IST 4 Information and Logic

Transcription

1 IST 4 Informtion nd Logic

2 T = tody x= hw#x out x= hw#x due mon tue wed thr fri 31 M1 1 7 oh M1 14 oh 1 oh 2M2 21 oh oh 2 oh Mx= MQx out Mx= MQx due 28 oh M2 oh oh = office hours oh 3 T 4 oh oh 19 oh oh oh 5 midterms oh oh oh oh

3 Everything is 0-1 (Two Vlued)????

4 Logic to Physics Clude Shnnon s

5 Shnnon Shnnon s Bckground Clude Elwood Shnnon ws born in Petoskey, Michign, on April 30, 1916 In 1932 (16) he entered the University of Michign, where he took course tht introduced him to the work of George Boole He grduted d in 1936 (20) with two bchelor 's degrees, one in electricl engineering nd one in mthemtics Joined MIT in 1936, received the msters in electricl engineering nd doctorte in Mthemtics, t the 1940 (24) commencement

6 Shnnon s Inspirtion Joined MIT in 1936 Shnnon Vnnevr Bush Smuel Cldwell The differentil nlyzer t MIT (1931) ws the first generl eqution solver It could hndle sixth-order differentil equtions

7 Shnnon Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 Hitchcock Shnnon s dvisor both MSc nd PhD mthemticin 76 yers go

8 Hitchcock Bush Sutherlnd ws fculty t Cltech from 1974 to 1978 (lso MS degree) Served s the founding chir of the CS Deprtment t Cltech Shnnon Ivn Sutherlnd 1938-

9 Shnnon Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 Beginning: g No mention of computers... they did not exist A concept tht is missing in the text?

10 Logic to Physics The lnguge of lines s

11 Shnnon meets Boole Physics meets Logic A lnguge for synthesizing lrge physicl mchines for COMPUTING

12 Boolen Clculus nd Physicl Circuits Single Lines nd Composition The lnguge of lines: A line cn hve only two possible colors: blue or red Two lines cn be composed in two possible wys In prllel: In series:

13 Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn be composed in two possible wys In prllel:

14 Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn be composed in two possible wys In series:

15 Boolen Clculus nd Physicl Circuits Composition In prllel: Compositions of lines cn be composed in two different wys, using their endpoints b b

16 Boolen Clculus nd Physicl Circuits Composition In prllel: Compositions of lines cn be composed in two different wys, using their endpoints

17 Boolen Clculus nd Physicl Circuits Composition Compositions of lines cn be composed in two different wys, using their endpoints In series: b b

18 Boolen Clculus nd Physicl Circuits Composition In series: Compositions of lines cn be composed in two different wys, using their endpoints

19 Boolen Clculus nd Physicl Circuits Color of Composition Wht is the color of composition? color = blue color = red???

20 Boolen Clculus nd Physicl Circuits Color of Composition Wht is the color of composition? color = blue color = red The color of composition is red if there is red pth between the endpoints Otherwise, the color is blue

21 Is the two-color line composition 0-1 Boolen lgebr? Wht do we need to do??

22 Boolen Algebr Algebric system: set of elements B, two binry opertions + nd B hs t lest two elements (0 nd 1) If the following xioms re true then it is Boolen Algebr: A1. identity A2. complement A3. commuttive A4. distributive

23 Two-Colored Line Composition nd 0-1 Boolen Algebr Algebric system: set of elements B, two binry opertions + nd B hs t lest two elements (0 nd 1) Elements: 0 1

24 Two-Colored Line Composition nd 0-1 Boolen Algebr Algebric system: set of elements B, two binry opertions + nd B hs t lest two elements (0 nd 1) Elements: Opertions: Compose in prllel: The color of composition is red if there is red pth between the endpoints 0 1 Otherwise, the color is blue + Compose in series:

25 Two-Colored Line Composition nd 0-1 Boolen Algebr Algebric system: set of elements B, two binry opertions + nd B hs t lest two elements (0 nd 1) Elements: Opertions: Compose in prllel: blue prllel configurtion? Compose in series: red series configurtion?

