Class Participation Instructions

• Put your full name and the date at the top of your card.
• Make sure your writing is legible.

Additional Skills for Programming Success

• Analytical and mathematical skills.
• Algorithm development.
• Debugging.
• Focus, determination, and teamwork.
• Reading and understanding code and documentation.
• Asking for help.

Good Practices in Programming

• Follow naming conventions.
• Outline objectives clearly.
• Ensure modularity and readability.
• Apply good practices habitually.

To Attain the Learning Objectives

• Complete the readings before class.
• Practice the exercises
• Writing is the key to understanding, you must write code to succeed.
• If you are still confused then write down a muddy point during class or attend office hours.

Learning Objectives

• Define bits and bytes.
• Explain why and how computers use bits to represent information.
• Describe integral types.
• Perform the following C operations:
  • Bit-level operations
  • Logic operations
  • Shift and mask operations

Everything is Bits

• Bits: 0 or 1.
• Computers encode instructions, numbers, sets, strings, etc., using bits.
• Why bits? Electronic implementation:
  • Bistable elements for storage.
  • Reliable transmission over noisy channels.

Binary and Number Representation

• 15, 213₁₀ == 11101101101101₂
• 1.20₁₀ == $1.\overline{0011}_2$
• 1.5213₁₀ * 10⁴ == 1.1101101101101₂ * 2¹³

Encoding Byte Values

Bits, Bytes, and Integers

• Representing information as bits.
• Bit-level manipulations:
  • Unsigned and signed integers.
  • Conversion, casting, expanding, truncating.
  • Arithmetic operations (addition, negation, shifting).
• Representations in memory, pointers, strings

Practice Conversion

• Convert 0x39A7F8 to binary:
• Convert 1100100101111011₂ to hexadecimal:

Practice Conversion

• Convert 0x39A7F8 to binary:

  3	9	A	7	F	8
  0011	1001	1010	0111	1111	1000

• Convert 1100100101111011 to hexadecimal:

Practice Conversion

• Convert 0x39A7F8 to binary:

  3	9	A	7	F	8
  0011	1001	1010	0111	1111	1000

• Convert 1100100101111011 to hexadecimal:

  1100	1001	0111	1011
  C	9	7	B

Boolean Algebra Basics

• True = 1, False = 0.
• Operators:
  • AND (A & B = 1 if both A = 1 and B = 1).
  • OR (A | B = 1 if either A = 1 or B = 1).
  • NOT (~A = 1 if A = 0).
  • XOR (A ^ B = 1 if either A = 1 or B = 1, but not both).

Boolean Algebra Operations

• Operators apply bitwise:

  01101001 & 01010101 = 01000001
  01101001 | 01010101 = 01111101
  01101001 ^ 01010101 = 00111100
  ~01010101 = 10101010

• All of the properties of boolean algebra apply (p. 52)

Boolean Algebra

• Boolean operations |, &, and ~ operating on bit vectors of length w form a Boolean Algebra for integer w > 0
• Boolean Algebra has many properties of arithmetic over integers.
  • E.g. & distributes over |
  • a & (b | c) = (a & b) | (a & c)
  • Just as multiplication distributes over addition
• Boolean | distributes over & but in integer arithmetic, addition does not distribute over multiplication, i.e. a + (b * c) is not = to (a + b) * (a + c)

Boolean Rings

• Operations (^, &, ~) on bit vectors form Boolean rings.
• Many properties of Boolean rings also hold on integer arithmetic
• In integer arithmetic, every integer x has an additive inverse; the equivalent to addition in Boolean is ^
• Some interesting results arise from this.

Why Boolean Forms Matter

Why Boolean Forms Matter

• Understanding logic and set representations.
• Facilitating bitwise manipulation in programming.

Learning Objectives

• Define bits and bytes.
• Explain why and how computers use bits to represent information.
• Describe integral types.
• Perform the following C operations:
  • Bit-level operations
  • Logic operations
  • Shift and mask operations

Integral Types

• integral means “of or denoted by an integer”
• Simplest machine data types are ones that represent integral types
• You should already be familiar with the basics of this, including in C
• Characters are just 8-bit integers
• Used for booleans (0 = false, non-zero = true, implementation)
• Basic boolean logic – AND, OR, XOR, NOT, shifts and masks
• These can be used for interesting things in C
• In general, in assignments (Datalab, for example) you may not use high-level logical operations.

Learning Objectives

• Define bits and bytes.
• Explain why and how computers use bits to represent information.
• Describe integral types.
• Perform the following C operations:
  • Bit-level operations
  • Logic operations
  • Shift and mask operations

Example: Representing & Manipulating (finite) Sets

• Representation

  • bit vector of length w, used to represent subsets of {0, …, w–1}

  • a_j = 1 if j ∈ A

  • { 0, 3, 5, 6 }

    0	1	1	0	1	0	0	1
    7	6	5	4	3	2	1	0

  • { 0, 2, 4, 6 }

    0	1	0	1	0	1	0	1
    7	6	5	4	3	2	1	0

Example: Representing & Manipulating (finite) Sets

• 01101001 { 0, 3, 5, 6 }

• 01010101 { 0, 2, 4, 6 }

• set operations performed as logical operations over bits

  op	name	result
  &	Intersection	01000001 { 0, 6 }
  |	Union	01111101 { 0, 2, 3, 4, 5, 6 }
  ^	exclusive or	00111100 { 2, 3, 4, 5 }
  ~	Complement(of 2nd)	10101010 { 1, 3, 5, 7 }

Bit-Level Operations in C

• Operations &, |, ~, ^ available in C
  • Apply to any “integral” data type
  • Views arguments as bit vectors
• Examples (Char data type = 8 bits) Steps:
  • ~ 0x41 == 0xBE
    • ~ 01000001₂ == 10111110₂
      
  • ~ 0x00 == 0xFF
    • ~00000000₂ -> 11111111₂
  • 0x69 & 0x55 == 0x41
    • 01101001₂ & 01010101₂ == 01000001₂
  • 0x69 | 0x55 == 0x7D
    • 01101001₂ | 01010101₂ == 01111101₂

Contrast: Logic Operations in C (Sec. 2.1.8)

• Contrast to Logical Operators
  • &&, ||, !
    • View 0 as “False”
    • Anything nonzero as “True”
    • Always return 0 or 1
    • Early termination (see last e.g. below)
• Examples (char data type)
  • !0x41 == 0x00 (false) NOT(True) == False
  • !0x00 == 0x01 (true)
  • !!0x41 == 0x01 (true)
  • 0x69 && 0x55 == 0x01
  • 0x69 || 0x55 == 0x01
  • p && *p (avoids null pointer access)

Shift Operations (Sec. 2.1.9)

Practice: Shift and Mask

• For x in hex
  • convert to binary first, then perform the shift
  • x << 3 (provide result in binary and hex)
  • x >> 2 (logical)
  • x >> 2 (arithmetic)
• x = 0xC3, 0x75, 0x87, 0x66
• Problem 2.16 (p.58):

Known Muddy Points

• Ox is the prefix that is attached to a number in C, when you want to indicate the number is a hexadecimal value.
• A vector of bits is simply a sequence of bits of some length n.
• 1 byte = 8 bits