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## Chapter 3 – Digital Logic and Binary Numbers

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### 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, 2nd edition, 2000.

By Michael Weeks

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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)

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

0 0 1

0 1 2

1 0 3

1 1 4

Bit2 Bit1 Bit0 State

0 0 0 1

0 0 1 2

0 1 0 3

0 1 1 4

1 0 0 5

1 0 1 6

1 1 0 7

1 1 1 8

- With 2 bits, 4 states are possible (22 = 4)

- With 3 bits, 8 states are possible (23 = 8)

- With n bits, 2n states are possible

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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 1410 = 11102 is clear

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Bytes

- A group of 8 bits is a byte
- A byte can represent 28 = 256 possible states
- Registers are usually a multiple of bytes
- SPARC registers have 32 bits (4 bytes)
- 232 = 4,294,967,296

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Memory Addresses

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Hexadecimal

Bit3 Bit2 Bit1 Bit0 Symbol

0 0 0 0 0

0 0 0 1 1

0 0 1 0 2

0 0 1 1 3

0 1 0 0 4

0 1 0 1 5

0 1 1 0 6

0 1 1 1 7

1 0 0 0 8

1 0 0 1 9

1 0 1 0 a

1 0 1 1 b

1 1 0 0 c

1 1 0 1 d

1 1 1 0 e

1 1 1 1 f

- 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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Binary to Hexadecimal

- 01010110101100112 = ? in hex
- Group into 4 bits, from the right:
- 0101, 0110, 1011, 00112
- Now translate each (see previous table):

01012 => 5, 01102 => 6, 10112 => b, 00112 => 3

So this is 56b316

- What if there are not enough bits?
- Pad with 0’s on the left

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Hexadecimal to Binary

- f0e516 = ? in binary
- Translate each into a group of 4 bits:
- f16 => 11112, 016 => 00002, e16 => 11102, 516 => 01012
- So this is 11110000111001012

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Decimal to Binary

- Binary is base 2
- Example: convert 35 (decimal) to binary

Quotient Remainder

35 / 2 = 17 1

17 / 2 = 8 1

8 / 2 = 4 0

4 / 2 = 2 0

2 / 2 = 1 0

1 / 2 = 0 1

- So 3510 = 1000112

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Any Number Base to Decimal

- From right to left, multiply the digit of the number-to-convert by its baseposition
- Sum all results

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Binary to Decimal

- Binary is base 2
- Example: convert 10110 (binary) to decimal

101102 = 1x24 + 0x23 + 1x22 + 1x21 + 0x20

= 1x16 + 0x8 + 1x4 + 1x2 + 0x1

= 16 + 0 + 4 + 2 + 0

= 22

- So 101102 = 2210

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Hexadecimal to Decimal

- Hexadecimal is base 16
- Example: convert 16 (hex) to decimal

1616 = 1x161 + 6x160

= 1x16 + 6x1

= 16 + 6

= 22

- So 1616 = 2210
- Not surprising, since 1616 = 0001, 01102
- If one of the hex digits had been > 9, say c, then we would have used 12 in its place.

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

ASCII chart

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Bitwise Logical Operations

- There are several binary operations:
- NOT
- AND
- OR
- XOR
- NAND
- NOR

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

NOT

- The NOT operation simply complements a binary value
- not (a)
- a’

a not(a)

0 1

1 0

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

AND

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

a b a and b

0 0 0

0 1 0

1 0 0

1 1 1

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

OR

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

a b a or b

0 0 0

0 1 1

1 0 1

1 1 1

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

0 0 0

0 1 1

1 0 1

1 1 0

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

0 0 1

0 1 1

1 0 1

1 1 0

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

0 0 1

0 1 0

1 0 0

1 1 0

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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!

ab

some

function

c

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Logic Operations

- A 0011 Logical Sparc
- B 0101
- 0 0000 false
- 1 0001 a and b and
- 2 0010 a and (not b) andn
- 3 0011 a
- 4 0100 b and (not a)
- 5 0101 b
- 6 0110 a xor b xor
- 0111 a or b or
- 1000 a nor b
- 1001 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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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. 1 0 1

or 0 1 1

1 1 1

- compare this with addition

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Logic Instruction Examples

mov 0x47, %l0

and %l0, 0xca, %l1

mov 0x47, %l0

andn %l0, 0xca, %l1

mov 0x47, %l0

or %l0, 0xca, %l1

mov 0x47, %l0

orn %l0, 0xca, %l1

mov 0x55, %l0

not %l0

42

mov 0x21, %l0

and %l0, 0x3c, %l1

mov 0x21, %l0

or %l0, 0x3c, %l1

mov 0x55, %l0

xnor %l0, 0x3c, %l1

mov 0x55, %l0

xor %l0, 0x3c, %l1

20

5

3d

cf

ffffff96

ffffff77

69

ffffffaa

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

A Few More Logic Examples

In all the examples below, these registers have the following initial values:

%l0 = 0x12345678

%l1 = 0x9abcdef0

What are the values for %l1 after the instruction?

and %l0, %l1, %l1

or %l0, %l1, %l1

xor %l0, %l1, %l1

not %l0, %l1

123456780

88888888

edcba987

9abcdef8

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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)

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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.

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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 )

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

Testing Flags

This command will see if one or more flags is set

btst reg_or_imm, regrs1

it expands to:

andcc regrs1, 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:

Richard P. Paul, SPARC Architecture, Assembly Language Programming, and C

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