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Binary/hexadecimal Byte representations Boolean algebra Expressing in C Uaa f@Why don t computers use Base 10?fBase 10 number representation  Digit in many languages also refers to fingers (and toes) Of course, decimal (from latin decimus) , means tenth A position numeral system (unlike, say Roman numerals) Natural representation for financial transactions (problems?) Even carries through in scientific notation Implementing electronically Hard to store ENIAC (First electronic computer) used 10 vacuum tubes / digit Hard to transmit Need high precision to encode 10 signal levels on single wire Messy to implement digital logic functions Addition, multiplication, etc.<6@>+<6   @>+,sgBinary representationsBase 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 Straightforward implementation of arithmetic functions,Y8       Y  8zh!Byte-oriented memory organizationbPrograms refer to virtual addresses Conceptually very large array of bytes Actually implemented with hierarchy of different memory types SRAM, DRAM, disk Only allocate for regions actually used by program In Unix and Windows NT, address space private to particular  process Program being executed Program can manipulate 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, and heap In any case, all allocation within single virtual address space$eDFS.$eDFS  .i&How do we represent the address space?Hexadecimal notation Byte = 8 bits Binary 000000002 to 111111112 Decimal: 010 to 25510 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#Ui#       T       j Machine wordsZMachine has  word size Nominal size of integer-valued data Including addresses A virtual address is encoded by such a word Most current machines are 32 bits (4 bytes) Limits addresses to 4GB Becoming too small for memory-intensive applications High-end systems are 64 bits (8 bytes) Potentially address 1.8 X 1019 bytes Machines support multiple data formats Fractions or multiples of word size Always integral number of bytes$@,M'''D$@,M'        'Dk!Word-oriented memory organizationAddresses specify byte locations Address of first byte in word Addresses of successive words differ by 4 (32-bit) or 8 (64-bit)&"b"blData representationsSizes of C Objects (in Bytes) (Or any other pointer) Portability: Many programmers assume that object declared as int can be used to store a pointer OK for a typical 32-bit machine Not for AlphaU ZZ aSZ.Z  0      .{Nm Byte orderingHow to order bytes within multi-byte word in memory Conventions Sun s, Mac s are  Big Endian machines Least significant byte has highest address (comes last) Alphas, PC s are  Little Endian machines Least significant byte has lowest address (comes first) Example Variable x has 4-byte representation 0x01234567 Address given by &x is 0x100@'8*8M@'8*8    ,V\oReading byte-reversed ListingsFor most programmers, these issues are invisible Except with networking or disassembly Text representation of binary machine code Generated by program that reads the machine code Example fragment VW\W\  pExamining data representations_Code to print byte representation of data Casting pointer to unsigned char * creates byte arrayD*6*qshow_bytes execution example  rRepresenting stringsStrings in C Represented by array of characters Each character encoded in ASCII format Standard 7-bit encoding of character set Other encodings exist, but uncommon Character  0 has code 0x30 Digit i has code 0x30+i String should be null-terminated Final character = 0 Compatibility Byte ordering not an issue Data are single byte quantities Text files generally platform independent Except for different conventions of line termination character(s)! Ji! *C Ji  ! *  C$$ s!Machine-level code representationEncode 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 A fundamental concept: Programs are byte sequences too!