IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 T = today x= hw#x out x= hw#x due mon tue wed thr fri 3 M 7 oh M 4 oh oh 2M2 2 oh oh 2 oh T Mx= MQx out 28 oh M2 oh oh = office hours oh 3 4 oh oh midterms oh Mx= MQx due 9 oh oh oh 5 oh oh oh

3 Last lecture: Associative Memories, Yue Li William Scoville (96 984) Brenda Milner (98 - ) Suzanne Corkin Henry Molaison or H. M. (926-28) Alexander Luria (92 977) Sarnoff A. Mednick Joshua Foer

4 MQ2

5 MQ2 Memory Deadline Thursday 4/29/24 at pm You are invited to write short essay on the topic of the Magenta Question. Recommended length is 3 pages (not more) Submit the essay in PDF format to ta4@paradise.caltech.edu file name lastname-firstname.pdf No collaboration. No extensions Grading of MQ: 3 points (out of 3) 5% for content quality, 5% for writing quality Some students will be given an opportunity to give a short presentation for up to 3 additional points

6 A word that is associated with the following?

7 Life: Unique Origin 3.5 Bya Origins?? Memory DNA Novelty Editing? Mutation? Selection? New species?

8 Human brain: Unique Origin: Origins 2, people in Africa 6 Kya Memory Languages g Novelty Interaction, editing, selection New ideas/languages

9 Will be posted on the class website Funes the Memorious A short story by: Jorge Luis Borges With no effort, he had learned English, French, Portuguese and Latin. I suspect, however, that he was not very capable of thought. To think is to forget differences, generalize, make abstractions. In the teeming world of Funes, there were only details, almost immediate in their presence.

10 Babylonian Clay Tablets Greek Proofs... Memory of mathematical knowledge

11 MCMXX = 93

12 Some History on Roman Numerals Origins of roman numerals are believed to be Origins of roman numerals are believed to be in the form of notches on tally sticks, such as those used by European shepherds

13 The Roman Numeral Puzzle: Very slow impact of the Babylonians??? Positional number system influence Babylonia 5, ya Greek, India, Chinese 2,5 ya Persia, Arabs,5 ya Europe 5 ya

14 The Syntax of Roman Numerals

15 A Refresher on the Roman Numeral System Roman Number Large Numeral Roman Number I Numerals V 5 V 5, X X, L 5 L 5, C C, D 5 D 5, M M,, LCD Monitor

16 The Abacus: It s all About Syntax IIIII V V D L V XXXXX CCCCC MMMMM L D V VV X M C X I LL C DD M

17 IIIII XXXXX CCCCC MMMMM The Abacus: It s all About Syntax V L D V What is the number? V D L V VV X M C X I LL C DD M

18 IIIII XXXXX CCCCC MMMMM The Abacus: It s all About Syntax V L D V What is the number? V D L V VV X M C X I LL DD C M DCI 6

19 The Abacus: It s all About Syntax IIIII XXXXX CCCCC MMMMM Touch the middle: Yes or No V V D L V it is a binary mechanism L D V VV X M C X I LL DD C M DCI 6

20 The Abacus Calculating Machine are Based on Syntax The first actual calculating mechanism known to us is the abacus, which h is thought ht to have been invented nt by the Babylonians sometime between, BC and 5 BC The original concept referred to a flat stone covered with sand or dust, with pebbles being placed on lines drawn in the sand Source: Wikipedia

21 The Abacus Calculating Machine are Based on Syntax The original concept referred to a flat stone covered with sand or? dust, with pebbles being placed on lines drawn in the sand? In Phoenician the word abak means sand In Hebrew the word abhaq אָב ק means dust Calculus is Latin for pebble Source: Wikipedia

22 The Abacus: It s all About Syntax IIIII V V D L V XXXXX L CCCCC D MMMMM V VV X M C X I LL DD C M DCI 6

23 IIIII XXXXX CCCCC MMMMM The Abacus: It s all About Syntax V L D V What is the number? V D L V VV X M C X I LL DD C M CCCCLXXXVIIII 489

