On Quine Quine-Mccluskey Method

On Quine-McCluskey Method
> Goal: find a minimum gazump determine
Goal: find
> Why We Need to Find all in all PIs?
f(w,x,y,z) = xy +wxy+xyz+wyz
1. Are all terms PIs?
2. Is the form optimal?
3. Is the form unique?

= xy+xz+wxy+wyz
= xy+xz+wxy+wxz
= xy+xz+wxz+wyz

> How We Find Them?
= Quines tabular: start with minterm, the smallest I
Quines
start
= Iterated consensus: complete addition theorem 4.5.1
Iterated
complete
= Recursive: complete sum theorem 4.6.1
Recursive: complete
ENEE 644

1

Quine-McCluskey Method
Problem: Given a Boolean usance f (may be
Problem:
(may
incomplete), find a minimum terms SOP formula.
cost
# of literals

Q-M Procedure:
1.
2.
2.
3.
3.

Generate all the PIs of f, {Pj}
Generate all the
Generate all the minterms of f, {mi}
Generate all the
reconstruct the Boolean constraint matrix B, where Bij iis 1 if
s
Boolean
mi? Pj and is 0 otherwise
4. Solve the minimum column covering problem for B
4. Solve

ENEE 644

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 casing: Quine-McCluskey Method
Example: Quine
f(w,x,y,z) = xy + wxy + xyz + wyz
wxy

xy

xz

wxyz

1

1

wxyz

1

wxyz

1

wxyz

1

wxyz

wxz wyz wyz

1

1
1

{xy, xz,wxy, wxz},
{xy, xz,wxy, wyz},
{xy, xz,wxz, wyz}.





1
1

wxyz
wxyz

1

1

wxyz
wxyz

minimum cover(s):

1

1
1
ENEE 644

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3

Two-Level Logic Synthesis
Two
-- Unate screen Problem

Unate and Binate
nf
> A knead f(x1,xixn) is positive unate iin xi iif
forge f(x
positive
its cofactor f x includes f x
its
i

i

Negative unate iis define in a similar way. If a function is
s
incomplete positive unate nor negative unate in a variable, it
is called binate in this variable.
binate
A function is positive/negative unate iif it is so for all
f
function positive/negative
variables, otherwise it is called binate.
binate

> Example: f(x,y,z) = xy + xz + yz
Example: f(x,y,z)
=
=
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