1. Bits and Bytes Topics Why bits? Tell me Representing information as bits Binary/Hexadecimal Byte representations »numbers »characters and strings »Instructions Bit-level manipulations – READ BOOK Boolean algebra Expressing in C CS 105 bits.ppt CS 105 “Tour of the Black Holes of Computing!”

2. – 2 – CS 105 Binary Representations Base 2 Number Representation Represent 15213 10 as 11101101101101 2 Represent 1.20 10 as 1.0011001100110011[0011]… 2 Represent 1.5213 X 10 4 as 1.1101101101101 2 X 2 13 Electronic Implementation Easy to store with bistable elements - relays Reliably transmitted on noisy and inaccurate wires (media) Straightforward implementation of arithmetic functions 0.0V 0.5V 2.8V 3.3V 010

3. – 3 – CS 105 Byte-Oriented Mem Organization Programs Refer to Virtual Addresses Conceptually very large array of bytes Actually implemented with hierarchy of different memory types – SRAM, DRAM, Disk System provides address space private to particular “process” Program being executed Program can clobber its own data, but not that of others Compiler + Run-Time System Control Allocation Where different program objects should be stored Multiple mechanisms: static, stack, heap, etc. All allocation within single virtual address space 000FFF

4. – 4 – CS 105 Encoding Byte Values Byte = 8 bits Binary 00000000 2 to 11111111 2 Decimal: 0 10 to 255 10 Hexadecimal 00 16 to FF 16 Base 16 number representation Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ Write FA1D37B 16 in C as 0xFA1D37B »Or 0xfa1d37b Internet used term ‘Octet’ Why? 000000 110001 220010 330011 440100 550101 660110 770111 881000 991001 A101010 B111011 C121100 D131101 E141110 F151111 Hex Decimal Binary

5. – 5 – CS 105 Word-Oriented Memory Organization Addresses Specify Byte Locations Address of first byte in word Addresses of successive words differ by 4 (32-bit) or 8 (64-bit) 0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 32-bit Words BytesAddr. 0012 0013 0014 0015 64-bit Words Addr = ?? Addr = ?? Addr = ?? Addr = ?? Addr = ?? Addr = ?? 0000 0004 0008 0012 0000 0008

6. – 6 – CS 105 Data Representations Sizes of C Objects (in Bytes) C Data TypeTypical 32-bitIntel IA32 x86-64 char111 short222 int444 long448 long long888 float444 double888 long double810/1210/16 char *448 »Or any other pointer

7. – 7 – CS 105 Examining Data Representations Code to Print Byte Representation of Data Casting pointer to unsigned char * creates byte array typedef unsigned char *pointer; void show_bytes(pointer start, int len) { int i; for (i = 0; i < len; i++) printf("0x%p\t0x%.2x\n", start+i, start[i]); printf("\n"); } Printf directives: %p:Print pointer %x:Print Hexadecimal

8. – 8 – CS 105 show_bytes Execution Example int a = 15213; printf("int a = 15213;\n"); show_bytes((pointer) &a, sizeof(int)); Result (Linux): int a = 15213; 0x11ffffcb80x6d 0x11ffffcb90x3b 0x11ffffcba0x00 0x11ffffcbb0x00

9. – 9 – CS 105 Representing Integers int A = 15213; int B = -15213; long int C = 15213; Decimal:15213 Binary: 0011 1011 0110 1101 Hex: 3 B 6 D 6D 3B 00 IA32, x86-64 A 3B 6D 00 Sun A 93 C4 FF IA32, x86-64 B C4 93 FF Sun B Two’s complement representation (Covered later) 00 6D 3B 00 x86-64 C 3B 6D 00 Sun C 6D 3B 00 IA32 C

10. – 10 – CS 105 Byte Ordering How should bytes within multi-byte word be ordered in memory? Religious Issue Conventions Suns, PowerPCs are “Big Endian” machines Least significant data byte has highest address Most significant data byte has lowest address Alphas, PCs are “Little Endian” machines Least significant byte has lowest address

11. – 11 – CS 105 Byte Ordering Example Big Endian Least significant byte has highest address Little Endian Least significant byte has lowest addressExample Variable x has 4-byte representation 0x01234567 Address given by &x is 0x100 0x1000x1010x1020x103 01234567 0x1000x1010x1020x103 67452301 Big Endian Little Endian 01234567 452301

12. – 12 – CS 105 Reading Byte-Reversed Listings Disassembly Text representation of binary machine code Generated by program that reads the machine code Example Fragment AddressInstruction CodeAssembly Rendition 8048365:5b pop %ebx 8048366:81 c3 ab 12 00 00 add $0x12ab,%ebx 804836c:83 bb 28 00 00 00 00 cmpl $0x0,0x28(%ebx) Deciphering Numbers Value: 0x12ab Pad to 4 bytes: 0x000012ab Split into bytes: 00 00 12 ab Reverse: ab 12 00 00