24 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

25 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

26 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

27 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

28 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

29 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

30 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

31 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

32 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

33 The Abacus: It s all About Syntax V D L V M C X I DCI 6 + CCCCLXXXVIIII 489

34 The Abacus: It s all About Syntax What is the decimal representation? V D L V MLXXXX?? M C X I DCI 6 + CCCCLXXXVIIII 489

35 The Abacus: It s all About Syntax What is the decimal representation? V D L V MLXXXX 9 What s wrong with this picture? M C X I DCI 6 + CCCCLXXXVIIII 489

36 Roman Numerals and Base Systems MLXXXX Roman numerals used for number Representation V D The representation in the abacus is a positional base representation 9 L V For calculation: we used the abacus M C X I

37 From Physical (abacus) to Symbols Algorizms

38 Algorizmi A positional number system is a key enabler for efficient arithmetic operations Operations are done on syntax Muhammad ibn Mūsā al-khwārizmī محمد بن موسى ورزي الخوارزمي 78-85AD85AD A Persian mathematician, who wrote on Hindu-Arabic numerals and was among the first to use zero as a place holder in positional base notation. The word algorithm derives from his name. His book Kitab al-jabr w'al-muqabala gives us the word algebra Source: Wikipedia

39 The Beginning of the Algebra Book by Algorizmi Everything requires computation...

40 The Beginning of the Algebra Book by Algorizmi Positional: order is important; from to infinity...

41 Example from the Algebra Book by Algorizmi computation = syntax manipulation?? It is rhetorical (words) no symbols

42 Algorithms and Algebra in Europe Leonardo Fibonacci 7-25AD Leonardo was born in Pisa, his father directed a trading post in Bugia, a port east of Algiers in North Africa, as a young boy Leonardo traveled there to help him. This is where he learned about the Arabic numeral system Perceiving that arithmetic with Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time, returning ng around 2. In 22, at age 32, he published what he had learned in Liber Abaci, or Book of Calculation Source: Wikipedia

43 Liber Abaci First Chapter Introduction of the syntax; from to infinity...

44 Liber Abaci First Chapter Positional: order is important

45 al-khwarizmi 78-85AD Dear Caltech students an algorizm is a procedure for syntax (language) processing (composition) Dear Mr. Algorizmi, how would you define an algorithm??? Thank you!

46 In first grade: We use our BRAIN for remembering Algorizmi s syntax

47 In first grade: We use our BRAIN for remembering Algorizmi s syntax Avoiding headaches! Use syntax-boxes for remembering Algorizmi s syntax

48 Syntax Boxes composition and magic

49 Syntax Manipulation with Boxes inputs a b a b o S-Box outputs o

50 Syntax Manipulation with Boxes inputs a b o outputs

51 Syntax Manipulation with Boxes inputs a b o outputs

52 Syntax Manipulation with Boxes inputs 2 a b o 2 2 outputs 2

53 Syntax Manipulation with Boxes inputs 2 a b o 2 2 outputs 2 2 2

54 Syntax Manipulation with Boxes inputs 2 a b o 2 2 outputs

55 Syntax Manipulation with Boxes Can we compute max (a,b,c) with this s-box? inputs outputs a b? o o = max (a,b) a b o

56 Syntax Manipulation with Boxes Can we compute max (a,b,c) with this s-box? o = max (a,b) a b

57 Syntax Manipulation with Boxes Can we compute max (a,b,c) with this s-box? o = max (a,b) a b Composition: build big s-boxes from small s-boxes c

