IST4_lec14

# IST4_lec14 - 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 Old/Odd Machines that Compute Please send a pdf file named yourname-mq2.pdf to T
Symmetric Functions Definition: A Boolean function f is symmetric if for an arbitrary permutation Theorem: A Boolean function f(X) is symmetric if and only if it is a function of the number of 1’s in X, namely |X|

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Number of Symmetric Functions Theorem: The number of symmetric functions of n variables is: Proof: f symmetric f a function of |X| can be 0 or 1 * functions Q 0 1 2 n * * * * Symmetric function table We know that
Implementing a 2-bit Adder 2 symbol adder digit 1 digit 2 carry sum carry

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0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 0 1 2 3 XOR 0 = 1 =
0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 0 1 2 3 XOR 0 = 1 =

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0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 0 1 2 3 XOR 0 = 1 = XOR
0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 0 1 2 3 XOR 0 = 1 = XOR

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0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 1 3 XOR 0 = 1 = XOR This two node are equivalent
0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 0 1 1 0 2 1 3 XOR 0 = 1 = XOR

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0 1 2 Implementing a 2-bit Adder 3 1 1 0 0 0 = 1 = XOR
Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 0 0 0 1 2 3 MAJ 0 = 1 = 0 1 1 2

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Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 0 0 0 1 2 3 MAJ 0 = 1 = 0 1 1 2
Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 0 0 2 3 MAJ 0 = 1 = 1 1 >1 There is no reason to check c

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Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 0 0 2 3 MAJ 0 = 1 = 1 1 >1 There is no reason to check c
Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 0 0 MAJ 0 = 1 = 1 1 >1 2

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Implementing a 2-bit Adder 0 1 2 3 0 1 0 1 MAJ 0 = 1 =
Implementing a 2-bit Adder Summary 0 = 1 = XOR 8 relays 6 relays MAJ Combination of the two functions?

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Implementing a 2-bit Adder Summary 0 = 1 = XOR 8 relays 6 relays MAJ Combination of the two functions?
Relay Circuits for the Sum and the Carry Functions 1 MAJ XOR sum: carry: 2 symbol adder c s d 1 d 2 c How many relays for a 2 symbol adder? 10

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Analysis vs. Synthesis
Key to this Progress Abstractions in Information Systems Physics = relay circuits Syntax = Boolean algebra reasoning

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Abstractions in Biological Information Systems ???? Biology = ????? Syntax = ????? reasoning
A G E G D F F E D D C B C B A k k k k k ⎯→ + + ⎯→ ⎯→ + ⎯→ + ⎯→ + 5 4 3 2 1 Chemical Reactions Networks 1 2 5 4 3

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A G E G D F F E D D C B C B A k k k k k ⎯→ + + ⎯→ ⎯→ + ⎯→ + ⎯→ + 5 4 3 2 1 Chemical Reactions Networks 1 2 5 4 3
A G E G D F F E D D C B C B A k k k k k ⎯→ + + ⎯→ ⎯→ + ⎯→ + ⎯→ + 5 4 3 2 1 1 2 5 4 3 Chemical Reactions Networks

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Abstractions in Biological Information Systems ???? Biology =
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IST4_lec14 - IST 4 Information and Logic IST 4 Planned...

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