13. – 13 – CS 105 Representing Pointers int B = -15213; int *P = &B; // p is pointer, address values are different on different machines. FF 7F 00 0C 89 EC FF x86-64 P Different compilers & machines assign different locations to objects FB 2C EF FF Sun P FF BF D4 F8 IA32 P

14. – 14 – CS 105 Machine-Level Code Representation Encode Program as Sequence of Instructions Each simple operation Arithmetic operation Read or write memory Conditional branch Instructions encoded as bytes Alpha’s, Sun’s, Mac’s use 4 byte instructions »Reduced Instruction Set Computer (RISC) PC’s use variable length instructions »Complex Instruction Set Computer (CISC) Different instruction types and encodings for different machines Most code not binary compatible Programs are Byte Sequences Too - von Neumman!

15. – 15 – CS 105 Representing Instructions int sum(int x, int y) { return x+y; return x+y;} Different machines use totally different instructions and encodings 00 30 42 Alpha sum 01 80 FA 6B E0 08 81 C3 Sun sum 90 02 00 09 For this example, Alpha & Sun use two 4-byte instructions Use differing numbers of instructions in other cases PC uses 7 instructions with lengths 1, 2, and 3 bytes Same for NT and for Linux NT / Linux not fully binary compatible E5 8B 55 89 PC sum 45 0C 03 45 08 89 EC 5D C3

16. – 16 – CS 105 Boolean Algebra – a review Developed by George Boole in 19th Century Algebraic representation of logic Encode “True” as 1 and “False” as 0And A&B = 1 when both A=1 and B=1 Not ~ A = 1 when A=0 Or A|B = 1 when either A=1 or B=1 Exclusive-Or (Xor) A^B = 1 when either A=1 or B=1, but not both

17. – 17 – CS 105 Bit-Level Operations in C follow Boolean Algebra Operations &, |, ~, ^ Available in C Apply to any “integral” data type long, int, short, char View arguments as bit vectors Arguments applied bit-wise Examples (Char data type) ~0x41 --> 0xBE ~01000001 2 -->10111110 2 ~0x00 --> 0xFF ~00000000 2 -->11111111 2 0x69 & 0x55 --> 0x41 01101001 2 & 01010101 2 --> 01000001 2 0x69 | 0x55 --> 0x7D 01101001 2 | 01010101 2 --> 01111101 2

18. – 18 – CS 105 Contrast: Logic Operations in C Contrast to Logical Operators &&, ||, ! View 0 as “False” Anything nonzero as “True” -- key point Always return 0 or 1 Early termination Examples (char data type) !0x41 --> 0x00 !0x00 --> 0x01 !!0x41 --> 0x01 0x69 && 0x55 --> 0x01 0x69 || 0x55 --> 0x01 p && *p ( avoids null pointer access) ??

19. – 19 – CS 105 Shift Operations Left Shift: x << y Shift bit-vector x left y positions Throw away extra bits on left Fill with 0’s on right Right Shift: x >> y Shift bit-vector x right y positions Throw away extra bits on right Logical shift Fill with 0’s on left Arithmetic shift Replicate most significant bit Useful with two’s complement integer representation 01100010 Argument x 00010000<< 3 00011000 Log. >> 2 00011000 Arith. >> 2 10100010 Argument x 00010000<< 3 00101000 Log. >> 2 11101000 Arith. >> 2 00010000 00011000 00010000 00101000 11101000 00010000 00101000 11101000

20. – 20 – CS 105 Cool Stuff with Xor void funny(int *x, int *y) { *x = *x ^ *y; /* #1 */ *x = *x ^ *y; /* #1 */ *y = *x ^ *y; /* #2 */ *y = *x ^ *y; /* #2 */ *x = *x ^ *y; /* #3 */ *x = *x ^ *y; /* #3 */} // x, y not equal Bitwise XOR is form of addition With extra property that every value is its own additive inverse A ^ A = 0 BA Begin BA^B 1 (A^B)^B = AA^B 2 A(A^B)^A = B 3 AB End *y*x

21. – 21 – CS 105 Main Points It’s All About Bits & Bytes Numbers Programs Text Different Machines Follow Different Conventions Word size Byte ordering Representations Boolean Algebra is Mathematical Basis Basic form encodes “false” as 0, “true” as 1 General form like bit-level operations in C Good for representing & manipulating sets

22. – 22 – CS 105 Practice Problems 2.52.5 2.82.8 2.142.14 2.232.23