Boolean Arithmetic. From Nand to Tetris Building a Modern Computer from First Principles. Chapter 2

Transcription

1 From Nand to Tetris Building a Modern Computer from First Principles Chapter 2 Boolean Arithmetic These slides support chapter 2 of the book The Elements of Computing Systems By Noam Nisan and Shimon Schocken MIT Press Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 1

2 Usage Notice You are welcome to use this presentation, or any part of it, in Nand to Tetris courses, or in other courses, as you see fit. Usage is free provided that it supports instruction in an educational, non-profit setting. Feel free to use the slides as-is, or modify them as needed. We ll appreciate it if you will give credit somewhere to Nand to Tetris, and put a reference to You are welcome to remove this slide from the presentation. If you make extensive changes to the slides, you can remove the copyright notice also. Happy teaching! Noam Nisam / Shimon Schocken Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 2

3 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 3

4 bits 8 possibilities N bits 2 N possibilities Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 4

5 Representing numbers Binary Decimal Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 5

6 Representing numbers s 10s 1s ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 6

7 Representing numbers s 10s 1s ten = = 789 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 7

8 Binary Decimal s 2s 1s two Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 8

9 Binary Decimal s 2s 1s two = = 5 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 9

10 Binary Decimal Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 10

11 Binary Decimal b n b n-1 b n-2 b 1 b 0 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 11

12 Binary Decimal b n b n-1 b n-2 b 1 b 0 = n b i 2 i i=0 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 12

13 Binary Decimal b n b n-1 b n-2 b 1 b 0 = n b i 2 i i=0 Maximum value represented by k bits: k-1 = 2 k -1 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 13

14 Fixed word size Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 14

15 Fixed word size We will use a fixed number of bits. Say 8 bits Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 15

16 Fixed word size We will use a fixed number of bits. Say 8 bits = 256 values Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 16

17 Representing signed numbers We will use a fixed number of bits. Say 8 bits positive values negative values Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 17

18 Representing signed numbers We will use a fixed number of bits. Say 8 bits positive values That s one possible representation We ll use a better one, later negative values Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 18

19 Decimal Binary 87 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 19

20 Decimal Binary 87 ten =???????? two Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 20

21 Decimal Binary 87 ten = 64 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 21

22 Decimal Binary 87 ten = Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 22

23 Decimal Binary 87 ten = Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 23

24 Decimal Binary 87 ten = Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 24

25 Decimal Binary 87 ten = Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 25

26 Decimal Binary 87 ten = =???????? two Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 26

27 Decimal Binary 87 ten = = two Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 27

28 Decimal Binary 87 ten = = two 32 8 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 28

29 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 29

30 Boolean arithmetic Addition Subtraction Comparison (<,>,=) Multiplication Division implement get for free get for free postpone to software postpone to software Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 30

31 Addition ???????? Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 31

32 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 32

33 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 33

34 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 34

35 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 35

36 Addition ???? Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 36

37 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 37

38 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 38

39 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 39

40 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 40

41 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 41

42 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 42

43 Addition ???????? Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 43

44 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 44

45 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 45

46 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 46

47 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 47

48 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 48

49 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 49

50 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 50

51 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 51

52 Addition Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 52

53 Overflow Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 53

54 Overflow ???????? Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 54

55 Overflow Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 55

56 Building an Adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 56

57 Building an Adder Half adder: adds two bits Full adder: adds three bits Adder: adds two integers Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 57

58 Half adder carry bit a b sum carry sum bit Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 58

59 Half adder a b sum carry HalfAdder.hdl /** Computes the sum of two bits. */ CHIP HalfAdder { IN a, b; OUT sum, carry; } PARTS: // Put your code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 59

60 Full adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 60

61 Full adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 61

62 Full adder carry bit sum bit Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 62

63 Full adder FullAdder.hdl /** Computes the sum of three bits. */ CHIP HalfAdder { IN a, b, c; OUT sum, carry; } PARTS: // Put your code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 63

64 Multi-bit Adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 64

65 Multi-bit Adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 65

66 Multi-bit Adder Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 66

67 Multi-bit Adder Add16.hdl /* Adds two 16-bit, two s-complement values. * The most-significant carry bit is ignored. */ CHIP Add16 { IN a[16], b[16]; OUT out[16]; } PARTS: // Put you code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 67

68 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 68

69 Representing numbers (using 4 bits) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 69

70 Representing numbers (using 4 bits) Using n bits, we can represent the positive integers in the range 0 2 n -1 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 70

71 Representing negative numbers Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 71

72 Possible solution: use a sign bit Use the left-most bit to represent the sign, -/+ Use the remaining bits to represent a positive number Complications: -0? x + (-x) is not 0 more complications Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 72

73 Two's Complement Represent the negative number -x using the positive number 2 n - x Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 73

74 Two's Complement (16-8) (16-9) (16-10) (16-11) (16-12) (16-13) (16-14) (16-15) Represent the negative number -x using the positive number 2 n - x Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 74

