some elemetary logic gates are show i Figure 3.5. A logic diagram is a graphical represetatio of a logic circuit that shows coectios betwee logic gates. A logic gate with more complicated fuctioality ca be implemeted by combiig ad itercoectig some elemetary logic gates. For example, Figure 3.6 shows the implemetatios of XOR, XNOR, NOR, ad NAND gates usig AND, OR, ad NOT gates. A XOR gate ca be implemeted with a OR gate, two AND gates, ad two NOT gates. Multi-iput gates ca also be made by combiig gates of the same type with less iputs. Figure 3.7 shows how a 3-iput AND gate ca be made out of 2-iput AND gates. As metioed before, a word is a group of bits that are hadled as a uit by the computer. Thus, computer hardware is typically desiged to hadle multiple bits simultaeously. A bus is a collectio of two or more related sigal lies each of which represets a bit. To avoid drawig multiple wires for a bus i logic circuit diagrams, we use the bus otatio. For example, the followig diagram shows a 4-bit bus: 4 To refer to idividual bits i a -bit bus amed data, we use the otatio

29 34 Logic Circuits x x y XOR f y f x y (a) x y XNOR f f (b) x y NOR f x y f (c) x y NAND f x y f (d) Figure 3.6: Implemetatios of XOR, XNOR, NOR, ad NAND gates with AND, OR, ad NOT gates: (a) the logic diagram of XOR; (b) the logic diagram of XNOR; (c) the logic diagram of NOR; (d) the logic diagram of NAND. data[i] to refer to the bit located at positio i (the LSB is data[]). I additio, to idicate a set of cosecutive bits i data, we use the otatio data[i : j] for j i + cosecutive bits from positio i to j iclusively. For example, the followig diagram shows that X[4] coects to Y [2] ad X[5] coects to Y [3]: X[4:5] Y[2:3] Like Boolea operatios, a set of logic gates that ca implemet ay Boolea fuctio is called a complete set of logic gates. Each of {AND, OR, NOT}, {NAND}, ad {NOR} is a complete set of logic gates. A uiversal gate is a gate that ca implemet ay Boolea fuctio without eed to use ay other gate type. Thus, NAND or NOR is a uiversal gate. Sice the implemetatio of NAND or NOR gate requires fewer trasistors ad is faster tha that of AND or OR gates, logic desigers prefer to use NAND or NOR gate to implemet their logic circuits.

30 3.2 Logic Gates 35 x y z f y z x f (b) (b) Figure 3.7: A implemetatio of a 3-iput AND gate with two 2-iput AND gates: (a) its Logic diagram; (b) its graphic symbol. x y x y x - y - (a) z z z - X Y (b) Z Figure 3.8: A implemetatio of a -bit AND gate with 2-iput AND gates: (a) its logic diagram; (b) its graphic symbol. A multi-bit(-bit) logic gate that performs a bit-wise Boolea operatio is implemeted by a array of gates each operatig separately o each bit positio of the operads. For example, as show i Figure 3.8, to implemet a -bit AND gate that performs a bit-wise AND operatio o two -bit biary values, we ca build a array of two-iput AND gates each operatig separately o a pair of bits from the two operads. x y carry sum x y sum carry (a) (b) Figure 3.9: A half adder: (a) its truth table; (b) its logic diagram.

31 36 Logic Circuits x y ci sum cout x y c i sum c out (a) (b) x y c i x sum Half adder y carry x sum Half adder y carry sum c out (c) Figure 3.: A full adder: (a) its truth table; (b) its logic diagram; (c) a implemetatio with two half adders. x - y - x -2 y -2 x -3 y -3 x y c x y Full adder c out c i c - x y Full adder c out c i x y Full adder... c -2 c out c i c -3 x y Full adder c out c i c c Overflow s - s -2 s -3 s Figure 3.: A -bit ripple carry adder. 3.3 Combiatioal Logic Circuits A combiatioal logic circuit cosists of logic gates ad performs a operatio that ca be specified by a set of Boolea fuctios. Its outputs are totally depedet o the curret iput values ad determied by combiig the iput values usig Boolea operatios. I cotrast, the outputs of a sequetial logic

32 3.3 Combiatioal Logic Circuits 37 circuit deped ot oly o the curret iput values but also o the past iputs. I additio to usig logic gates, it employs memory ad its outputs are a fuctio of the curret iput values ad the data stored i memory. Cosequetly, a sequetial logic circuit has states. The Half Adder A half adder adds two oe-bit biary umbers x ady. It has two outputs: sum ad carry. The Boolea fuctios for the sum ad carry are give as follows: sum = x y carry = x ^ y Figure 3.9 shows the truth table ad logic circuit diagram of a half adder. Whe we add two multi-bit umbers, the half adder has little practical value. All but LSB colums eed to cosider the carry from the ext least sigificat positio i additio to the two bits from the operads. Oly the LSB ca be added with a half adder. The Full Adder A full adder adds three oe-bit biary umbers x, y, ad a carry (c i ). It has two outputs: sum ad carry (c out ). The Boolea fuctios for the sum ad carry are give as follows: sum = (x y) c i carry = (x ^ y) _ (c i ^ (x y)) Two half adders ca be combied with a OR gate to make a full adder. The oe-bit full adder s truth table ad logic diagram are show i Figure 3.. We ca implemet a -bit biary adder by cascadig full adders. The carry output c out of the full adder at the ext least sigificat positio is coected to the carry iput c i of the curret full adder. We call this type of -bit adder as a ripple carry adder because each carry ripples to the ext full adder. The -bit ripple carry adder adds two -bit biary umbers X ad Y. The outputs are sum ad carry (c ) from the MSB. Figure 3. shows the logic diagram of a -bit ripple-carry-adder. I the worst case, the carry propagates from the LSB through the MSB. Gate delay (also kow as propagatio delay) is the time delay betwee the chages whe a iput chage causes a output chage. The ripple carry adder requires oe propagatio delay time for each bit positio of the operads. Thus, it is ot appropriate for high speed computig systems. The overflow detectio logic for two s complemet additio is also show i Figure 3.. Whe addig two two s complemet biary umbers, as metioed before i Chapter 2, detectig overflow is doe by comparig the carry out of the MSB with the carry ito the MSB: Overflow = c c