Csci 2021 Class Web Pages. Data Representa?on: Bits, Bytes, and Integers. Binary Representa?ons. Bits, Bytes, and Integers

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1 Csci 1 Class Web Pages Departmental Class Web Page: Pblic access withot login Corse syllabs, schedle, lectre notes, other sefl info h7p://www- sers.cselabs.mn.ed/classes/spring- 1/csci1/ Yo shold read corse syllabs careflly for all corse policies, personnel info, and corse logis?cs. Moodle. Web page: Use yor niversity x5 accont to access Class forms, assignment sbmission, exam info h7ps://ay15.moodle.mn.ed/corse/view.php?id13 Data Representa?on: Bits, Bytes, and CSci 1 - Machine Architectre and Organiza?on Professor Pen- Chng Yew With sides from Randy Bryant and Dave O Hallaron 1 Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary Binary Representa?ons Each bit is or 1 Why bits? Electronic Implementa?on Easy to store with bistable elements, e.g. switches, transistors, Reliably transmi7ed on noisy wires By encoding/interpre?ng sets of bits in varios ways Compters determine what to do (instrcuons), and represent and maniplate nmbers, sets, strings, etc 1.1V.9V 1 3.V.V Base- nmber representa?on Decimal to Binary Represent 15 1 as 111 Represent 1. 1 as 1.11[11] (binary repetend) Represent 1.51 X 1 as 1.11 X 7 Binary to Decimal Represent 111 as 1 Represent as Represent X 3 as 1.35 X 1 1 Encoding Byte Vales Byte bits Binary to Decimal: 1 to 551 Hexadecimal: 1 to 1 Base 1 nmber representauon Use characters to 9 and A to F Write FA1D37BE1 in C as xfa1d37be xfa1d37be xfa1d37be F A 1 D 3 7 B E Hex Decimal Binary A 1 11 B C 1 11 D E F

2 Example Data Representa?ons Today: Bits, Bytes, and C Data Type Typical 3- bit Typical - bit x- char short int long float doble long doble 1/1 pointer RepresentaUon: nsigned and signed Conversion, casung Expanding, trncaung AddiUon, negauon, mluplicauon, divide with shic Smmary 7 Boolean Algebra Developed by George Boole in 19th Centry Algebraic representauon of logic Encode Tre as 1 and False as And Or n A&B 1 when both A1 and B1 n A B 1 when either A1 or B1 B B A A Not Exclsive- Or (Xor) n ~A 1 when A n A^B 1 when either A1 or B1, bt not both B Applica?on of Boolean Algebra Applied to Digital Systems by Clade Shannon 1937 MIT Master s Thesis Reason abot networks of relay switches Encode closed switch as 1, open switch as A ~A A&~B ~B B ~A&B Connection when A&~B ~A&B A^B A A 9 1 General Boolean Algebras Represen?ng & Manipla?ng Sets Operate on Bit Vectors OperaUons applied bitwise 1111 & ^ ~ All of the Proper?es of Boolean Algebra Apply 11 Representa?on Width w bit vector represents sbsets of {,, w 1} a j 1 if j A 1111 A {, 3, 5, } A {,,, } 7531 Opera?ons & IntersecUon 11 {, } Union {,, 3,, 5, } ^ Symmetric difference 1111 {, 3,, 5 } ~ Complement 1111 { 1, 3, 5, 7 } 1