75 Two's Complement (16-8) (16-9) (16-10) (16-11) (16-12) (16-13) (16-14) (16-15) positive numbers range: 0 2 n-1-1 negative numbers range: -1-2 n - 1 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 75

76 Addition using two's complement Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 76

77 Addition using two's complement Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 77

78 Addition using two's complement Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 78

79 Addition using two's complement = 27 ten 1011 = 11 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 79

80 Addition using two's complement = 27 ten 1011 = 27 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 80

81 Addition using two's complement Two s complement rationale: representation is modulu 2 n addition is modulu 2 n = 27 ten 1011 = 27 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 81

82 Computing -x Input: x Output: -x (in two's complement) Insight: if we solve this we ll know how to subtract: y x = y + (-x) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 82

83 Computing -x Input: x Output: -x (in two's complement) Idea: 2 n x = 1 + (2 n - 1) - x two (some x example) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 83

84 Computing -x Input: x Output: -x (in two's complement) Idea: 2 n x = 1 + (2 n - 1) - x two (some x example) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 84

85 Computing -x Input: x Output: -x (in two's complement) Idea: 2 n x = 1 + (2 n - 1) - x two (some x example) (flip all the bits) Now add 1 to the result Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 85

86 Computing -x (example) Input: 4 Output: should be 12 (representing -4 in two s compelemt) Input: 0100 Flip the bits: 1011 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 86

87 Computing -x (example) Input: 4 Output: should be 12 (representing -4 in two s compelemt) Input: 0100 Flip the bits: Add one: 1 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 87

88 Computing -x (example) Input: 4 Output: should be 12 (representing -4 in two s compelemt) Input: 0100 Flip the bits: Add one: 1 Output: 1100 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 88

89 Computing -x (example) Input: 4 Output: should be 12 (representing -4 in two s compelemt) Input: 0100 Flip the bits: Add one: 1 To add 1: Flip all the bits from right to left, stop when the first 0 flips to 1 Output: 1100 = 12 ten Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 89

90 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 90

91 Von Neumann Architecture Computer System CPU Input Memory ALU Output Contro l Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 91

92 The Arithmetic Logical Unit Computer System CPU Input Memory ALU Output Contro l Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 92

93 The Arithmetic Logical Unit The ALU computes a function on the two inputs, and outputs the result input1 f f : one out of a family of pre-defined arithmetic and logical functions input2 ALU f (input1, input2) Arithmetic functions: integer addition, multiplication, division,... logical functions: And, Or, Xor, Which functions should the ALU perform? A hardware / software tradeoff. Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 93

94 The Hack ALU Operates on two 16-bit, two s complement values Outputs a 16-bit, two s complement value Which function to compute is set by six 1-bit inputs out 0 Computes one out of a family of 18 functions 1 Also outputs two 1-bit values (to be discussed later). -1 x y zx nx zy ny f no!x!y -x -y x+1 y+1 x 16 bits x-1 ALU out y-1 y 16 bits x+y 16 bits x-y y-x x&y x y zr ng Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 94

95 The Hack ALU out x y zx nx zy ny f no!x!y -x -y x+1 y+1 x 16 bits x-1 ALU out y-1 y 16 bits x+y 16 bits x-y y-x x&y x y zr ng Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 95

96 The Hack ALU x y To cause the ALU to compute a function, set the control bits to one of the binary combinations listed in the table. control bits zx nx zy ny f no out x y zx nx zy ny f no !x !y x y x y+1 16 bits x-1 ALU out y-1 16 bits x+y 16 bits x-y y-x x&y x y zr ng Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 96

97 The Hack ALU in action: compute y-x 2. Evaluate the chip logic Load tools/builtinchips/alu.hdl 3. Inspect the ALU outputs 1. Set the ALU s inputs and control bits to some test values ( codes output y-x ) The built-in ALU implementation has some GUI side-effects Built-in ALU implementation Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 97

98 The Hack ALU in action: compute x & y Set to binary I/O format Inspect the ALU outputs Set the ALU s inputs and control bits to some test values ( codes compute x&y ) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 98

99 Opening up the Hack ALU black box control bits x y out = f(control bits, x, y) Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 99

100 The Hack ALU operation pre-setting the x input pre-setting the y input selecting between computing + or & post-setting the output Resulting ALU output zx nx zy ny f no out if zx then x=0 if nx then x=!x if zy then y=0 if ny then y=!y if f then out=x+y else out=x&y if no then out=!out out(x,y)= zx nx zy ny f no x y 16 bits 16 bits ALU 16 bits out zr ng Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 100

101 The Hack ALU operation pre-setting the x input pre-setting the y input selecting between computing + or & post-setting the output Resulting ALU output zx nx zy ny f no out if zx then x=0 if nx then x=!x if zy then y=0 if ny then y=!y if f then out=x+y else out=x&y if no then out=!out out(x,y)= x y !x !y x y x y x y x+y x-y y-x x&y x y Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 101