+UZ=aZ.a(Z&a(ZAZ a9U+=.  (  &(A 9  GRepresenting instructions 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 j>67A>67A  uBoolean algebraDeveloped by George Boole in 19th Century Algebraic representation of logic Encode  True as 1 and  False as 0 N*"$*"$ XvApplication of Boolean Algebra4Applied 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 N-A,-A, wInteger & Boolean algebraInteger Arithmetic Z, +, *,  , 0, 1 forms a mathematical structure called  ring Addition is  sum operation Multiplication is  product operation  is additive inverse 0 is identity for sum 1 is identity for product Boolean Algebra {0,1}, |, &, ~, 0, 1 forms a mathematical structure called  Boolean algebra Or is  sum operation And is  product operation ~ is  complement operation (not additive inverse) 0 is identity for sum 1 is identity for product ZZZZZ      H<Boolean Algebra Integer Ring$ I<Boolean Algebra Integer Ring2 "{Properties of & and ^Boolean ring {0,1}, ^, &, I, 0, 1 Identical to integers mod 2 I is identity operation: I (A) = A A ^ A = 0 Property: Boolean ring Commutative sum A ^ B = B ^ A Commutative product A & B = B & A Associative sum (A ^ B) ^ C = A ^ (B ^ C) Associative product (A & B) & C = A & (B & C) Prod. over sum A & (B ^ C) = (A & B) ^ (B & C) 0 is sum identity A ^ 0 = A 1 is prod. identity A & 1 = A 0 is product annihilator A & 0 = 0 Additive inverse A ^ A = 0 X O     O#|Relations between operationsDeMorgan s Laws Express & in terms of |, and vice-versa A & B = ~(~A | ~B) A and B are true if and only if neither A nor B is false A | B = ~(~A & ~B) A or B are true if and only if A and B are not both false Exclusive-Or using Inclusive Or A ^ B = (~A & B) | (A & ~B) Exactly one of A and B is true A ^ B = (A | B) & ~(A & B) Either A is true, or B is true, but not both *(9: -(9 : - $$(((((( w$}General Boolean algebras We can extend the four Boolean operations to also operate on bit vectors Operations applied bitwise All of the Properties of Boolean Algebra Apply Resulting algebras: Boolean algebra: {0,1}(w), |, &, ~, 0(w), 1(w) Ring: {0,1}(w), ^, &, I, 0(w), 1(w) IF[IF           %~ Representing & manipulating setsUseful application of bit vectors  represent finite sets Representation Width w bit vector represents subsets of {0, & , w 1} aj = 1 if j A 01101001 represents { 0, 3, 5, 6 } 01010101 represents { 0, 2, 4, 6 } Operations & Intersection 01000001 { 0, 6 } | Union 01111101 { 0, 2, 3, 4, 5, 6 } ^ Symmetric difference 00111100 { 2, 3, 4, 5 } ~ Complement 10101010 { 1, 3, 5, 7 }IFF I6  F   ~&Bit-level operations in COperations &, |, ~, ^ available in C Apply to any  integral data type long, int, short, char View arguments as bit vectors Arguments applied bit-wise Examples (Char data type) ~0x41 --> 0xBE ~010000012 --> 101111102 ~0x00 --> 0xFF ~000000002 --> 111111112 0x69 & 0x55 --> 0x41 011010012 & 010101012 --> 010000012 0x69 | 0x55 --> 0x7D 011010012 | 010101012 --> 011111012 ("9$%("9                          Q,'TLogic operations in C  not quite the samejContrast to logical operators &&, ||, ! View 0 as  False Anything nonzero as  True Always return 0 or 1 Early termination (if you can answer looking at first argument, you are done) Examples (char data type) !0x41 --> 0x00 !0x00 --> 0x01 !!0x41 --> 0x01 0x69 && 0x55 --> 0x01 0x69 || 0x55 --> 0x01l d   d(Shift operationsLeft 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 on right Useful with two s complement integer representation $5%_$5  %_) Main pointsdIt s all about bits & bytes Numbers Programs Text Different machines follow different conventions Word size Byte ordering Representations Boolean algebra is mathematical basis Basic form encodes  false as 0,  true as 1 General form like bit-level operations in C Good for representing & manipulating sets0(&Y*0(  &Y*/*+,-. / 0 1 345679:;?@ABCDE F!J"K#L$O   0` .T3f` T3f3f` 999MMM` lff3f3޲` eoHff33Ҷ` ff!