RTL Design Flow - Computation Structures Group

Algebraic vs. Boolean Methods
Algebraic techniques view equations as
polynomials and attempt to factor equations or
“divide” them
Do not exploit Boolean identities e.g., a a = 0
In algebraic substitution (or division) if a function
f = f(a, b, c) is divided by g = g(a, b), a and b
will not appear in f / g
Algebraic division: O(n log n) time
Boolean division: 2-level minimization required
Strong (or Boolean) Division
Given a function f to be strong divided by g
Add an extra input to f corresponding to g,
namely G and obtain function h as follows
h DC = Gg+ Gg
h ON = f ON − h DC
Minimize h using two-level minimizer
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Comparison
f = ab + ac+ ba+ bc + ca + cb
Algebraic factorization procedures
f = a( b + c ) + a ( b + c ) + bc + cb
Boolean factorization produces
f = ( a + b + c ) ( a + b + c )
l = ( bf+ bf ) ( a + e)+
ae( b f + bf )
r = ( bf+ b f ) ( a + e)+
ae( b f + bf )
Algebraic substitution of l into r fails
Boolean substitution
r = a( el + el ) + a( el + el )
l = a( er + e r ) + a( er + er)
Strong Division Example
f=a bc +abc +abc +a b c
g = ab+ab
bc
00
01
11
10
Ga
00 01 11 10
1
x
x
x
x x
Minimization gives h = G c + G c
1
h DC = G (a b + a b) + G (a b + a b)
h ON = f ON − h DC
1
x
x
x
1
Function h
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