General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

x1 x2 x3 x4 monomials
F F F F x1 0 x2 0 x3 0 x4 0
F F F T x1 0 x2 0 x3 0 x4 1
F F T F x1 0 x2 0 x3 1 x4 0
F T F T x1 0 x2 1 x3 0 x4 1
T F T F x1 1 x2 0 x3 1 x4 0
T F T T x1 1 x2 0 x3 1 x4 1
T T T F x1 1 x2 1 x3 1 x4 0
T T T T x1 1 x2 1 x3 1 x4 1
Note: the monomials on the right hand side are only for illustration
Idea: do the resolution directly on the left hand side
Find rows that differ only by a single entry. (first two rows)
resolve: replace them by one, where that entry has an X (canceled literal)
Example 295 〈F, F, F, F〉 and 〈F, F, F, T〉 resolve to 〈F, F, F, X〉.
c○: Michael Kohlhase 167
A better Mouse-trap for QMC1: optimizing the data structure
One step resolution on the table
x1 x2 x3 x4 monomials
F F F F x1 0 x2 0 x3 0 x4 0
F F F T x1 0 x2 0 x3 0 x4 1
F F T F x1 0 x2 0 x3 1 x4 0
F T F T x1 0 x2 1 x3 0 x4 1
T F T F x1 1 x2 0 x3 1 x4 0
T F T T x1 1 x2 0 x3 1 x4 1
T T T F x1 1 x2 1 x3 1 x4 0
T T T T x1 1 x2 1 x3 1 x4 1
x1 x2 x3 x4 monomials
F F F X x1 0 x2 0 x3 0
F F X F x1 0 x2 0 x4 0
F X F T x1 0 x3 0 x4 1
T F T X x1 1 x2 0 x3 1
T T T X x1 1 x2 1 x3 1
T X T T x1 1 x3 1 x4 1
X F T F x2 0 x3 1 x4 0
T X T F x1 1 x3 1 x4 0
Repeat the process until no more progress can be made
x1 x2 x3 x4 monomials
F F F X x1 0 x2 0 x3 0
F F X F x1 0 x2 0 x4 0
F X F T x1 0 x3 0 x4 1
T X T X x1 1 x3 1
X F T F x2 0 x3 1 x4 0
This table represents the prime implicants of f
c○: Michael Kohlhase 168
A complex Example for QMC (QMC1)
The PIT:
FFFF FFFT FFTF FTFT TFTF TFTT TTTF TTTT
x1 x2 x3 T T F F F F F F
x1 x2 x4 T F T F F F F F
x1 x3 x4 F T F T F F F F
x2 x3 x4 F F T F T F F F
x1 x3 F F F F T T T T
x1 x2 x3 is not essential, so we are left with
FFFF FFFT FFTF FTFT TFTF TFTT TTTF TTTT
x1 x2 x4 T F T F F F F F
x1 x3 x4 F T F T F F F F
x2 x3 x4 F F T F T F F F
x1 x3 F F F F T T T T
here x2, x3, x4 is not essential, so we are left with
FFFF FFFT FFTF FTFT TFTF TFTT TTTF TTTT
x1 x2 x4 T F T F F F F F
x1 x3 x4 F T F T F F F F
x1 x3 F F F F T T T T
all the remaining ones (x1 x2 x4, x1 x3 x4, and x1 x3) are essential
87