Boolean Expression Simplification
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1 Principles Of Digital Design Boolean Epression Simplification Literal Minimization Boolean Factoring Implicants and Prime Implicants K Maps Don t Care Conditions
2 Epression Minimization Goal? Literal minimization E: y + y = (y + y ) = E: yz + yz +y z + y z = (yz + yz + y z + y z ) = How? Visualization (few variables) K- Maps (2-6 variables) Reality? CAD tools (many variables) E: 64-bit adder (64 variables, 2 64 terms) Copyright 2-2 by Daniel D. Gajski 2 EECS 3/CSE 3, University of California, Irvine
3 Boolean Functions and Factors Each Boolean function of n variables can be represented by a truth table where each raw represents a minterm Each subset of n-m literals, l l 2 l n m, is called a factor iff l l 2 l n m (any minterm of m variables) is a -minterm y z F y z F y z = ( + ) y z y = y(z + z) z = ( y + y + y + y)z Copyright 2-2 by Daniel D. Gajski 3 EECS 3/CSE 3, University of California, Irvine
4 Boolean Functions and Factors For any Boolean function a prime implicant is a factor not contained in any other prime implicant An essential prime implicant is a factor that contains a - minterm that is not included in any other prime implicant y z F y y z z F y z = ( + ) y z y = y(z + z) z = ( y + y + y + y)z F = y + z Copyright 2-2 by Daniel D. Gajski 4 EECS 3/CSE 3, University of California, Irvine
5 Map Representation K-maps define Boolean functions Map representation is equivalent to truth tables, Boolean epressions Maps aid in visually identifying prime implicants and essential prime implicants in each Boolean function Maps are used for manual optimization of Boolean functions with few variables Copyright 2-2 by Daniel D. Gajski 5 EECS 3/CSE 3, University of California, Irvine
6 Karnaugh Maps for n =, 2, 3, and 4 y m m m 2 m 3 n = n = 2 y zw yz m m m 3 m 2 m m m 3 m 2 m 4 m 5 m 7 m 6 m 4 m 5 m 7 m 6 m 2 m 3 m 5 m 4 m 8 m 9 m m n = 3 n = 4 Copyright 2-2 by Daniel D. Gajski 6 EECS 3/CSE 3, University of California, Irvine
7 2 variable K-Map y y y y Factor Factor y Map Organization y 2 3 y 2 3 Factor Eample of -factors Eample: y AND OR XOR Truth Table y y y Copyright 2-2 by Daniel D. Gajski AND OR XOR 7 EECS 3/CSE 3, University of California, Irvine
8 Three variable K-Map yz 3 2 y z y z yz yz y z y z yz yz yz Map Organization 3 2 Factor z Factor z Factor Eample of 2-factors yz 3 2 Factor y Factor yz Factor z Copyright 2-2 by Daniel D. Gajski 8 Eample of -factors EECS 3/CSE 3, University of California, Irvine
9 Maps for Carry and Sum Functions c i i y i c i + s i i y i c i 3 2 i y i c i 3 2 Truth Table Carry Function c i + Sum Function s i c i + = i y i + c i i + c i y i s i = i y i c i + i y i c i + i y i c i + i y i c i Copyright 2-2 by Daniel D. Gajski 9 EECS 3/CSE 3, University of California, Irvine
10 Four variable K-Map zw y 3 2 y z w y z w y zw y zw yz w yz w yzw yzw yz w yz w yzw yzw 8 9 y z w y z w y zw y zw Map Organization y zw 3 2 Factor y w y zw 3 2 Factor Factor y Factor w Factor z Eample of 2-factors Copyright 2-2 by Daniel D. Gajski Eample of 3-factors EECS 3/CSE 3, University of California, Irvine
11 Simplification of Comparison Functions y y Greater Than Equal Copyright 2-2 by Daniel D. Gajski Less Than Truth Table y y y y Greater-than Function Less-than Function G = y + y y + y L = y + y + y y EECS 3/CSE 3, University of California, Irvine
12 Boolean Simplification with Maps Truth table, canonical form or standard form Generate map Determine prime implicants Select essential prime implicants Find minimal cover Standard form Copyright 2-2 by Daniel D. Gajski 2 EECS 3/CSE 3, University of California, Irvine
13 Boolean Simplification with Maps Eample: Maps method Problem: Using the map method, simplify the Boolean function F = w y z + wz + yz + w y w yz w yz Map Organization PI List: EPI List: Cover List: Prime Implicants in the Map w z, wz, yz, w y w z, wz () w z, wz, yz (2) w z, wz, w y Copyright 2-2 by Daniel D. Gajski 3 EECS 3/CSE 3, University of California, Irvine
14 Boolean Simplification with Maps Eample: Map method Problem: Simplify the Boolean function F = w yz + w y + wz + w y + w y z w yz PI List: EPI List: Cover List: w z, w y, wz, w y, y z, wy z, yz, w yz empty () w z, w y, wz, w y (2) y z, wy z, yz, w yz Copyright 2-2 by Daniel D. Gajski 4 EECS 3/CSE 3, University of California, Irvine
15 Don t Care Conditions Completely specified functions have a value assigned for every minterm Incompletely specified functions do not have values assigned for some minterms which are called don t care minterms (d minterms) or don t care conditions Don t care minterms can be assigned any value during simplifications in order to simplify Boolean epressions Copyright 2-2 by Daniel D. Gajski 5 EECS 3/CSE 3, University of California, Irvine
16 Don t Care Conditions Eample: Don t care conditions Problem: Derive Boolean epressions for the 9 s complement of a BCD digit Digits Nine s Complements Decimal BCD BCD Decimal 3 2 y 3 y 2 y y Nine s Complement Table Copyright 2-2 by Daniel D. Gajski X X X X 8 9 X X X X X X 8 9 y 3 = 3 2 y 2 = 2 X X X X X X EECS 3/CSE 3, University of California, Irvine X X X X X X y = K Maps y = X X
17 Summary Simplification of Boolean epressions by K-Map method (visual for 2-6 variables)) Prime implicants Essential prime implicants Minimal cover Copyright 2-2 by Daniel D. Gajski 7 EECS 3/CSE 3, University of California, Irvine
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