58 Syntax Manipulation with Boxes Can we compute z with the max s-box?? a b z a b a b o 2 YES 2 o o = max (a,b)

59 Syntax Manipulation with Boxes Can we compute w with the max s-box?? a b w a b a b o 2 NO 2 Why not? o o = max (a,b)

60 Syntax Manipulation with Boxes Can we compute w with the max s-box? a? b w NO Why not? a b The output of the big s-box must be bigger or equal to its inputs Composition: build big s-boxes from small s-boxes The output of every small s-box is bigger or equal to its inputs Big s-box w

61 Is there a finiteuniversalit i set of building blocks? Can construct everything. The most important idea in Information ABCDE... DNA

62 A Magic (Universal) Box A binary s-box that can compute any binary s-box? a b m a b m HW#

63 A Magic Box min(x,y) Can you compute the following with the magic box? a b a b m x y o m

64 Hint : A Magic Box min(x,y) Can you compute the following with the magic box? a b a b m x y o m

65 Hint 2: A Magic Box min(x,y) Can you compute the following with the magic box? a b a b m x y o m

66 A Magic Box min(x,y) x y a b m x y o o

67 A Magic Box min(x,y) x y a b m x y o o

68 HINT A Magic Box max(x,y) Can you compute the following with the m-box? a b a b m x y o m

69 A Magic Box max(x,y) x y a b m x y o o

70 A Magic Box max(x,y) x y a b m x y o o

71 A Magic Box max(x,y) x y a b m x y o o

72 A Magic Box max(x,y) x y a b m x y o o

73 m-box: A two input binary syntax box that can compute any (two input) binary syntax box? How many different binary 2-input s-boxes? 2 4 = 6 a b m a b x y o How will you prove it? * * * * m

74 Syntax Boxes proof of universality

75 4 Useful Boxes min(x,y) -y x y o x y o max(x,y)

76 -y y a b m x y o o

77 y -y a b m x y o o

78 4 Useful Boxes min(x,y) -y So what? ( ) max(x,y) Need to prove: any (two input) binary syntax box can be computed by the 4 Useful Boxes

79 An Arbitrary Two Input Box Two -input boxes! x y o x= then x= then * * * * What are the possible values of

80 An Arbitrary Two Input Box min(x,y) -y max(x,y) y What are the possible values of x y o Can we compute it with the m-box? * * * *

81 An Arbitrary Two Input Box x= then x y o x= then * How can you compute this box? * * * x o

82 x= then x= then -x x min(a,b) min(a,b) max(a,b) o

83 x= then x= then min(a,b) min(a,b) max(a,b) o

84 x= then x= then min(a,b) min(a,b) max(a,b) o QED

85 Does the magic continue? Given a 2-input binary box that can compute any 2-input binary box Can it compute any 3-input binary box? a b x y z m-box o o

86 How many different binary 3-input s-boxes? 2 8 = input binary s-box x y z o * * * * * * * *

87 3-input binary s-box Two 2-input boxes! x y z o * * z= then * * z= then * * * *

88 x y x y x y z o * * * * * * * * z z= then o z= then

89 ???? z o -z min(a,b) z= then z= then z min(a,b) x y z o * * * * * * * * max(a,b) o

90 x y z o * * * * * min(a,b) min(a,b) * * * max(a,b) o

91 x y z o * * * * * min(a,b) min(a,b) * * * max(a,b) o

92 Does the magic continue? Given a magical box for any 2-input binary box We proved that it is magical for any 3-input binary box! Is it magical for any n-input binary box???? Proof by induction on the number of inputs a b YES!!!!... m-box o o

93 z= then z= then Are tables with n- variables -z z min(a,b) min(a,b) max(a,b) o

94 We need a language for S-boxes!! Questions about building blocks? Feasibility Given a set of building blocks: What can/cannot be constructed? Efficiency and complexity Given a set of building blocks and a description of a structure: Size: If feasible, how many blocks are needed? Time: How long will it take to complete the construction?

95 A word that is associated with the following? Face

96 A word that is associated with the following? Face

97 You have one week!

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100 Difference in approximation - 7 digits it base 6

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