On Quine Quine-Mccluskey Method

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

= x’y’+x’z’+wxy+wy’z
= x’y’+x’z’+wxy+wxz
= x’y’+x’z’+wxz+wyz’

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

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Quine-McCluskey Method
Problem: Given a Boolean function f (may be
Problem:
(may incomplete), find a minimum cost 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
Build 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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Example: Quine-McCluskey Method
Example: Quine f(w,x,y,z) = x’y’ + wxy + x’yz’ + wy’z wxy x’y’

x’z’

wx’y’z’

1

1

w’x’y’z

1

w’x’y’z’

1

wxyz

1

wxyz’

wxz wyz’ wy’z

1

1
1

{x’y’, x’z’,wxy, wxz},
{x’y’, x’z’,wxy, wy’z},
{x’y’, x’z’,wxz, wyz’}.

1
1

w’x’yz’ wx’y’z 1

1

wx’yz’ wxy’z minimum cover(s):

1

1
1
ENEE 644

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Two-Level Logic Synthesis
Two
-- Unate Covering Problem

Unate and Binate nf > A function f(x1,•••xi•••xn) is positive unate iin xi iif function f(x positive its cofactor f x includes f x ' its i

i

Negative unate iis defined in a similar way. If a function is s neither 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,