3 Bit- Level Opera?ons in C Opera?ons &,, ~, ^ Available in C Apply to any integral data type long, int, short, char, nsigned View argments as bit vectors Argments applied bit- wise Examples (char data type) (1 byte) ~x1 xbe ~ ~x x ~ x9 & x55 x & x9 x55 x7d Contrast: Logic Opera?ons in C Contrast to Logical Operators &&,,! View as False Anything nonzero as Tre Always retrn or 1 Examples (char data type)!x1 x (different from ~ operauon)!x x1!!x1 x1 x9 && x55 x1 x9 x55 x Contrast: Logic Opera?ons in C Contrast to Logical Operators &&,,! View as False Anything nonzero as Tre Always retrn or 1 Early terminauon Examples (char data type)!x1 x!x x1!!x1 x1 Watch ot for && vs. & (and vs. ) one of the more common oopsies in C programming x9 && x55 x1 x9 x55 x1 ShiO Opera?ons LeO ShiO: x << y ShiO bit- vector x leo y posi?ons Throw away extra bits on leo Fill with s on right Right ShiO: x >> y ShiO bit- vector x right y posi?ons Throw away extra bits on right Logical shio Fill with s on leo Arithme?c shio Replicate most significant bit on leo Undefined Behavior ShiO amont < or word size Argment x 111 << 3 1 Log. >> 11 Arith. >> 11 Argment x 111 << 3 1 Log. >> 11 Arith. >> Annoncement 1/5/1 Programming Assignment #1 (Data Lab) will be issed today, de in weeks on //1 11:55pm Monday Download Data Lab from class Moodle page This lab asks yo to maniplate bit strings sing restricted instrcuon types and conts Each problem in the assignment has different degrees of difficlty, start with easier ones first. Feel free to se class Form to ask qesuons and get informauon from TA or other stdents TA will discss Data Lab in Thr recitauon sessions. Today: Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary Smmary

4 Encoding Unsigned and Signed Unsigned (BU) w 1 BU(X) x i i i short int x 1513; short int y -1513; Two s Complement (BT) w BT(X) x w 1 w 1 + x i i i Sign Bit For s complement, most significant bit indicates sign for nonnega?ve 1 for nega?ve In C, data type short is bytes long Decimal Hex Binary x B D y C Sign Bit 19 Encoding Example (Binary to Unsigned BU ) x 1513: y -1513: Weight Sm w 1 BU(X) x i i i w BT(X) x w 1 w 1 + x i i i Binary Weight Table x 1 K 1 +1 x 1 K x 1 K 1 +1 x 1 1K x 1 3K 1 +1 x 1 K x 1 1K 1 +1 x 1 5K x 1 51K x 1 1M 1 Conver?ng Unsigned to Binary (UB) Dividing the nmber repeatedly by n?l the nmber becomes 9? - 9? BU(X) w 1 x i i i Same algorithm to ANY nmber system Divide by Nmber Remainder x x + x 3 + x + x x 9 Nmeric Ranges Unsigned Vales UMin UMax w Ranges for w 1, i.e. bytes (short) Decimal Hex Binary UMax TMax 377 7F TMin Two s Complement Vales Tmin w 1 1 Tmax w Other Vales Mins 1 w w BT(X) x w 1 w 1 + x i i i w 1 BU(X) x i i i Vales for Different Word Sizes w"! "" 1"(short)" 3"(int) "(long)" UMax" 55! 5,535!,9,97,95! 1,,7,73,79,551,15! TMax" 17! 3,77!,17,3,7! 9,3,37,3,5,775,7! TMin" -1! -3,7! -,17,3,! -9,3,37,3,5,775,!!! Observa?ons TMin TMax + 1 Asymmetric range UMax * TMax + 1 C Programming #inclde <limits.h> Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Vales plalorm specific Need to know - > to avoid overflow 3

5 Unsigned & Signed Integer Vales X BU(X) BT(X) Eqivalence Same encodings for nonnegauve/ negauve integer vales Uniqeness Every bit pa7ern represents niqe integer vale Each representable integer has niqe bit encoding Can Invert Mappings UB(x) BU - 1 (x) Bit pa7ern for nsigned integer TB(x) BT - 1 (x) Bit pa7ern for two s comp integer Today: Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary 5 Mapping Between Signed & Unsigned Signed (Two s Complement) x Unsigned x TU TB X BU Maintain Same Bit Paoern UT UB BT X Maintain Same Bit Paoern Unsigned x Signed (Two s Complement) x Mappings between nsigned and two s complement nmbers: keep bit representa?ons and reinterpret 7 Mapping Signed Unsigned Bits Signed TU UT Unsigned Mapping Signed Unsigned Bits Signed /- 1 Unsigned Rela?on between Signed & Unsigned Two s Complement w 1 x x Large nega?ve weight becomes Large posi?ve weight x TU TB X BU Maintain Same Bit Paoern x Unsigned x x x x + w x < 3 5