26 Two-Colored Line Composition nd 0-1 Boolen Algebr 0 The color of the composition equls the color of + Compose in prllel The color of the composition i Compose in series equls the color of 1 The color of composition is red if there is red pth between the endpoints Otherwise, the color is blue

27 Two-Colored Line Composition nd 0-1 Boolen Algebr The color of the composition is red = 1 The color of the composition is blue = 0 + Compose in prllel Compose in series The color of composition is red if there is red pth between the endpoints. Otherwise, the color is blue

28 Two-Colored Line Composition nd 0-1 Boolen Algebr By the definition of the color of composition + Compose in prllel Compose in series The color of composition is red if there is red pth between the endpoints. Otherwise, the color is blue

29 Two-Colored Line Composition nd 0-1 Boolen Algebr b Two disjoint pths: The top one is determined by,the bottom one is bc b c c Includes the two pths nd bc Two other pths re b nd c determined d by so re redundnt

30 Is the two-color line composition 0-1 Boolen lgebr?

31 circuits = lgebr In Shnnon s words: We re now in position to demonstrte the equivlence of this clculus with certin elementry prts of the clculus of propositions. The lgebr of logic originted by George Boole, is symbolic method of investigting logicl reltionships. The symbols of Boolen lgebr dmit of two logicl interprettions. If interpreted in terms of clsses, the vribles re not limited to the two possible vlues 0 nd 1. E. V. Huntington' gives the following set of postultes for symbolic logic:

32 Shnnon Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 Rely on the edge controlled by 0-1 vrible

33 Shnnon Connection Between Boolen Clculus nd Physicl Circuits Shnnon The vlue of circuit is 1 if there is connected pth between the endpoints Otherwise, it is The color of composition is red if there is red pth between the endpoints Otherwise, the color is blue

34 Rely Circuits nlysis nd synthesis s

35 nlysis of circuits..ny circuit is represented by set of equtions, The terms of the equtions corresponding to the vrious relys nd switches in the circuit. A A clculus is developed for mnipulting these equtions by simple mthemticl processes most of which re similr to ordinry lgebric lgorisms. This clculus is shown to be exctly nlogous to the clculus of propositions used in the symbolic study of logic.

36 synthesis of circuits For the synthesis problem the desired chrcteristics re first written s system of equtions, nd the equtions re then mnipulted into the form representing the simplest circuit. The circuit my then be immeditely drwn from the equtions. desired chrcteristic system of equtions simplified set of equtions simple circuit

37 synthesis of circuits For the synthesis problem the desired chrcteristics re first written s system of equtions, nd the equtions re then mnipulted into the form representing the simplest circuit. The circuit my then be immeditely drwn from the equtions. desired chrcteristic Logic design is born! system of equtions Complexity? simplified set of equtions simple circuit

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39 The First Book on Switching Circuits Keister, Ritchie nd Wshburn, Willim Keister

40 SpinOut Keister Willim Keister Keister Willim Keister ws pioneer in switching theory nd design t Bell Lbs When he retired in 1972, he ws director of Bell Lbs' Computing Technology Center t Holmdel, New Jersey. Keister begn working in his spre time to prove tht puzzles could be solved using Boolen lgebr U S Ptent (1972): SpinOut U.S. Ptent (1972): SpinOut U.S. Ptent (1972): The Hexdeciml Puzzle

41 The First Book on Switching Circuits Keister, Ritchie nd Wshburn, 1951 C nd Unix Dennis Ritchie Son of

42 The First Book on Switching Circuits Keister, Ritchie nd Wshburn, 1951 Being recognized by the president with co-inventor Ken Thompson C nd Unix Dennis Ritchie Son of

43 The First Book on Switching Circuits Keister, Ritchie nd Wshburn, 1951

44 Rely Circuits nlysis s

45 A rely circuit it corresponds to formul rely circuits Boolen functions Boolen sum of ll the pths between endpoints

46 Anlysis of Rely Circuits Exmple 1: - series-prllel - independent pths between endpoints d b c e b c d e

47 Exmple 1: - series-prllel - independent pths between endpoints b d c e b c d e

48 Exmple 1: - series-prllel - independent pths between endpoints b d c e b c d e

49 Exmple 1: - series-prllel - independent pths between endpoints b d c e b c d e

50 Exmple 1: - series-prllel - independent pths between endpoints b d c e b c d e

51 Anlysis of Rely Circuits Exmple 2: - non series-prllel - dependent pths between endpoints b c d e

52 Anlysis of Rely Circuits b c d e

53 Anlysis of Rely Circuits b c d e

54 Anlysis of Rely Circuits b c d e

55 Anlysis of Rely Circuits b c d e

56 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) How mny functions? b c d e

57 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

58 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

59 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

60 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

61 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

62 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

63 Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) b c d e

64 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths between endpoints Q: how mny FORWARD pths?

