IST4_lec16

IST4_lec16 - IST 4 Information and Logic IST 4: Planned...

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IST 4 Information and Logic
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1 M1 31 6 2 6 5 26 19 M2 5 4 12 4 5 3 M2 28 3 2 21 2 1 14 M1 7 fri thr wed tue mon IST 4: Planned Schedule – Spring 2008 x= hw#x out x= hw#x due = today T Mx= MQx out Mx= MQx due T D
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“The Big Picture” ‘Universal Language/Syntax Dream’ calculus, binary systems, logic. .. ‘Algebra to Physics’ automatic syntax processing with relay circuits ‘The Babylonian Knew Everything’ masters of syntax base-60 positional system, abacus,. .. ‘Logic to Algebra’ Boolean algebra for syntax processing. .. Circuit complexity – cost/performance of computing: -how many gates / relays? size/cost -how long it takes to compute? depth/performance
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Some History Warren McCulloch 1899 - 1969 Walter Pitts 1923 - 1969 Neurophysiologist, MD Their collaboration let to the 1943 seminal neural networks paper: A Logical Calculus of Ideas Immanent in Nervous Activity Logician, Autodidact Neural networks and Logic Time Memory Threshold Logic State Machines
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Neuron – Neural Gate Linear Threshold (LT) gate
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Neuron – Neural Gate Linear Threshold (LT) gate We proved that LT gates are functionally complete Can compute {AND, OR, NOT} with LT gates
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The Comparison Function Linear Threshold (LT) gate Computing the COMPARISON function with LT gates? ? ? 20 22 24 23
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Linear Threshold (LT) gate How many LT gates? ist The Comparison Function Computing the COMPARISON function with LT gates?
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XOR Function: Size of LT vs AON in Depth 2 5 4 2 AON LT-l LT-nl * * * * = it is optimal Exponential gap in size Next
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More Questions on Symmetric Functions Q: What are the symmetric functions that can be realized by a single LT gate ? Definitions: (1) SYM = the class of Boolean symmetric functions (2) LT 1 = the class of Boolean functions that can be realized by a single LT gate. Q: How is SYM related to LT 1 ??
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SYM and LT 1 Theorem: SYM LT1 TH
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The Class TH - Single Transition 0 1 1 1 TH 3 0 0 1 1 TH 2 0 0 0 1 TH 1 0 0 0 0 TH 0 1 1 1 1 3 0 1 1 1 2 0 0 1 1 1 0 0 0 1 0 TH 3 TH 2 TH 1 TH 0 |X| Q: what is |TH| ? the number TH functions. .. A: 2n+2 = a transition
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More Layers Definition: Q: How is SYM related to LT 2 ?? 2n+2 functions of SYM are in LT 1 , what about the rest of SYM ? LT 2 = the class of Boolean functions that can be realized by a two layer circuit of LT gates, size is bounded by poly(n) , for example: where k is a constant independent of n
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XOR Function: Size of LT vs. AON in Depth 2 5 4 2 AON LT-l LT-nl Symmetric functions
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Linear Threshold (LT) gate How many LT gates? Idea: compute the difference inputs weights A single LT gate ist The Comparison Function Computing the COMPARISON function with LT gates?
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Alexey Pajitnov Developed the computer game Tetris in 1985 while working as a computer engineer in Moscow (USSR) Claim to fame?
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Playing TH etris only horizontal collection of k blocks
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Playing TH etris complete last row disappears -down
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Playing TH etris incomplete last row stays
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Playing TH etris incomplete last row stays
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