6 Conversion Visalized Signed Unsigned Ordering Inversion NegaUve Big PosiUve Warning: Can case a lot of confsion and bgs in C!!! TMax UMax UMax 1 TMax + 1 TMax Unsigned Range Annoncement 1/7/1 This Thrsday s recita?on sessions will cover both Data Lab assignment, and more C programming langage featres. s Complement Range 1 TMin 31 3 Review: Signed vs. Unsigned in C Constants By defalt are considered to be signed integers Unsigned if have U as sffix U, 99759U Cas?ng (i.e. conversion) Explicit cas?ng between signed & nsigned same as UT and TU int tx, ty; nsigned x, y; tx (int) x; y (nsigned) ty; Implicit cas?ng also occrs via assignments and procedre calls tx x; y ty; 33 Review: Cas?ng Srprises Expression Evala?on If there is a mix of nsigned and signed in single expression, signed vales implicitly cast to nsigned Inclding comparison operauons <, >,, <, > Examples for w 3: TMIN -,17,3,, TMAX,17,3,7 Constant 1 Constant Rela?on Evala?on U U nsigned < signed U U > nsigned > signed 1737U < nsigned > signed (nsigned) > nsigned U < nsigned (int) 173U > signed 3 Smmary Cas?ng Signed Unsigned: Basic Rles Bit paoern is maintained (i.e. kept the same) Bt reinterpreted Can have nexpected effects: adding or sbtrac?ng w Expression containing nsigned and signed int int is cast to nsigned!! Today: Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary 37 3

7 Sign Extension Task: Given w- bit signed integer x Convert it to w+k- bit integer with same vale Rle: Make k copies of sign bit: X ʹ x w 1,, x w 1, x w 1, x w,, x k copies of MSB X w X ʹ k w 39 Sign Extension Example short int x 1513; int ix (int) x; short int y -1513; int iy (int) y; Conver?ng from smaller to larger integer data type C atoma?cally performs sign extension Decimal Hex Binary x B D ix B D y C iy C Smmary: Expanding, Trnca?ng: Basic Rles Expanding (e.g., short to int) Unsigned: zeros added Signed: sign extension Both yield expected reslt Trnca?ng (e.g., nsigned to nsigned short) Unsigned/signed: high- ordered bits are trncated Reslt reinterpreted Unsigned: mod operauon Signed: similar to mod For small nmbers yields expected behavior Today: Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary 1 Nega?on: Complement & Increment Claim: Following Holds for s Complement ~x + 1 -x A Shortct to Calclate s Complement ObservaUon: ~x + x x ~x w BT(X) x w 1 w 1 + x i i i Complement & Increment Examples x 1513 x Ranges for short ( bytes) Decimal Hex Binary x B D ~x -151 C ~x C y C Decimal Hex Binary ~ ~+1 Decimal Hex Binary UMax TMax 377 7F TMin What abot Tmin? Tmin - Tmax

8 Unsigned Addi?on Operands: w bits Tre Sm: w+1 bits + v + v Discard Carry: w bits UAdd w (, v) Standard Addi?on Fnc?on Ignores carry otpt (overflow is not signaled as errors in C langage) Implements Modlar Arithme?c s UAdd w (, v) ( + v) mod w + v + v < w UAdd w (,v) + v w + v w Integer Addition Visalizing (Mathema?cal) Integer Addi?on Integer Addi?on - bit integers, v Compte tre sm Add (, v) Vales increase linearly with and v Forms planar srface Add (, v) v Visalizing Unsigned Addi?on Wraps Arond If tre sm w Overflow UAdd (, v) Two s Complement Addi?on Operands: w bits + v Tre Sm: w+1 bits + v Discard Carry: w bits TAdd w (, v) Tre Sm w+1 w Overflow Modlar Sm v TAdd and UAdd have Iden?cal Bit- Level Behavior Signed vs. nsigned addiuon in C: int s, t,, v; s (int) ((nsigned) + (nsigned) v); t + v Will give s t 7 TAdd Overflow Visalizing s Complement Addi?on Fnc?onality Tre sm reqires w+1 bits Drop off Most Significant Bit (MSB) Treat remaining bits as s comp. integer (w) 1 (w 1) (w 1) - 1 Tre Sm w PosOver NegOver TAdd Reslt Vales - bit two s comp. Range from - to +7 Wraps Arond If sm w 1 Becomes negauve If sm < w 1 Becomes posiuve NegOver TAdd (, v) v PosOver 9 5

