IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 HW4 will be returned today Average is 28/3~=93%

3 T = today x= hw#x out x= hw#x due mon tue wed thr fri 3 M 6 oh M oh 3 oh oh 2M2M 2 oh oh 2 Mx= MQx out 27 oh M2 oh oh = office hours 4 3 oh 3 4 oh oh midterms oh Mx= MQx due 8 oh oh oh 5 T oh oh oh

4 MQs. Everyone has a gift! (Tuesday) 2. Memory (Thursday)

5 Tuesday, 6/2, 2:3pm. Christopher Haack: The gift of resilience 2. Joon Lee: Settling is not an option 3. Spencer Strumwasser: The gift of dyslexia 4. Richard Zhu: The gift of memory 5. Ah Ashwin Hari: The gift of musical composition 6. Jessica Nassimi: Evolution a gift in disguise 7. Serena Delgadillo: The gift of self-expression 8. Megan Keehan: Gift of motherliness 9. Zane Murphy: Grandmother and the piano

6 Thursday, 6/4, 2:3pm. Connor Lee: Memory is a fickle thing blessing or curse 2. Pallavi Aggarwal: The wonders of human memory 3. Peter Kundzicz and Anshul Ramachandran: Muscle memories 4. Siva Gangavarapu: A cultural retrospection 5. Philip Liu: The light of other days 6. Jason Simon: Math and Broadway 7. Yujie Xu: Memory v.s. ESL 8. Celia Zhang: When memory sours

7 Last Lecture X TH TH2 TH+TH2-2 Saburo Muroga size at most n+, depth-2 circuit for symmetric functions

8 Input: initial state Feedback Networks Computing with Dynamics Associative Memory The Leibniz-Boole Machine Output: stable state Feedback Network

9 The Three Cases Cycle lengths mode W symmetric antisymmetric Example # serial? fully-parallel,

10 An Inspiring Paper Warren McCulloch Walter Pitts Neurophysiologist, MD Logician, Autodidact Their collaboration let to the 943 seminal neural networks paper: p A Logical Calculus of Ideas Immanent in Nervous Activity Neural networks and Logic Memory Time Threshold Logic Feedback Networks State Machines

11 Today State Machines Summary

12 State t Machines definition Honoré de Balzac Honoré de Balzac

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14 State Diagram / a b / / / states labeled vertices transitions ii directed d edges inputs labels on edges that correspond to the symbols that trigger the transitions outputs labels on edges that outputs labels on edges that correspond to the symbols that are generated by the transitions

15 State t Machines stream of bits in stream of bits out

16 State Diagram / a b / / / state machine symbols in symbols out

17 State Diagram starting state / a b / / / state machine a

18 State Diagram starting state / a b / / / state machine ab

19 State Diagram starting state / a b / / / state machine ba

20 State t Machines The nosy professor

21 State Diagram for??? starting state / a b / / / a a b b b a b b b b a a

22 State Diagram for??? starting state / odd even a b / / / a a b b b a b b b b a a XOR of the incoming sequence.. output if saw an odd number of s

23 State Diagram for XOR starting state / even a b odd / / / Q: How to implement it using a syntax box / logic circuit?

24 State t Machines synthesis

25 An Architecture for a State Machine starting state / How do we represent a and b? even a b / / odd / inputs logic circuit state outputs functions of inputs and current state

26 An Architecture for a State Machine starting state / even a= b= odd / / / inputs logic circuit state outputs functions of inputs and current state

27 State Machine for XOR starting state / odd even a= b= / / / Computing the next state? inputs current state What is the function? XOR

28 State Machine for XOR starting state / odd even a= b= / / / Computing the output? inputs current state What is the function? XOR

29 State Machine for XOR starting state / odd even a= b= / / / Computing the output Computing the next state inputs current state XOR XOR

30 State Machine for XOR starting state / even a= b= odd / / / inputs logic circuit XOR output state

31 State Machine for XOR starting state / even a= b= odd / / / inputs logic circuit XOR output

32 State Machine for XOR starting state / even a= b= odd / / / inputs logic circuit XOR output

33 State Machine for XOR starting state / even a= b= odd / / / inputs logic circuit XOR output

34 State Machine for XOR starting state / even a= b= odd / / / inputs logic circuit XOR output

35 State t Machines synthesis of an adder d d2 c 2 symbol adder c s

36 digit digit 2 carry 2 symbol adder carry sum

37 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder + = digit digit 2 carry 2 symbol adder carry sum

38 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder + = digit digit 2 carry 2 symbol adder carry sum

39 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

40 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

41 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

42 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

43 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

44 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder digit digit 2 + = 2 symbol adder sum

45 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder + = digit digit 2 What is the new ingredient in the box??? 2 symbol adder memory sum

46 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder + = digit digit 2 one bit = carry How many bits do we need to remember? 2 symbol adder memory sum

47 2 symbol adder 2 symbol adder 2 symbol adder 2 symbol adder + = digit digit 2 one bit = carry The STATE of the box is represented by the bits of the memory Two states carry = carry = 2 symbol adder memory sum

48 State Diagram for Addition The STATE of the MACHINE is represented by the bits of the Memory Possible inputs Two states carry = carry =?? 2 symbol adder

