15 213 The Class That Gives CMU Its Zip Bits Bytes and Integers Topics Representing information as bits Bit level manipulations Boolean algebra Expressing in C Representations of Integers Basic properties and operations Implications for C class02 ppt 15 213 F 08 Binary Representations Base 2 Number Representation Represent 1521310 as 111011011011012 Represent 1 2010 as 1 0011001100110011 0011 2 Represent 1 5213 X 104 as 1 11011011011012 X 213 Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires 0 1 0 3 3V 2 8V 0 5V 0 0V 2 15 213 S 09 Encoding Byte Values Byte 8 bits Binary 000000002 Decimal 010 to to 111111112 25510 First digit must not be 0 in C Hexadecimal 0016 to FF16 Base 16 number representation Use characters 0 to 9 and A to F Write FA1D37B16 in C as 0xFA1D37B Or 0xfa1d37b 3 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 15 213 S 09 Byte Oriented Memory Organization Programs Refer to Virtual Addresses Conceptually very large array of bytes Actually implemented with hierarchy of different memory types 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 4 Where different program objects should be stored All allocation within single virtual address space 15 213 S 09 Machine Words Machine Has Word Size Nominal size of integer valued data Including addresses Most current machines use 32 bits 4 bytes words Limits addresses to 4GB Becoming too small for memory intensive applications High end systems use 64 bits 8 bytes words Potential address space 1 8 X 1019 bytes x86 64 machines support 48 bit addresses 256 Terabytes Machines support multiple data formats Fractions or multiples of word size Always integral number of bytes 5 15 213 S 09 Word Oriented Memory Organization 32 bit 64 bit Words Words Addresses Specify Byte Locations Address of first byte in word Addresses of successive words differ by 4 32 bit or 8 64 bit Addr 0000 Addr 0000 Addr 0004 Addr 0008 Addr 0012 6 Addr 0008 Bytes Addr 0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 15 213 S 09 Data Representations Sizes of C Objects in Bytes C Data Type Typical 32 bit Intel IA32 x86 64 1 2 4 4 8 4 8 8 4 1 2 4 4 8 4 8 10 12 4 1 2 4 8 8 4 8 10 16 8 char short int long long long float double long double char Or any other pointer 7 15 213 S 09 Byte Ordering How should bytes within multi byte word be ordered in memory Conventions Big Endian Sun PPC Mac Internet Least significant byte has highest address Little Endian x86 Least significant byte has lowest address 8 15 213 S 09 Byte Ordering Example Big Endian Least significant byte has highest address Little Endian Least significant byte has lowest address Example Variable x has 4 byte representation 0x01234567 Address given by x is 0x100 Big Endian 0x100 0x101 0x102 0x103 01 Little Endian 45 67 0x100 0x101 0x102 0x103 67 9 23 45 23 01 15 213 S 09 Reading Byte Reversed Listings Disassembly Text representation of binary machine code Generated by program that reads the machine code Example Fragment Address 8048365 8048366 804836c Instruction Code 5b 81 c3 ab 12 00 00 83 bb 28 00 00 00 00 Assembly Rendition pop ebx add 0x12ab ebx cmpl 0x0 0x28 ebx Deciphering Numbers 10 Value Pad to 32 bits Split into bytes Reverse 0x12ab 0x000012ab 00 00 12 ab ab 12 00 00 15 213 S 09 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 11 15 213 S 09 show bytes Execution Example int a 15213 printf int a 15213 n show bytes pointer a sizeof int Result Linux int a 15213 12 0x11ffffcb8 0x6d 0x11ffffcb9 0x3b 0x11ffffcba 0x00 0x11ffffcbb 0x00 15 213 S 09 Representing Integers int A 15213 int B 15213 long int C 15213 IA32 x86 64 A 6D 3B 00 00 IA32 x86 64 B 93 C4 FF FF 13 Sun A 00 00 3B 6D Sun B FF FF C4 93 Decimal 15213 Binary 0011 1011 0110 1101 3 Hex IA32 C 6D 3B 00 00 B 6 D x86 64 C Sun C 6D 3B 00 00 00 00 00 00 00 00 3B 6D Two s complement representation Covered later 15 213 S 09 Representing Pointers int B 15213 int P B Sun P IA32 P x86 64 P EF FF FB 2C D4 F8 FF BF 0C 89 EC FF FF 7F 00 00 Different compilers machines assign different locations to objects 14 15 213 S 09 Representing Strings Strings in C char S 6 15213 Represented by array of characters Each character encoded in ASCII format Standard 7 bit encoding of character set Character 0 has code 0x30 Digit i has code 0x30 i String should be null terminated Final character 0 Compatibility 15 Linux Alpha S Sun S 31 35 32 31 33 00 31 35 32 31 33 00 Byte ordering not an issue 15 213 S 09 Boolean Algebra Developed by George Boole in 19th Century Algebraic representation of logic Encode True as 1 and False as 0 And Or A B 1 when both A 1 and B 1 0 1 0 0 0 1 0 1 Not A 1 when A 0 0 1 1 0 16 A B 1 when either A 1 or B 1 0 1 0 0 1 1 1 1 Exclusive Or Xor A B 1 when either A 1 or B 1 but not both 0 1 0 0 1 1 1 0 15 213 S 09 Application of Boolean Algebra Applied to Digital Systems by Claude Shannon 1937 MIT Master s Thesis Reason about networks of relay switches Encode closed switch as 1 open switch as 0 A B A Connection when B A B A B A B A B 17 A B 15 213 S 09 General Boolean Algebras Operate on Bit Vectors Operations applied bitwise 01101001 01010101 01000001 01000001 01101001 01010101 01111101 01111101 01101001 01010101 00111100 00111100 01010101 10101010 10101010 All of the Properties of Boolean Algebra Apply 18 15 213 S 09 Representing Manipulating Sets Representation Width w bit vector represents subsets of 0 w 1 aj 1 if j A 01101001 0 3 5 6 76543210 01010101 76543210 0 2 4 6 Operations 19 Intersection Union Symmetric difference Complement 01000001 01111101 00111100 10101010 0 6 0 2 3 …
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