9 Characterizing TAdd Unsigned Ml?plica?on in C Fnc?onality Tre sm reqires w+1 bits Drop off MSB Treat remaining bits as s comp. integer TAdd(, v) > v < NegaUve Overflow < > PosiUve Overflow + v + w 1 + v < TMin w (NegOver) TAdd w (,v) + v TMin w + v TMax w + v w 1 TMax w < + v (PosOver) Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v Standard Ml?plica?on Fnc?on Ignores high order w bits Implements Modlar Arithme?c UMlt w (, v) v mod w * v UMlt w (, v) 51 5 Signed Ml?plica?on in C Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v Standard Ml?plica?on Fnc?on Ignores high order w bits Some of which are different for signed vs. nsigned mluplicauon Lower bits are the same as nsigned mluplicauon * v TMlt w (, v) Ml?plica?on Comp?ng Exact Prodct of w- bit nmbers x, y Either signed or nsigned Ranges Unsigned: Up to w bits Reslt range: x * y ( w 1) w w Two s complement min (negauve) : Up to w 1 bits Reslt range: x * y ( w 1 )*( w 1 1) w + w 1 Two s complement max (posiuve): Up to w bits, bt only for (TMin w ) Reslt range: x * y ( w 1 ) w Maintaining Exact Reslts Wold need to keep expanding word size with each prodct compted Done in socware by arbitrary precision arithmeuc packages 53 5 Power- of- Ml?ply with ShiO Opera?on << k gives * k Both signed and nsigned Operands: w bits Tre Prodct: w+k bits k k 1 Discard k bits: w bits UMlt w (, k ) TMlt w (, k ) Examples << 3 * << 5 - << 3 ( * 3) ( * ) * Most machines shic and add faster than mluply Compiler generates this code atomaucally * k

10 Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary Unsigned Power- of- Divide with ShiO Qo?ent of Unsigned by Power of >> k gives / k ( floor operauons) Uses logical right shic k Operands: / k 1 Division: Reslt: / k / k Division Compted Hex Binary x B D x >> D B x >> B x >> B Binary Point 57 5 Signed Power- of- Divide with ShiO Qo?ent of Signed by Power of x >> k gives x / k Uses arithme?c right shic Ok when >, bt ronds wrong direcuon when < (shold rond toward ) k x Binary Point Operands: / k 1 Division: x / k. Reslt: RondDown(x / k ) Division Compted Hex Binary y C y >> E y >> FC y >> C Correct Power- of- Divide Qo?ent of Nega?ve Nmber by Power of Want x / k (Rond toward, i.e. se ceiling opera?on) Compte as (x+ k -1)/ k In C: (x + (1<<k)-1) >> k Biase dividend toward Case 1: No ronding k Dividend: 1 + k Divisor: / k / k Biasing has no effect Binary Point Correct Power- of- Divide (Cont.) Case : Ronding Dividend: Divisor: k x 1 + k / k x / k 1 Incremented by Binary Point Today: Bits, Bytes, and RepresentaUon: nsigned and signed Conversion, casung Expanding, trncaung AddiUon, negauon, mluplicauon, divide with shic Smmary Biasing adds 1 to final reslt Incremented by 1 1 1