49 State Diagram for Addition Possible inputs Two states carry = carry = 2 symbol adder

50 State Diagram for Addition Possible inputs Two states carry = carry = 2 symbol adder??

51 State Diagram for Addition Possible inputs Two states carry = carry = 2 symbol adder

52 State Diagram for Addition Two states carry = carry = 2 symbol adder Possible inputs????

53 State Diagram for Addition Two states carry = carry = 2 symbol adder Possible inputs

54 State Diagram for Addition Two states carry = carry = 2 symbol adder Possible inputs / What is the output? / / / / / / /

55 State Diagram for Addition / / / / / / / / Q: How to implement it using a logic circuit?

56 State Machine for Addition / / / / / / / / inputs logic circuit state output functions of inputs and current state

57 State Machine for Addition / / / / / / / / Computing the next state inputs current state MAJority 2 symbol adder

58 State Machine for Addition / / / / / / / / inputs logic circuit state output MAJority

59 State Machine for Addition / / / / / / / / Computing the output inputs current state XOR 2 symbol adder

60 State Machine for Addition / / / / / / / / inputs logic circuit XOR output state MAJority functions of inputs and current state

61 State Machine for Addition 2 symbol adder inputs logic circuit XOR output state MAJority functions of inputs and current state

62 State Machine for Addition ymbol adder 2 sy inputs logic circuit XOR output state MAJority functions of inputs and current state

63 Playing with State t Machines

64 State Diagram for??? start t

65 Does this sequence generate a? NO start t

66 Does this sequence generate a? YES start t

67 State Diagram for??? detect t start t

68 Playing with State t Machines One more...

69 State Diagram for??? starting state / a b / / / / / / c d /

70 a c d b d c a b a c ist starting state / a b / / / / / / c d /

71 Even or odd number of s Even or odd number of s starting state / a b / / / / / / c d /

72 Even or odd number of s Even or odd number of s starting state / a b / / / / / even number of s and odd number of s c / / d

73 Key to this Progress Abstractions in Information Systems We know how to synthesize: going from left to right reasoning George Boole Syntax = Boolean algebra Claude Shannon 96-2 Physics = relay circuits

74 Key to this Progress Abstractions in Information Systems We know how to synthesize: going from left to right A clear problem definition coupled Systematic process... reasoning Syntax = state diagrams Physics = state machines

75 Key to this Progress Abstractions in Information Systems We know how to synthesize: going from left to right However, it is hard for us to analyze: going from right to left reasoning Syntax = state diagrams Physics = state machines

76 Equal number of red and blue balls? Can we compute / recognize any sequence? NO Finite state is a limitation...

77 Turing Machine finite state machine Alan Turing infinite memory (tape) It can be shown that a single special machine of that type can be made to do the work of all. It could in fact be made to work as a model of any other machine. The special machine may be called the universal machine In On Computable Numbers, with an Application to the Entscheidungsproblem" (submitted 936)

78 State t Machines history...

79 History on FSM (Finite State Machines) Warren McCulloch Walter Pitts : A Logical Calculus of Ideas Immanent in Nervous Activity Stephen C. Kleene David A. Huffman : Representation of Events in Nerve Nets and Finite Automata 953: The Synthesis of Sequential Switching Circuits George H. Mealy, 955: A Method for Synthesizing Sequential Circuits Edward F. Moore, 956: Gedanken-Experiments on Sequential Machines

80 It is all about PEOPLE Shannon 96-2 McCarthy 927-2

81 John McCarthy B.S. in Mathematics, Caltech, 948 Ph.D. in Mathematics, Princeton, 95 Claude Shannon John McCarthy /29/24, Palo Alto, CA

82 John von Neumann Warren McCulloch Claude Shannon Hixon Symposium at Caltech, 948 Cerebral Mechanisms in Behavior John McCarthy 927-2

83 von Neumann s lecture: The central and logical Theory of Automata -comparisons between computing machines and living organisms In the discussion following the lecture. McCulloch: I confess that there is nothing I envy Dr. von Neumann more than the fact that the machines with which he has to cope are those for which he has, from the beginning, a blueprint of what the machine is supposed to do and how it is supposed to do it. Unfortunately for us in the biological sciences-or, at least, in psychiatry - we are presented with an alien, or enemy's, machine. We do not know exactly what the machine is supposed to do and certainly we have no blueprint of it.

84 A Final Question... almost... Can a brain be simulated by a computer? Answers: Of course it can! It is a stupid question... Not sure... probably yes... I thought that computers can compute anything... Not my brain seriously, I think that it is a valid question...

85 Gottfried Leibniz Leibniz Information and Logic Characteristica Universalis: Leibniz s goal was to develop an alphabet of human thought, a universal symbolic language (characteristic) to describe nature 3 year later, it is still a dream...

86 The appearance of life is the first Information Megamorphosis The appearance of the human brain is the second Information Megamorphosis The Final Question! What will be the third Information Megamorphosis?

87 Progress happens with the introduction of new languages DNA brain molecular l switches associations spoken language evolution state machines written language memory stochastic relays number systems analysis probability bilit mathematics proofs syllogism algebra abacus algorizms synthesis abstractions axioms syntax boxes neural gates relays AON gates Boolean algebra

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89 The Babylonians knew everything! Everyone has a gift! Mother s day is the most important day for the year! Smile!