系统级程序设计 - chapter2

Representing information as bits

• Each bitis 0 or 1（每位是0或1）

• By encoding/interpreting sets of bits in various ways
  – Computers determine what to do (instructions)
  – … and represent and manipulate numbers, sets, strings, etc…

• Why bits? Electronic Implementation
  – Easy to store with bistable elements（双稳态成分）
  – Reliably transmitted on noisy and inaccurate wires
  

Bit - level manipulations

Boolean Algebra （布尔算法）

Example: Representing & Manipulating Sets

Representation

– Width w bit vector represents subsets of {0, …, w–1} 序号从0开始然后从右开始
– a_j = 1 if j ∈ A

• 01101001 {0,3,5,6}
  76543210
• 01010101 {0,2,4,6}
  76543210

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 }

Contrast: Logic Operations in C

Contrast to Logical Operators

• &&, ||, ! （两个表示短）
  – View 0 as “False”
  – Anything nonzero as “True” （非0都当作正确）
  – Always return 0 or 1
  – Early termination（早结束）

Shift Operations（移动操作）

Left Shift: x << y

• Shift bit-vector x left y positions
  –- Throw away extra bits on left and fill with 0’s on right（丢丢丢全丢掉）
  Right Shift: x >> y
• Shift bit-vector x right y positions
  – Throw away extra bits on right
• Logical shift 逻辑移动
  – Fill with 0’s on left
• Arithmetic shift 算数移动
  – Replicate most significant bit on left  ---- Undefined Behavior
  – Shift amount < 0 or ≥ word size

Integers

Representation: unsigned and signed

Numeric Ranges

Unsigned Values
无符号的讨论

• UMin = 0
  – 000…0
• UMax = 2^w – 1
  –111…1
  Two’s Complement Values
• TMin = -2^w-1
• 10…0
• TMax = 2^w-1-1
  – 011…1

Values for W = 16

Conversion, casting

Mapping Signed & Unsigned

Conversion Visualized

Casting Surprises

• Expression Evaluation
  – If there is a mix of unsigned and signed in single expression, signed values implicitly cast to unsigned
• Including comparison operations <,>,==,<=,>=
• Examples for W = 32:
  – TMIN = -2,147,483,648
  – TMAX = 2,147,483,647
  

Summary

Casting Signed ↔ Unsigned: Basic
Rules

• Bit pattern is maintained
• But reinterpreted
• Can have unexpected effects: adding or subtracting 2w
• Expression containing signed and unsigned int
  – int is cast to unsigned!!

Expanding, truncating

Task:

• Given w-bit signed integer x
• Convert it to w+k-bit integer with same value
  Expanding, Truncating: Basic Rules - Expanding (e.g., short int to int)
  – Unsigned: zeros added
  – Signed: sign extension
  – Both yield expected result
• Truncating (e.g., unsigned to unsigned short)
  – Unsigned/signed: bits are truncated
  – Result reinterpreted
  – Unsigned: mod operation
  – Signed: similar to mod
  – For small numbers yields expected behavior

Addition, negation, multiplication, shifting

Unsigned Addition

Multiplication

• Goal: Computing Product of w-bit numbers x, y
  – Either signed or unsigned
• But, exact results can be bigger than w bits
  – Unsigned: up to 2w bits
  Result range:0≤x * y≤(2^w –1)2 = 2^2w –2^w+1 +1
  – Two’s complement min (negative): Up to 2w-1 bits
  Result range: x * y ≥ (–2^w–1)*(2^w–1–1) = –2^2w–2 + 2^w–1
  – Two’s complement max (positive): Up to 2w bits, but only for (TMin_w)²
  Result range: x * y ≤ (–2w–1) 2 = 2^2w–2

Summary

Addition:

Unsigned/signed: Normal addition followed by truncate, same operation on bit level

• Unsigned: addition mod 2w
  – Mathematical addition + possible subtraction of 2w
• Signed: modified addition mod 2w (result in proper range)
  – Mathematical addition + possible addition or subtraction of 2w

Multiplication:

• Unsigned/signed: Normal multiplication followed by truncate, same operation on bit level
• Unsigned: multiplication mod 2w
• Signed: modified multiplication mod 2w (result in proper range)

Representations in memry, pointer, strings

Machine Words

Any given computer has a “Word Size”

Byte Ordering

Example

• Variable x has 4-byte value of 0x01234567
• Address given by &xis 0x100
  大端地址
  – 大端地址放低字节（低位）
  小端地址
  – 小端地址放低字节（低位）
  

Complement & Increment Examples