11 Arithme?c: Basic Rles Arithme?c: Basic Rles Addi?on: Normal addiuon followed by trncate, same operauon on bit level Unsigned: addiuon mod w MathemaUcal addiuon & Possible sbtracuon of w Signed: modified addiuon mod w (reslt in proper range) MathemaUcal addiuon & Possible addiuon or sbtracuon of w Ml?plica?on: (prodct has w bits) Normal mluplicauon followed by trncate, same operauon on bit level Unsigned: mluplicauon mod w (reslt in proper range) Signed: modified mluplicauon mod w (reslt in proper range) LeO shio (Ml?plica?on) Unsigned/signed: mluplicauon by k Always logical leo shio Right shio (Division) Unsigned: logical right shic, div (division + rond toward zero) by k Signed: arithmeuc right shic Posi?ve nmbers: div (division + rond toward zero) by k Nega?ve nmbers: div (division + rond away from zero) by k Use biasing to fix 3 Integer C Pzzle Tmin and Tmax make a good conter example in many cases 1. x < ((x*) < ) False: TMin. x > Tre: UMin 3. x & 7 7 (x<<3) < Tre: x 1 1. x > -1 False: 5. x > y -x < -y False: -1, TMin. x * x > False: 3 7. x > && y > x + y > False: TMax, TMax. x > -x < Tre: TMax < 9. x < -x > False: TMin Today: Bits, Bytes, and Representa?on: nsigned and signed Expanding, trnca?ng Addi?on, nega?on, ml?plica?on, divide with shio Smmary Data representa?on in memory, pointers, character strings int x foo(); int y bar(); nsigned x x; Ini?aliza?on nsigned y y; 9 Byte- Oriented Memory Organiza?on F System provides a private address space to each process Think of a process as a program being exected A program can clobber its own data, bt not that of others Machine Words Any given machine has a Word Size Nominal size of integer- valed data Inclding addresses Many crrent machines se 3 bits ( bytes) words Limits addresses to GB Becoming too small for memory- intensive applicauons 1 1 mega giga tera peta exa zeoa- 1 1 yooa- Programs refer to data by address Conceptally, envision it as a very large array of bytes In reality, it s not, bt can think of it that way An address is like an index into that array and, a pointer variable stores an address Most high- end systems se bits ( bytes) words PotenUal address space 1 X 1 1 bytes (exabytes) Machines spport mluple data formats (data types) FracUons or mluples of word size Always integral nmber of bytes

12 Word- Oriented Memory Organiza?on Addresses specify byte loca?ons Address of first byte in a word Addresses of sccessive words differ by (3- bit) or (- bit) Need to dis?ngish the no?on of address vs. data stored in that address 3-bit Words Addr?? Addr?? Addr?? Addr 1?? -bit Words Addr?? Addr?? Bytes Addr Byte Ordering How shold bytes within a ml?- byte word be ordered in memory? Conven?ons Big Endian: Sn, PPC Mac, Internet Most significant byte first Least significant byte has highest address, i.e Liole Endian: x, ARM processors rnning Android, ios and Windows Least significant byte first Least significant byte has lowest address, i.e. 73 Annoncement /1/1 Homework Assignment #1 has been issed today Download from Moodle class web page De date before class Wednesday /1/1 (check class schedle) Data Lab de next Monday //1 Shold start asap, if not yet Check Data Lab Form for common Q&As, or post yor problem there. Review: Byte Ordering Example Big Endian Most significant byte first i.e. Least significant byte has highest address Liole Endian Least significant byte first i.e. Least significant byte has lowest address Example Variable x has - byte representauon x7531 Address given by &x is x1 Big Endian Little Endian x1 x11 x1 x x1 x11 x1 x Represen?ng int A 1513; IA3, x- D 3B IA3, x- 93 C Sn 3B D int B -1513; Sn C 93 Decimal: 1513 Binary: Hex: 3 B D long int C 1513; IA3 D 3B x- D 3B Two s complement representation Sn 3B D 7 Examining Data Representa?ons Code to Print Byte Representa?on of Data CasUng pointer to nsigned char * allows treatment as a byte array typedef nsigned char *pointer; void show_bytes(pointer start, size_t len){ size_t i; for (i ; i < len; i++) printf( %p\tx%.x\n",start+i, start[i]); printf("\n"); } Printf directives: %p: print pointer (i.e. address) %x: print Hexadecimal \t : tab \n: new line 79 1

13 show_bytes Exec?on Example Represen?ng Pointers int a 1513; printf("int a 1513;\n"); show_bytes((pointer) &a, sizeof(int)); int B -1513; int *P &B; Sn IA3 x- Reslt (Linx x-): int a 1513; x7fffb7f71dbc d x7fffb7f71dbd 3b x7fffb7f71dbe x7fffb7f71dbf EF FB C D F BF C 9 EC 7F Different compilers & machines assign different locauons to objects Even get different reslts each Ume rn program 1 Represen?ng Strings char S[] "13"; Strings in C Represented by array of characters Each character encoded in ASCII format Linx/x- Sn Standard 7- bit encoding of character set Character has code x3 3 3 Digit i has code x3+i 3 3 String shold be nll- terminated 3 3 Final character # invisible Compa?bility Byte ordering not an isse Move on to Floa?ng Point Nmbers ASCII: American Standard Code for InformaUon Interchange Unicode Consor?m 3 13