IST 4 Information and Logic

Transcription

1 IST 4 Informtion nd Logic

2 mon tue wed thr fri sun T = tody 3 M1 oh 1 x= hw#x out 10 oh M1 17 oh oh 1 2 M2 oh oh x= hw#x due 24 oh oh 2 oh = office hours oh 1 oh M2 8 3 oh midterms oh oh Mx= MQx out 15 oh T 3 4 oh Mx= MQx due 22 oh oh 4 5 CP1 CP = chllenge prolem 29 CP2 5 oh 5 oh oh CP oh oh

3 Chllenge Prolems They re optionl nd cn e completed nytime efore the due dtes, to e eligile for presenttion on Thursdy 6/1: Due dtes: First prolem 5/26, Second prolem 5/29. Credit of 3 points for those not lredy presenting their MQs Credit of 1 point for those presenting MQs. Collorte with friends (mx 2-3) nd/or sk for help from the TAs. We re excited to see wht you will come up with!

4 Chllenge Prolem 1 due 5/26 The first coding chllenge is to implement function tht retrieves the minimum numer of steps it tkes to generte string using only tndem duplictions from one of the following seed strings: {0, 1, 01, 10, 101, 010}; s listed in prolem 1, HW1. Once you hve completed this you will relize tht for lrge strings it tkes lot of time for your progrm to grow s the serch spce ecomes much lrger. The chllenge is to think of wys to optimize this genertion process nd rgue why your solution is good. In your solution, plese descrie your lgorithm nd pproch. Discuss pros nd cons nd wht things you like/dislike out your lgorithm. We might run competition in clss

5 Chllenge Prolem 2 due 5/29 In homework 3, you found circuit for computing prity of 2 nd 3 vriles with 4 nd 8 m-oxes respectively. By using the sme ide, you cn compute prity of 4 vriles with 12 m- oxes. Is 12 is the smllest size of circuit of m-oxes for computing the prity of 4 vriles? Wht is the smllest circuit for prity of 4 vriles? You cn use forml proof or n lgorithm/code to check ll possile circuit configurtions.

6 Lst time DNF is wy to express syntx ox using formul lger meets syntx oxes Ide: construct DNF y dding the norml terms tht correspond to the norml ssignments (f=1) m(,)

7 Lst time DNF is wy to express syntx ox using formul XOR(,) XOR(,)

8 Lst time Boolen Alger lnguge for inry mgic oxes m Cn relize the opertions of the lger cn compute ny DNF mgicl for Binry

9 HW#3, perspective:?? with 5 m-oxes?

10 Leiniz d1 d2 c 2 symol dder c s

11 When ws the first inry dder uilt? Shnnon Leiniz Boole d1 d2 c 2 symol dder c s sum crry

12 The First Digitl Adder George Stiitz, Ford Model T Grduted with PhD from Cornel in 1930 He worked t Bell Ls in New York In the fll of 1937 Stiitz used surplus relys, tin cn strips, flshlight uls, nd other common items to construct his "Model K" K stnds for kitchen...

13 Bell Ls Model Bell Ls Model 1 Complex Clcultor Computed with complex numers 450 relys Remote opertion... vi telegrph Drtmouth college to New York Invented the term (pulse) digitl s opposed to nlog

14 Logic to Physics Clude Shnnon s

15 Shnnon Clude Elwood Shnnon ws orn in Petoskey, Michign, on April 30, The first sixteen yers of Shnnon's life were spent in Gylord, Michign

16 Shnnon Shnnon s Bckground Clude Elwood Shnnon ws orn 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 in 1936 (20) with two chelor'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

17 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

18 Shnnon Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 Hitchcock Shnnon s dvisor oth MSc nd PhD mthemticin 79 yers go

19 Hitchcock Bush Shnnon Institute for Advnced Study : Bell Ls : MIT

20 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 Institute for dvnced studies : Bell Ls : MIT

21 Bush Shnnon , MIT 1924, MIT Posted on the clss wesite Termn Buss Cumming Goodmn 1989, Stnford 1940, Stnford 1955, Stnford 1963, Stnford Bruck

22 Bush Leiniz Posted on the clss wesite Two centuries go Leiniz invented clcul/ng mchine which emodied most of the essen5l fetures of recent keyord devices, ut it could not then come into use. The economics of the situ5on were ginst it: the lor involved in construc/ng it, efore the dys of mss produc5on, exceeded the lor to e sved y its use, since ll it could ccomplish could e duplicted y sufficient use of pencil nd pper.

23 Logic to Physics The lnguge of lines s

24

25 Boolen Clculus nd Physicl Circuits Single Lines nd Composition The lnguge of lines: A line cn hve only two possile colors: lue or red Two lines cn e composed in two possile wys In prllel: In series:

26 Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn e composed in two possile wys In prllel:

27 Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn e composed in two possile wys In series:

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

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

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

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

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

33 Boolen Clculus nd Physicl Circuits Color of Composition Wht is the color of composition? color = lue color = red The color of composition is red if there is red pth etween the endpoints Otherwise, the color is lue

34 The two-color line composition is 0-1 Boolen lger! How cn we prove it?

35 Boolen Alger Algeric system: set of elements B, two inry opertions + nd B hs t lest two elements (0 nd 1) If the following xioms re true then it is Boolen Alger: A1. identity A2. complement A3. commuttive A4. distriutive