102 ALU operation example: compute!x pre-setting the x input pre-setting the y input selecting between computing + or & post-setting the output Resulting ALU output zx nx zy ny f no out if zx then x=0 if nx then x=!x if zy then y=0 if ny then y=!y if f then out=x+y else out=x&y if no then out=!out out(x,y)= x y !x !y Example: compute!x x x: y y: x y Following 1 pre-setting: 1 0 x x: y y: x+y x-y Computation 1 and 1 post-setting: 1 y-x x&y: x&y 0 1 0!(x&y): (!x) 1 x y Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 102

103 ALU operation example: compute y-x pre-setting the x input pre-setting the y input selecting between computing + or & post-setting the output Resulting ALU output zx nx zy ny f no out if zx then x=0 if nx then x=!x if zy then y=0 if ny then y=!y if f then out=x+y else out=x&y if no then out=!out out(x,y)= Example: 1 compute 1 y-x x x: (2) y: (7) y !x Following 1 pre-setting: !y x x: y y: x y+1 Computation and post-setting: x-1 x+y: y-1!(x+y): (5) x+y x-y y-x x&y x y Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 103

104 The Hack ALU output control bits zx nx zy ny f no x y 16 bits 16 bits ALU 16 bits out zr ng if (out == 0) then zr = 1, else zr = 0 if (out < 0) then ng = 1, else ng = 0 These two control bits will come into play when we build the complete computer s architecture. Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 104

105 Perspective The Hack ALU is: Simple Elegant To implement it this ALU, you only need to know how to: Set a 16-bit value to Set a 16-bit value to Negate a 16-bit value (bit-wise) Compute plus or And on two 16-bit values That s it! Simplicity is the ultimate sophistication. Leonardo da Vinci Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 105

106 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 106

107 Project 2 Given: All the chips built in Project 1 Goal: Build the following chips: HalfAdder FullAdder Add16 Inc16 ALU A family of combinational chips, from simple adders to an Arithmetic Logic Unit. Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 107

108 Half Adder a b sum carry HalfAdder.hdl /** Computes the sum of two bits. */ CHIP HalfAdder { IN a, b; OUT sum, carry; Implementation tip Can be built using two very elementary gates. } PARTS: // Put your code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 108

109 Full Adder a b c sum carry FullAdder.hdl /** Computes the sum of three bits. */ CHIP HalfAdder { IN a, b, c; OUT sum, carry; } PARTS: // Put your code here: Implementation tips Can be built using two half-adders. Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 109

110 16-bit adder Add16.hdl Implementation tips An n-bit adder can be built from n full-adder chips The carry bit is piped from right to left The MSB carry bit is ignored. /* * Adds two 16-bit, two s-complement values. * The most-significant carry bit is ignored. */ CHIP Add16 { IN a[16], b[16]; OUT out[16]; } PARTS: // Put you code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 110

111 16-bit incrementor Implementation tip The single-bit 0 and 1 values are represented in HDL as false and true. Inc16.hdl /* * Outputs in + 1. * The most-significant carry bit is ignored. */ CHIP Inc16 { IN in[16]; OUT out[16]; } PARTS: // Put you code here: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 111

112 ALU Implementation tips Building blocks: Add16, and some gates built in project 1 Can be built with ~20 lines of HDL code ALU.hdl /** The ALU. */ // Manipulates the x and y inputs as follows: // if (zx == 1) sets x = 0 // 16-bit true constant // if (nx == 1) sets x =!x // bitwise Not // if (zy == 1) sets y = 0 // 16-bit true constant // if (ny == 1) sets y =!y // bitwise Not // if (f == 1) sets out = x + y // int. 2's-complement addition // if (f == 0) sets out = x & y // bitwise And // if (no == 1) sets out =!out // bitwise Not // if (out == 0) sets zr = 1 // 1-bit true constant // if (out < 0) sets ng = 1 // 1-bit true constant... Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 112

113 Project 2 Resources All the necessary project 2 files are available in: nand2tetris / projects / 02 Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 113

114 More resources HDL Survival Guide Hardware Simulator Tutorial nand2tetris Q&A forum All available in: Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 114

115 Best practice advice Try to implement the chips in the given order If you don t implement some of the chips required in project 2, you can still use them as chip-parts in other chips. Just rename the given stub-files; this will cause the simulator to use the built-in versions of these chips You can invent new, helper chips ; however, this is not required: you can build any chip using previously-built chips only Strive to use as few chip-parts as possible You will have to use chips implemented in Project 1 For efficiency and consistency s sake, use their built-in versions rather than your own implementation. Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 115

116 Chapter 2: Boolean arithmetic Binary numbers Binary addition Negative numbers Arithmetic Logic Unit Project 2 overview Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 116

117 From Nand to Tetris Building a Modern Computer from First Principles Chapter 2 Boolean Arithmetic These slides support chapter 2 of the book The Elements of Computing Systems By Noam Nisan and Shimon Schocken MIT Press Nand to Tetris / / Chapter 2 / Copyright Noam Nisan and Shimon Schocken Slide 117