65 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths between endpoints Q: how mny FORWARD pths? 3X3X3X3 =81

66 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths between endpoints Red = vrible Blue = complement of b b b d c c d e

67 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths between endpoints Red = vrible Blue = complement of b b b d c c d e Q: Is (=0, b=1, c=1, d=1, e=1) stisfying ssignment? NO

68 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths between endpoints Red = vrible Blue = complement of b b b d c c d e Q: Is (=1, b=1, c=1, d=1, e=1) stisfying ssignment? YES

69 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths between endpoints Red = vrible b Blue = complement of b b c b c Q: Is there stisfying ssignment?? NO

70 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths between endpoints Red = vrible b Blue = complement of b b c b c Q: Is there stisfying ssignment? b must be 1

71 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths between endpoints Red = vrible b Blue = complement of b b c b c Q: Is there stisfying ssignment? b must be 0 Contrdiction!

72 Questions on stisfying (SAT) ssignments? Efficient lgorizms? Is given ssignment stisfying? Is there stisfying ssignment? YES NO

73 Anlysis of Rely Circuits it lgorithms nd complexity P vs NP s

74 Complexity clss NP There is n efficient lgorizm for verifying given SAT solution b b b d c c d e

75 Complexity clss P There is n efficient i lgorizm for finding SAT ssignment b b b c b c

76 NP: There is n efficient lgorizm for verifying given SAT solution P: There is n efficient lgorizm for finding SAT ssignment Q: How re NP nd P relted? NP P NO P NP YES

77 Finding stisfying ssignment is computtionlly difficult problem computtionlly difficult is formlly clled NP-Complete

78 NP-complete problem for width 3 However, there is n efficient lgorizm for width 2 b b b d c c d e b b b c b c

79 NP-Complete The lnguge of hrd problems An NP-complete problem is Mgic box for ny problem in NP Cn solve it cn solve ll!!

80 Finding stisfying ssignment is computtionlly difficult problem I cn't find n efficient lgorizm, I guess I'm just too dumb... Source: Computers nd Intrctbility, by Grey nd Johnson

81 Finding stisfying ssignment is computtionlly difficult problem I cn't find n efficient lgorizm, becuse no such lgorizm is possible! Source: Computers nd Intrctbility, by Grey nd Johnson

82 Finding stisfying ssignment is computtionlly difficult problem I cn't find n efficient lgorizm, but neither cn ll these fmous people Source: Computers nd Intrctbility, by Grey nd Johnson

83 Finding stisfying ssignment is computtionlly difficult problem I showed it is NP-Complete And ll of these smrt people encountered other NPcomplete problems... And could not solve it, yet... Source: Computers nd Intrctbility, by Grey nd Johnson

84 Finding stisfying ssignment is computtionlly difficult problem Stephen Cook NP-complete complete, My 1971 I cn't find n efficient lgorizm, but neither cn ll these fmous people Source: Computers nd Intrctbility, by Grey nd Johnson

85 The most importnt ide in Informtion Finite it Universlity Lnguge of Cn construct everything from finite set of building blocks

86 Is there finite universl set of building blocks? Cn construct everything. DNA ABCDE... +, -, x, / M-boxes

87 Wht is the function?

88 Wht Anlysis is the of Rely function? Circuits Red = vrible b Blue = complement of c b c d b c d b c

89 Wht Anlysis is the of Rely function? Circuits Red = vrible Blue = complement of odd prity even prity The key: 1 cuses switch in prity 0 keeps the prity the sme

90 Wht Anlysis is the of Rely function? Circuits Red = vrible Blue = complement of odd prity even prity The key: 1 cuses switch in prity 0 keeps the prity the sme

91 Wht Anlysis is the of Rely function? Circuits Red = vrible Blue = complement of odd prity even prity The key: 1 cuses switch in prity 0 keeps the prity the sme

92 Importnt note Shnnon Used the Dul Nottion In this HW set use the nottion from clss! 1 = closed circuit 0 = open circuit

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