` T3f3fffq` T3f3fff[>?" dd@ ?nPd@ d " @ ` n?" dd@   @@``PR   @ ` `$p>>  C; ,(  , , s *Llj "`0  T Click to edit Master title style! ! , c $8ʉ "@0  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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H :8c?4 <00150  / H=8c?,0 H 64-bit Words 0   0 <!8c?\ | Addr = ??< 0  1 <pF8c? \l  | Addr = ??< 0  2 <XM8c?Pl n Addr = ??. 0  3 <hP8c?Pl  | Addr = ??< 0  4 <U8c?P l  | Addr = ??< 0  5 <[8c?P l | Addr = ??< 0 4 6 S ~ajJ11?"`p<,$D 0 D0000$0 Z4 7 S ~$fjJ11?"`p< ,$D 0 D0004$0 Z4 8 S ~jjJ11?"`p < ,$D 0 D0008$0 Z4 9 S ~|njJ11?"`p<,$D 0 D0012$0 Z4 : S ~rjJ11?"`,$D 0 D0000$0 Z4 ; S ~(mjJ11?"` P ,$D 0 D0008$0 ZH  0޽h ? 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GGG}  " 0 P/0 q(   r  S ,`0   r  S 짶,@0    c $8 p   (int sum(int x, int y) { return x+y; }) )H  H8c?pW }CDifferent machines use totally different instructions and encodingsD0 DF p     ^pa \T pp   #      B8c?p0  >00 0    BL8c?0   >00 0    Bt8c?   >30 0    B8c? p  >42 0    HĶ8c?p  m a Alpha sum< 0 \T pp   #      Bɶ8c?p0  >01 0    BͶ8c?0   >80 0    B`Ѷ8c?   >FA 0    BԶ8c? p  >6B 0 F     .1 \T `   #     Bض8c?` P  >E0 0    B`׶8c?`P   >08 0    Bܷ8c?` >81 0    B8c?`  >C3 0    Hl8c?m aSun sum>0 \T pp   #      Bd8c?p0  >90 0    B߶8c?0   >02 0    BX8c?   >00 0    B8c? p  >09 0 0 8 PP  0 PP \T `   # D   BD8c?` P  >E5 0  !  B8c?`P   >8B 0  "  B|8c?` >55 0  #  BT 8c?`  >89 0  $  N88c?P| pPC sum (Linux and NT)>0 \T pp  % # D  &  B8c?p0  >45 0  '  B|8c?0   >0C 0  (  B8c?   >03 0  )  B8c? p  >45 0 \T pp  * #  N  +  Bx"8c?p0  >08 0  ,  B&8c?0   >89 0  -  B\*8c?   >EC 0  .  B.8c? p  >5D 0  /  H28c? NP  >C3 0 H  0޽h ? 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GGG  0   d v (  d r d S p,`0   r d S T,@0   d H8c?0dJ n 01101001 & 01010101 010000012!0  dB d <D8c?^` ^ d H8c? J n 01101001 | 01010101 011111012!0  dB d <D8c?^ ^ d Hx%8c?0 dJ n 01101001 ^ 01010101 001111002!0  dB  d <D8c?^ P^  d H+8c?J f ~ 01010101 1010101020  dB  d <D8c?^^  d H/8c?0dz,$ 0 J 01000001$ 0    d H(38c?p z,$ 0 H01111101$ 0   d H78c? dz,$ 0 H00111100$ 0   d HX;8c?z,$ 0 H10101010$ 0  H d 0޽h ? GGG  ___PPT10 +bD '  = @B DU ' = @BA?%,( < +O%,( < +D]' =%(D' =%(D' =A@BBBB0B@=%(D' =1:Bvisible*o3>+B#style.visibility<* d %(JD]' =%(D' =%(D' =A@BBBB0B@=%(D' =1:Bvisible*o3>+B#style.visibility<* d %(JD]' =%(D' =%(D' =A@BBBB0B@=%(D' =1:Bvisible*o3>+B#style.visibility<*d %(JD]' =%(D' =%(D' =A@BBBB0B@=%(D' =1:Bvisible*o3>+B#style.visibility<*d %(J++0+ d 0 ++0+ d 0 ++0+d 0 ++0+d 0 +  0  %;h (  h r h S K,`0   r h S L,@0   h  p 6h  #"   4h  # l!1? p I0 @` 2h  # l/1?  I1 @` -h  # l$1?`p I6 @` +h  # lTm 1?` I1 @` (h  # la 1?` p I5 @` &h  # l 1?`  I1 @` #h  # l 1? p I4 @` !h  # l' 1?  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f33f3 ] 0 P(  X  C d     S yd9 3    H  0ηo~ ? 3380___PPT10. e,Uf 0 p e(   X  C d     S d9 3   g'Mayans use a vigesimal system (base 20)$  H  0ηo~ ? 3380___PPT10.zlPv9 g 0  (   X  C d     S \d9 3    H  0ηo~ ? 3380___PPT10.zl< h 0  (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zl< i 0  (   X  C d     S |̯d9 3    H  0ηo~ ? 3380___PPT10.zl0 > j 0  (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zl1G k 0  (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zlpLM l 0  (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zlfS m 0  (   X  C d     S `d9 3    H  0ηo~ ? 3380___PPT10.zlfS o 0  (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zl0tV p 0  (   X  C d     S Xd9 3    H  0ηo~ ? 3380___PPT10.zl0tV q 0   (   X  C d     S T8d9 3    H  0ηo~ ? 3380___PPT10.zl0tV r 0 0 (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zlW s 0 @ (   X  C d     S d9 3    H  0ηo~ ? 3380___PPT10.zlW u 0 ` (   X  C d     S ld9 3    H  0ηo~ ? 3380___PPT10.zl0b v 0 p (   X  C d     S dd9 3    H  0ηo~ ? 3380___PPT10.zl/d w 0  (   X  C d     S @d9 3    H  0ηo~ ? 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