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Universality, Gates, and Logic (ppt)

# Universality, Gates, and Logic (ppt) - ESE534 Computer...

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Penn ESE534 Spring2010 -- DeHon 1 ESE534: Computer Organization Day 2: January 20, 2010 Universality, Gates, Logic Work Preclass Exercise

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Penn ESE534 Spring2010 -- DeHon 2 Last Time Computational Design as an Engineering Discipline Importance of Costs
Penn ESE534 Spring2010 -- DeHon 3 Today Universality Simple abstract computing building blocks gates, Boolean Equations RTL Logic (at least the logic part) Logic in Gates optimization properties Costs

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Preclass 1 Do the Case 1 circuits calculate the same thing? Case 2? Penn ESE534 Spring2010 -- DeHon 4
General How do we define equivalence? How do we determine if two circuits are equivalent? Penn ESE534 Spring2010 -- DeHon 5

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Penn ESE534 Spring2010 -- DeHon 6 Model: Stateless Functions (Combinational Logic) Compute some “ function – f(i 0 ,i 1 ,…i n ) o 0 ,o 1 ,…o m Each unique input vector implies a particular, deterministic, output vector
Boolean Equivalence Two functions are equivalent when They have the same outputs for every input vector i.e. , they have the same truth table There is a canonical specification for a Boolean function its Truth Table Penn ESE534 Spring2010 -- DeHon 7

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Penn ESE534 Spring2010 -- DeHon 8 Implementation in Gates Gate: small Boolean function Goal : assemble gates to cover our desired Boolean function Collection of gates should implement same function I.e. collection of gates and Boolean function should have same Truth Table
Penn ESE534 Spring2010 -- DeHon 9 Netlist Netlist: collection of interconnected gates A list of all the gates and what they are connected to

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Implementation How can I implement any Boolean function with gates? Penn ESE534 Spring2010 -- DeHon 10
Implementation Single output {0, 1} Use inverters to produce complements of inputs For each input case (minterm) If output is a 1 Develop an AND to detect that case » Decompose AND into gates OR together all such minterms Decompose OR into gates Multiple outputs Repeat for each output Penn ESE534 Spring2010 -- DeHon 11

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Universal set of primitives What primitives did I need to support previous implementation set? Conclude: can implement any Boolean function by a netlist of gates selected from a small set. Homework (B.1): How small can set be? Penn ESE534 Spring2010 -- DeHon 12
Penn ESE534 Spring2010 -- DeHon 13 Boolean Equations o=/a*/b*c+/a*b*/c+a*b*/c+a*/b*c Another way to express Boolean functions a b c o 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

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Penn ESE534 Spring2010 -- DeHon 14 Boolean Equations o= /a*/b*c + /a*b*/c + a*b*/c + a*/b*c Another way to express Boolean functions a b c o 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0
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