Last Time. AND/OR/NOT DeMorgan s Law
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1 Last Time AND/OR/NOT DeMorgan s Law
2 And, Or, Not The most important logical operations are and, or, and not. x and y is True only if both x and y are True. x or y is True only if either x or y are True. not x is True only if x is False. A lot like their English meanings, but unambiguous.
3 Relating And/Or/Not Note: not (not x) = x DeMorgan s Law: not flips ands and ors not (x and y) = (not x) or (not y) not (x or y) = (not x) and (not y)
4 DeMorgan s TV Schedule x: The show airs on Tuesdays y: The show airs on Mondays. not (x or y) The show does not air on Mondays and the show does not air on Tuesdays. The show does not air on Mondays or Tuesdays.
5 Clicker Questions
6 Chapter 1: Nuts and Bolts CS105: Great Insights in Computer Science
7 Multiple Switches A = False B = True C = True D = False
8 Multiple Switches A = False B = True Switches A and B are wired in parallel: either will light the bulb. C = True D = False
9 Multiple Switches A = False B = True Switches A and B are wired in parallel: either will light the bulb. C = True D = False Switches C and D are wired in series: both are needed to light the bulb.
10 Multiple Switches A = False B = True light1 = A or B Switches A and B are wired in parallel: either will light the bulb. C = True D = False Switches C and D are wired in series: both are needed to light the bulb.
11 Multiple Switches A = False B = True light1 = A or B Switches A and B are wired in parallel: either will light the bulb. C = True D = False light2 = A and B Switches C and D are wired in series: both are needed to light the bulb.
12 Multiple Circuits A B Special switches allow a single mechanical switch to control two circuits simultaneously.
13 Multiple Circuits A B light1 = A or B Special switches allow a single mechanical switch to control two circuits simultaneously.
14 Multiple Circuits A B light1 = A or B light2 = A and B Special switches allow a single mechanical switch to control two circuits simultaneously.
15 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
16 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
17 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
18 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
19 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
20 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
21 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
22 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
23 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
24 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
25 mininim There s a pile of objects, say 10. On her turn, a player can take away either one or two objects. Players alternate. The player to take the last object wins.
26 Nim5Bot: Game Tree 5 left Let s start by considering the 5-object version. takeone taketwo We ll design a strategy for the computer C to beat the user U. 3 left 2 left takeone taketwo takeone taketwo C C C U
27 Further Considerations To win mininim: if possible, remove pieces to leave opponent with a multiple of 3. Why does it work? We win if opponent has 3; if opponent has a multiple of 3, can leave her with next smaller multiple of 3. What if goal is to not take the last object? What if we can take 1, 2, or 3 objects per round? 2 or 3? 1 or 3? Is there a general rule?
28 Complete Nim5 Logic fiveleft = True threeleft = takeone 1 twoleft = taketwo 1 C-Win = (threeleft and (takeone 2 or taketwo 2 )) or (twoleft and takeone 2 ) U-Win = twoleft and taketwo 2
29 Nim5 Circuit fiveleft threeleft projects/cs105/35736 taketwo 2 U-Win takeone 1 taketwo 1 takeone 2 twoleft C-Win
30 What a Headache! Encoding Nim10 with switches is a nightmare. Why? Since some switches are used in multiple places, needs more than a double-throw switch. Since values are reused, hard to keep track of different circuits. Appears to need a separate circuit for each output: Gets too complex too fast.
31 Bits, Inside and Out Inputs: Switches A = False Outputs: Lights light1on = False B = True light2on = True
32 Bits, Inside and Out Inputs: Switches A = False Inside: Logic AND OR Outputs: Lights light1on = False B = True AND AND AND OR OR OR NOT light2on = True NOT
33 Need a New Approach Let s consider an alternate way of building and and or logic. Makes simple things more complex. Makes complex things much simpler! That s a tradeoff we can deal with.
34 Electricity Activated Switches Easy to make and and or out of switches, but we need to make switches that other switches can switch!
35 Switch Switcher A = False A = True lighton = True lighton = False Before, we used switches to control the flow of current; now, we will use current (via electromagnetism) to control switches (which control the flow of current)!
36 Relay Circuit: A=F, B=F A = False B = False lighton = False
37 Relay Circuit: A=F, B=T A = False B = True lighton = False
38 Relay Circuit: A=T, B=F A = True B = False lighton = False
39 Relay Circuit: A=T, B=T A = True B = True lighton = True
40 What Is This Thing? A B C False False False False True False True False False True True True A power source B C
41 What Is This Thing? A B C False False False True True False False False False True True True A. or gate B. and gate C. not gate D. none of these A power source B C
42 And One For Not We saw several slides ago that a single relay inverts its input signal, turning a True to a False and vice versa. These output summaries are known as truth tables. A False True B = not A A B B True False A NOT GATE B
43 Abstraction: The Black Box Our new relay-based and and not gates take current, not switches, as input. As a result, they are easier to chain together. The original switch-based scheme did not support this kind of modularity. A NOT GATE AND GATE NOT GATE C B NOT GATE
44 A Third Black Box A B C False False False False True True True False True True True True A NOT GATE B AND GATE NOT GATE NOT GATE C
45 A Third Black Box A B C False False False False True True False True A NOT GATE B NOT GATE False True True False True True True True True AND GATE False NOT GATE C
46 A Third Black Box A B C False False False False True True False True A NOT GATE B NOT GATE True False True False True True True True True AND GATE False NOT GATE C
47 A Third Black Box A B C False False False False True True True False A NOT GATE B NOT GATE False True True False True True True True True AND GATE False NOT GATE C
48 A Third Black Box A B C False False False False True True True False A NOT GATE B NOT GATE True False True False True True True True False AND GATE True NOT GATE C
49 A Third Black Box A B C False False False False True True True False A NOT GATE B NOT GATE True False True False True True True True It s an or gate : a black box built out of other black boxes! False AND GATE True NOT GATE C
50 Simplified Nim5 Circuit AND GATE U-Win True takeone 1 taketwo 1 takeone 2 taketwo 2 twoleft AND GATE OR GATE OR GATE fiveleft threeleft C-Win AND GATE
51 How to Make a Gate switches/relays hydraulic valves tinkertoys silicon: semiconductors/ transistors soap bubble DNA quantum material optics nanotubes neurons dominoes legos/marbles
52 Movie
53 NOT Gate (v3)
54 Or Gate (v4) Release bottom row first
55 Wire, transmitting bits (v1)
56 Could It Work? My domino or gate requires 24 dominoes. The first Pentium processor had 3.3M transistors, or roughly 800K gates. So, perhaps 19M dominoes needed. World record for domino toppling: 4M. Oh, and the Pentium did its computations 60M times a second, whereas dominoes might require a week to set up once.
57 But There Are Lots of Them ipod: 60Gb. 1Gb = one billion bytes, each of which is 8 bits. Format uses 128 Kbps (kilobits or 1000 bits per second of sound). So, 62,500 minutes of sound or 15,000 songs at 4 minutes per song. Screen: 320x240 pixels, each of which stores 1 byte each of R,G,B intensity (24 bits). That s 76,800 pixels and 1.8M bits. At 30 frames per sec., that s 55.3 million bits per second or 144 min. of (quiet) video.
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