Lecture17CircuitMinimization

Lecture17CircuitMinimization - Lecture 17 - CS 2603 Applied...

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 17 —CS 2603 Applied Logic for Hardware and Software Circuit Minimization using Karnaugh maps
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Boolean Functions new name for same old animal ± Boolean function — a special kind of predicate ² Domain of discourse: tuples of {True, False} values ² Boolean functions truth tables ± F(a, b) — two-variable Boolean function ² Domain of discourse: pairs of Boolean values ² Specifies a Boolean output for each pair of Boolean inputs ² Representations we’ve already seen for 2-var Boolean functions 9 Truth table with four rows (unique: one per Boolean function) 9 Propositional WFF with two variables (not a unique rep'n) 9 Combinational circuit with two input lines, one output line ± F(a, b, c) — three-variable Boolean function ² Domain of discourse: triples of Boolean values ² 8-line truth table, 3-variable WFF, 3-input/1-output circuit ± F(x 1 , x 2 , … x n ) — n-variable Boolean function ² Domain of discourse: n-tuples of Boolean values ² 2 n -line truth table, n-variable WFF, n-input/1-output circuit
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 sum of products ± Focus on places in truth-table where F is True ² Boolean tuple (b 1 , b 2 , … b n ), with F(b 1 , b 2 , … b n ) = True 9 That is, ( k. (x k = b k )) (F(x 1 , x 2 , … x n ) = True) ² Find a propositional WFF whose value is ( k. (x k = b k )) 9 WFF w, variables x 1 , x 2 , … x n , value of w = ( k. (x k = b k )) That is, w = True if ( k. (x k = b k )) = True w = False if ( k. (x k = b k )) = False ² w = (y 1 y 2 y n ) = ( k. (x k = b k )) y k =x k if b k = True y k = ¬ x k if b k = False known as a “minterm” Sum-of-Products Representation constructing a WFF for Boolean function F(x 1 , x 2 , …x n ) ± A WFF for F(x 1 , x 2 , … x n ) ² F(x 1 , x 2 , … x n ) = w 1 + w 2 + ⋅⋅⋅ w m —each±w k is a minterm ² m = number of 1’s in last column of truth table ² Minterms are usually written in EE notation 9 For example: x 1 x 2 x 3 , when x 1 = 1, x 2 = 0, x 3 = 1
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Example sum-of-products representation for 3-variable function Truth table representation of a Boolean function F x y z 1 1 1 1 0 0 1
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This note was uploaded on 04/08/2008 for the course CS 2603 taught by Professor Rexpage during the Spring '08 term at The University of Oklahoma.

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Lecture17CircuitMinimization - Lecture 17 - CS 2603 Applied...

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