Chapter 3 Digital Logic and Binary Numbers

Transcription

1 Chapter 3 Digital Logic and Binary Numbers THESE ARE LECTURE NOTES TO ACCOMPANY THE BOOK SPARC ARCHITECTURE, ASSEMBLY LANGUAGE PROGRAMMING, AND C, BY RICHARD P. PAUL, 2 ND EDITION, EDITED BY CHINUA UMOJA

2 Binary A computer is a bistable device A bistable device: Easy to design and build Has 2 states: 0 and 1 One Binary digit (bit) represents 2 possible states (0, 1)

3 With 2 bits, 4 states are possible (2 2 = 4) Bit 1 Bit 0 State Bit 2 Bit 1 Bit 0 State With 3 bits, 8 states are possible (2 3 = 8) With n bits, 2 n states are possible

4 Binary Coded Decimal (BCD) Why not use 4 bits to represent decimal? Let 0000 represent 0 Let 0001 represent 1 Let 0010 represent 2 Let 0011 represent 3, etc. This is called BCD Only uses 10 of the 16 possibilities

5 Binary Number System From left to right, the position of the digit indicates its magnitude (in decreasing order) E.g. in decimal, 123 is less than 321 In binary, 011 is less than 100 A subscript indicates the number s base E.g. is 100 decimal or binary? We don t know! But = is clear

6 Bytes A group of 8 bits is a byte A byte can represent 2 8 = 256 possible states Registers are usually a multiple of bytes SPARC registers have 32 bits (4 bytes) 2 32 = 4,294,967,296

7 Memory Addresses Memory addresses are in binary often 32 bits, these days if each memory address maps to 1 byte: 2 32 bytes = 4 GB K = kilo = thousand, but 1KB actually means 1024 bytes 1MB = 1024 x 1024 bytes 1GB = 1024 x 1024 x 1024 bytes

8 Octal and Hexadecimal It is difficult for a human to work with long strings of 0 s and 1 s Octal and Hexadecimal are ways to group bits together Octal: base 8 Hexadecimal: base 16

9 Hexadecimal With 4 bits, there are 16 possibilities Use 0, 1, 2, 3, 9 for the first 10 symbols Use a, b, c, d, e, and f for the last 6 Bit 3 Bit 2 Bit 1 Bit 0 Symbol a b c d e f

10 Binary to Hexadecimal =? in hex Group into 4 bits, from the right: 0101, 0110, 1011, Now translate each (see previous table): => => => b => 3 So this is 56b3 16 What if there are not enough bits? Pad with 0 s on the left

11 Hexadecimal to Binary f0e5 16 =? in binary Translate each into a group of 4 bits:

12 Hexadecimal to Binary f0e5 16 =? in binary Translate each into a group of 4 bits: f 16 => , 0 16 => , e 16 => , 5 16 => So this is

13 Decimal to Any Number Base Take the decimal number, and divide by the new number base Keep track of the quotient and remainder Repeat until quotient = 0 Read number from the bottom to the top

14 Decimal to Binary Binary is base 2 Example: convert 35 (decimal) to binary Quotient Remainder 35 / 2 = / 2 = / 2 = / 2 = / 2 = / 2 = 0 1 So =

15 Any Number Base to Decimal From right to left, multiply the digit of the numberto-convert by its base position Sum all results

16 Binary to Decimal Binary is base 2 Example: convert (binary) to decimal = 1x x x x x2 0 = 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = = 22 So = 22 10

17 Hexadecimal to Decimal Hexadecimal is base 16 Example: convert 16 (hex) to decimal = 1x x16 0 = 1x16 + 6x1 = = 22 So = Not surprising, since = 0001, If one of the hex digits had been > 9, say c, then we would have used 12 in its place.

18 ASCII American Standard Code for Information Interchange Use byte values to represent characters The assembler allows double-quotes mov 0x4d, %r3! Moves capital M to register 3 mov M, %r3! This command does the same

19 ASCII chart

20 Bitwise Logical Operations There are several binary operations: NOT AND OR XOR NAND NOR XNOR

21 NOT The NOT operation simply complements a binary value not (a) a a not(a)

22 AND The AND operation uses 2 binary values a and b a b a and b

23 OR The OR operation uses 2 binary values a or b a b a or b

24 XOR The XOR (exclusive-or) operation uses 2 binary values True when only one input is true. a xor b a b a xor b

25 NAND The NAND (Not-AND) operation uses 2 binary values Take the AND function, and complement it. a nand b a b a nand b

26 NOR The NOR (Not-OR) operation uses 2 binary values Take the OR function, and complement it. NAND and NOR are easy to make on a chip. Why? Take CSc 4250 and find out! a b a nor b

27 Possible Logic Functions Suppose you have 2 binary digits: a, b Imagine that some function operates on them to create c. What could this function be? There are only 16 possibilities And some of these are not useful! a b some function c

28 Logic Operations A 0011 Logical Sparc B false a and b and a and (not b) andn a b and (not a) b a xor b xor a or b or a nor b a xor (not b) xnor A 1010 not b B 1011 a or (not b) orn C 1100 not a D 1101 b or (not a) E 1110 a nand b F 1111 true

29 Bitwise Each of these logic functions is a bitwise operation, meaning that the result is independent of the bits to the left or right e.g or compare this with addition

30 Logic Instruction Examples mov 0x21, %l0 and %l0, 0x3c, %l1 20 mov 0x47, %l0 and %l0, 0xca, %l1 42 mov 0x21, %l0 or %l0, 0x3c, %l1 mov 0x55, %l0 xnor %l0, 0x3c, %l1 3d ffffff96 mov 0x47, %l0 andn %l0, 0xca, %l1 mov 0x47, %l0 or %l0, 0xca, %l1 5 cf mov 0x55, %l0 xor %l0, 0x3c, %l1 69 mov 0x47, %l0 orn %l0, 0xca, %l1 mov 0x55, %l0 not %l0 ffffff77 ffffffaa

31 A Few More Logic Examples In all the examples below, these registers have the following initial values: %l0 = 0x %l1 = 0x9abcdef0 What are the values for %l1 after the instruction? and %l0, %l1, %l or %l0, %l1, %l1 9abcdef8 xor %l0, %l1, %l not %l0, %l1 edcba987

32 SPARC Instruction Format These commands are in the form: command source register 1, source register 2, destination register command source register 1, immediate value, destination register command can be any of the following: and, andn, xor, or, xnor, orn andcc, andncc, xorcc, orcc, xnorcc, orncc the cc means set condition codes andn means a and (not b) orn means a or (not b)

33 SPARC Logical Instruction Example cmp %a_r, 0 ble next nop add %b_r, 1, %b_r next: This is equivalent to: if (a > 0) b++; %a_r and %b_r will be replaced by the actual registers, such as %r2 and %r3

34 Synthetic Instructions The cmp command is a synthetic one. It is a macro that uses %g0. The above cmp command will be expanded to: subcc %a_r, %g0, %g0 Also, the tst command compares a register to 0: tst %a_r which the assembler turns into: orcc %a_r, %g0, %g0 Since %g0 ignores any updates, only the condition codes are affected.

35 Flags Since individual bits are used to represent boolean flags, a word may contain 32 flags. Common flag operations and mnemonics set: bset ( done with or ) clear: bclr ( done with andn ) toggle: btog ( done with xor )

36 Testing Flags This command will see if one or more flags is set btst reg_or_imm, reg rs1 it expands to: andcc reg rs1, reg_or_imm, %g0 (notice how the operands are switched) example: test if flag 0x02 is set btst 0x02, %a_r be clear nop set: clear: