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Administrivia

Administrivia. New students? No class Thursday 9/27 Lab Problem Set Favorite browser and why Favorite search engine and why Questions??. Collaboration Policy. On Problem Sets / Labs: Allowed (actually, encouraged):

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Administrivia

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  1. Administrivia • New students? • No class Thursday 9/27 • Lab • Problem Set • Favorite browser and why • Favorite search engine and why • Questions??

  2. Collaboration Policy • On Problem Sets / Labs: • Allowed (actually, encouraged): • Talk with friends (or strangers) about problems, but write up on your own • Not Allowed: • Copying verbatim • On Exams: • Obviously, no collaboration allowed. • Exams will be open notes

  3. Big Ideas in Computer Science

  4. The secret lives of 0’s and 1’s

  5. A Simple Logic Puzzle • Frank will go to the party if Ed goes AND Dan does NOT. • Dan will go if Bob does NOT go OR if Carole goes. • Ed will go to the party if Alice AND Bob go. • Alice and Bob decide to go,but Carol stays home. • Will Frank go to the party?

  6. AND Gate (cont.) X W Z Y AND W X Y Z 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1

  7. OR Gate (cont.) OR W Z W X Y Z 0 0 0 ? 0 0 1 ? 0 1 0 ? 0 1 1 ? 1 0 0 ? 1 0 1 ? 1 1 0 ? 1 1 1 ? X Y

  8. NOT Gate NOT • “Yanni will go to the party if Xena does NOT go.” Y X Shorthand: Y X X Y 0 1 1 0 Truth Table

  9. AND AND OR Logic Puzzle Circuit Alice Ed Frank Bob Dan Carole

  10. Modified AND Gate AND A F A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 B C

  11. Can realize any Truth Table by the Universal Method AND AND AND A A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 B C A B C A B C

  12. Universal Method AND AND AND OR A B C A F B C A Combine AND gates with an OR gate. B C Build an AND gate for each 1 in the truth table

  13. Universality • For ANY Truth Table we can write down,we can make a circuit for it using only 3 Logic Gates: AND, OR, NOT • In real life, make AND/OR/NOT gatesout of Transistors

  14. Our First Abstract Tool Universal Method: Circuits for ANY Truth Table 0’s & 1’s Universal Method Computers We are here

  15. Next Time: Why are 0’s and 1’s allwe need?

  16. Why are 0’s and 1’s allwe need?

  17. Limits of the Universal Method • For how large a circuit can we realistically expect to use the Universal Method? • What do we do for larger circuits?

  18. Numeracy • How large do circuits get? • How large do truth tables get?

  19. Numeracy (cont.) X Y 0 1 1 0 1-input Truth Table 2 rows

  20. Numeracy (cont.) X Y Z 0 0 0 0 1 1 1 0 1 1 1 1 X Y 0 1 1 0 1-input Truth Table 2 rows 2-input Truth Table 4 rows

  21. Numeracy (cont.) A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 3-input Truth Table: 8 rows

  22. Numeracy (cont.)

  23. Numeracy (cont.) • Each input can be 0 or 1 : 2 possibilities. • So if there are 4 inputs, that is a total of 2 * 2 * 2 * 2 = 16 possible input values.

  24. Numeracy (cont.)

  25. Numeracy (cont.)

  26. Numeracy (cont.) • Each input can be 0 or 1 : 2 possibilities. • In general, if there are n inputs, there will be 2n possible input values.

  27. Numeracy (cont.)

  28. Powers of 2 • 2n comes up a lot in Computer Science. • Numbers to memorize: • 21 = 2, 22 = 4, 23 = 8, • 24 = 16, 25 = 32, 26 = 64, • 27 = 128, 28 = 256, 29 = 512, • 210 = 1024

  29. Powers of 2 (cont.) • 210 = 1,024 • 220 = 1,048,576 • 230 = 1,073,741,824 • 240 = 1,099,511,627,776 • 250 = 1,125,899,906,842,624 • 260 = 1,152,921,504,606,846,976

  30. Powers of 2 (cont.) • Some rough numbers: • 210 = 1,024 1,000 (103 ) • 220 = 1,048,576 1,000,000 (106 ) • 230  1,000,000,000 (109 ) • 240  1,000,000,000,000 (1012 ) • 250  1015 • 260  1018

  31. Powers of 2 (cont.) • Some rough numbers: • 210  1,000 (103 ) kilo • 220  1,000,000 (106 ) mega RAM • 230  109 giga disk • 240  1012 tera BIG disk • 250  1015 peta • 260  1018 exa knowledge

  32. Sidebar on Exabytes • It's taken the entire history of humanity through 1999 to accumulate 12 exabytes of information. By the middle of 2002 the second dozen exabytes will have been created • 1 exabytes = 50,000 times the library of congress • Floppys to make 1 exabyte would stack 24 million miles high.

  33. Powers of 2 (cont.) • 16 inputs means Truth Tablehas216 = 210 * 26 = 1,024 * 64 64,000 rows • Circuit could have 64,000 or more gates

  34. Universal Method (cont.) • What about 100 inputs: • Is it reasonable to use Universal Method on Truth Table with 100 inputs?

  35. Universal Method (cont.) • What about 100 inputs: • Is it reasonable to use Universal Method on Truth Table with 100 inputs? • 2100 1030 • Each AND/OR/NOT gate uses at least 1 transistor. • This is way beyond current technology,in fact . . .

  36. Numeracy (cont.) • State-of-the-art transistors are about.1 micrometer (mm) on a side: • Lined end-to-end, you could fit10,000,000 transistors in 1 m. ( 3 f.) • In a 1 m. by 1 m. square, you could fit100,000,000,000,000 transistors • That’s still 999,999,999,999,999, 900,000,000,000,000too few for 100 input Truth Table !

  37. No good? • How sad should we be?

  38. No good? • How sad should we be? Not very. • You just need to use many Truth Tables each having fewer inputs. • Can make an entire computer using only 16-input Truth Tables and the Universal Method! • On the other hand, must realize that in some cases, we need more efficientspecial purpose circuits than the UniversalMethod. (We won’t cover these.)

  39. Our First Abstract Tool Universal Method: Circuits for ANY Truth Table 0’s & 1’s Universal Method Computers We are here

  40. So what? • We can deal with 0’s and 1’s now, but why should we? • Answer: Because we canrepresent so many thingswith 0’s and 1’s.

  41. Meaning • In Logic, we thought of 0 and 1 as meaning True and False. • Now, we remove these connotations. • Definition: a bit is just a single variable that can be 0 or 1.

  42. Representing Information • Information in the world comes in manyforms – letters, numbers, pictures, sounds… • How can we represent these thingswith 0’s and 1’s? • Start with numbers.

  43. Representing Numbers • Before computers, in devices, numbers typically represented continuously. • e.g. Clocks:

  44. Binary Numbers • How do we count normally? • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …19, 20, …99, 100, … 999, 1000, … • Suppose we only had two numerals – 0 and 1. • Then how would we count?0, 1, 10, 11, 100, 101, 110, …

  45. Binary Numbers (cont.) 0 1 2 3 4 5 6 7 8 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 0 0

  46. Binary Numbers (cont.) • Addition : • 1 0 1 • + 1 1 1 • _______________

  47. Binary Numbers (cont.) • Addition – Our “basic” addition table is really easy now: • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 10

  48. Binary Numbers (cont.) • Addition – just like usual: • 1 0 1 • + 1 1 1 • _______________ 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10

  49. Binary Numbers (cont.) • Addition – just like usual: • 1 • 1 0 1 • + 1 1 1 • _______________ • 0 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10

  50. Binary Numbers (cont.) • Addition – just like usual: • 11 • 1 0 1 • + 1 1 1 • _______________ • 0 0 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10

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