36 Two-Colored Line Composition nd 0-1 Boolen Alger Algeric system: set of elements B, two inry opertions + nd B hs t lest two elements (0 nd 1) Elements: 0 1

37 Two-Colored Line Composition nd 0-1 Boolen Alger Algeric system: set of elements B, two inry 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 etween the endpoints 0 1 Otherwise, the color is lue + Compose in series:

38 Two-Colored Line Composition nd 0-1 Boolen Alger 0 The color of the composition equls the color of The color of the composition equls the color of 1 + Compose in prllel Compose in series The color of composition is red if there is red pth etween the endpoints Otherwise, the color is lue

39 Two-Colored Line Composition nd 0-1 Boolen Alger The color of the composition is red = 1 The color of the composition is lue = 0 + Compose in prllel Compose in series The color of composition is red if there is red pth etween the endpoints. Otherwise, the color is lue

40 Two-Colored Line Composition nd 0-1 Boolen Alger 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 etween the endpoints. Otherwise, the color is lue

41 Two-Colored Line Composition nd 0-1 Boolen Alger Two pths: nd c c c

42 Is the two-color line composition 0-1 Boolen lger?

43 circuits = lger In Shnnon s words: The lger of logic originted y George Boole, is symolic method of investigting logicl reltionships. The symols of Boolen lger dmit of two logicl interprettions. If interpreted in terms of clsses, the vriles re not limited to the two possile vlues 0 nd 1. E. V. Huntington gives the following set of postultes for symolic logic: We re now in position to demonstrte the equivlence of this clculus with certin elementry prts of the clculus of propositions.

44 Shnnon Shnnon used relys nd connected them in series-prllel circuits Rely on the edge controlled y 0-1 vrile

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

46 Rely Circuits nlysis nd synthesis s

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

48 In Shnnon s words: nlysis of circuits..ny circuit is represented y set of equtions, The terms of the equtions corresponding to the vrious relys nd switches in the circuit. A clculus is developed for mnipulting these equtions y simple mthemticl processes most of which re similr to ordinry lgeric lgorisms. This clculus is shown to e exctly nlogous to the clculus of propositions used in the symolic study of logic.

49 In Shnnon s words: synthesis of circuits For the synthesis prolem 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 e immeditely drwn from the equtions. desired chrcteristic system of equtions simplified set of equtions simple circuit

50 In Shnnon s words: synthesis of circuits For the synthesis prolem 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 e immeditely drwn from the equtions. desired chrcteristic The lnguge of system of equtions Logic Design is orn! simplified set of equtions simple circuit

51 Rely Circuits nlysis

52 A rely circuit corresponds to formul rely circuits Boolen functions Boolen sum of ll the pths etween endpoints

53 Anlysis of Rely Circuits Exmple 1: - series-prllel - independent pths etween endpoints d c e c d e 10 relys

54 Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d 10 relys

55 Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d 10 relys

56 Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d 10 relys

57 Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d 10 relys

58 Anlysis of Rely Circuits Exmple 2: - non series-prllel - dependent pths etween endpoints c d e 5 relys

59 Anlysis of Rely Circuits c d e 5 relys

60 Anlysis of Rely Circuits c d e 5 relys

61 Anlysis of Rely Circuits c d e 5 relys

62 Anlysis of Rely Circuits c d e 5 relys

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

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

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

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

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

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

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

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

71 Exmple 3: 6 functions multiple terminls c d e

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

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

74 Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of d c c d e

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

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

77 Anlysis of Rely Circuits Is there stisfying ssignment?

78 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment?? NO

79 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment? must e 1

80 Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment? must e 0 Contrdiction!

81 Efficient lgorizms? Questions on stisfying ssignments? The SAT prolem Is given ssignment stisfying? Is there stisfying ssignment?

82 There is n efficient lgorizm for verifying given SAT solution for ny structure! d c c d e It is relted to lgorizms for solving connectivity prolems grphs

83 Algorizms for finding SAT ssignment? c c If the circuit hs width 2: There is n efficient lgorizm for finding stisfying ssignment...

84 However, if the circuit hs width>2: No efficient lgorizm is known! It is likely, tht n efficient lgorizm does not exist d c c d e

85 Efficient lgorizms? Questions on stisfying (SAT) ssignments? Is given ssignment stisfying? Is there stisfying ssignment? Shnnon s connection etween computtion nd Boolen lger is t the core of Algorithms nd Complexity! P vs NP question

86 Wht is the function?

87 Wht Anlysis is the of Rely function? Circuits Red = vrile Blue = complement of c c d c d c

88 Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level Red = vrile Blue = complement of c d odd prity c d c even prity

89 Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level odd prity =1 d=1 =1 c=0 even prity

90 Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level odd prity =1 c=0 d=1 =0 even prity

91 Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level Red = vrile Blue = complement of odd prity even prity

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

93

94

95 One circuit with multiple terminls for mny functions

96 One circuit with multiple terminls for mny functions

97 One circuit with multiple terminls for mny functions f(,) f(,)=1 f(,)= 00? ??? f(,)=and f(,)=xor

98 Wit until fter the clss on Thursdy... c CQ(,,c)

99 Wit until fter the clss on Thursdy...

100 Wit until fter the clss on Thursdy...