George Boole

1
?????? ???????? - ?????? ??????
George Boole 1854 ? Boolean Algebra
Claud Shannon
1938 ? Daul Valued Boolean Algebra
(Information Theory)
????? ???? B - ????? (?????, ?? B
2 ? ??? Boolean Algebra ) ? - ???????
?????? S?S ? S
(????? S ??????? ?? ?????) ?????? ??
???? ?? ???? ?-S ????? ?????? S
??????? ?? ????? ???????? - ??????????? (X
Y) Z X (Y Z) ???????? -
?????????? X Y Y X ???? ????? 1
X X 1 X (For all X in S) ???????
For all X there exists Y such that X Y 1
? ?? ??????? ??????? ???????? ????????
?????????????? ??? ????????? ?
x?(yz) x?y x?z
?? ???
2
????? ?????????
B - ????? (2 B ?????? ??????) ???
????????? (OR ,?) ) AND, ?) 1. B
????? ???? ?"" ?-"" . 2. ?. ???? ???? ????
???? ?-"" x0 0x
x ?. ???? ???? ????? ???? ?-"" .
x1 1x x 3. ???
?????? xy yx xy
yx 4. ??????? ????????
x(yz) xy xz x(yz)
(xy) (xz) 5. ????? ??? x ???? x
(not(x), x) ?? xx 1 (x or
(not(x)) True) xx 0 (x and
(not(x)) False) 6. ?????? ????? ??? ??????
x,y?B ( B 2 ?????? ??????)
3
????? ??????
AND

OR
5v
???????????? ??????? ?? ????? ??????? (X Y)
ZX (YZ) ??? ?????? ?????? ???? ,
?? ?????? ??????? ???????? (??????)
?????? ?? ???? ??????? ???????? ??????
??????? ????????? ?? 2lt B ??????
???????? ?? ????? (?????) 1 , 0 B
NOT
OR
AND
X X
1 0 0 1
XY Y X
0 1 1 1 0 0 1 1 0 1 0 1
X.Y Y X
0 0 0 1 0 0 1 1 0 1 0 1
?? 6 ?????? ????????
4
????? ?????? ??????? ??"?

• ????? xx x
• ??? ?????? ?????? /?????? ??"? ????? ?-6 ??????
  (?????????) ???????? ?"?.
• ???? ?????? ?????? ?? ?"? ????? ????? ???????
  (Brute Force)
• ?????? ????????
• ?? ???? ???? ??"? ???? ?????? ????? ??? ???? ???
  ?"?
• x(yz)
  ?
• x(yz) 0
  ? 1
• ????? x 1 x ? x 0 x

5
?????? ???????

• xx x xx x
• x1 1 x0 0
• (x) x
• (xy)z x(yz) (xy)z x(yz)
  
• ???? ??- ?????
• (xy) xy
• (xy) xy
• ????? x(xy) x x (xy) x
• ???? x(xy) xy

(X1 X2 .. Xn) X1. X2 . Xn
XY X(X.Y) XY y X
0 1 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 1
6

• ???? x(xy) xy
• ?????
• ?. ???????? (??? 4) x(xy) (xx)
  (xy)
• ?. ??? ?????? (1 (xx
• (xx)(xy) 1(xy)
• ?. ???? ????? (x1 1x x)
• 1(xy) xy QED
• ?????? ????? ?????????? ????? ????? ?????
  ???????? (??? ?? ????? ??????).)
• ?????? ????????? 3
  2 1 0
  
• ( ) ?
  NOT ? AND ? OR
• ( )
  ? ,( ) ? , ? ? ?,
• ????? (x (y (z))) ? x y z

??????? ?? ???? ??????
7
???????? ?????????
?0,1 ? 0,1n
??????? ???????? ???? n ??????. ?? ?????
???? ?????? ???????. ???? ???? ???? 2n
?????? . ???? ?"? ????? ?????
x
xy
y
x
x
x
xy
y
x
x(yz)
x
y
z
8
????? ?? ???????
???? ???????? "?????" (????? ???? ????? ?????
ƒ(x,,z) 1 ), ???? ???? ?? ??????/ ?????? ?? ƒ
?????? ???? ?? ??????. ??? ??- ?????
De Morgan Rule (x y) x y
(x y) x y ???? ?????
?????????? ???????? ????? ???? ?????? ?????
?????? ??? ?? ???????.
(xyz
(tsqx))
(x (yz ( t sqx)))

x ( yz ( t sqx))
x ( yz
t (s q x))
xy xz xts xtq
xtx


???? ?- ??-?????
0
???? ??????
9
????? ???????
xyzxyzxyz
?? ??????? ???????? ????? ?????? ????? ??????
(xyz)
(xyz)

?????? ??????
??????
??????
????? ???? ????? ???? f z y x
M0 xyz m0 xyz 1 0 0 0
M1 xyz m1 xyz 0 1 0 0
M2 xyz m2 xyz 1 0 1 0
M3 xyz m3 xyz 1 1 1 0
M4 xyz m4 xyz 1 0 0 1
M5 xyz m5 xyz 0 1 0 1
M6 xyz m6 xyz 1 0 1 1
M7 xyz m7 xyz 1 1 1 1

• ?????? ???? ??? ?? ??????? f
• ????? ?? f ?????? ?????? ?"? ????? Mi ????? f0.
• ??
• 2) ????? ?? f ????? ?????? ?"? ????? mi ????? f1.

minterm
(xyz)
(xyz)
xyzxyz
10
????? ?????? ??????? ????? ????????
f2 f1 z y x
0 1 0 0 0 0
1 0 0 1 0 0
2 1 1 0 1 0
3 1 0 1 1 0
4 1 0 0 0 1
5 1 1 1 0 1
6 1 0 0 1 1
7 0 0 1 1 1

• ?????? ????? ?? ?????? ?? ???? ???? ??????
  ??????? ??? ?????? ??????? ?????????
  (???????).
• ???? ????? ????? ??????

11
????? ????? ????????
?????? ????? ???? ???? ???? ????? ??????. ????
??? ????? ???? ?? ?????? ?????
?????? ??????.
12
????????? ?????? ??????
?? ????????
????? ?? ??? ???????
F00
AND X Y F1xy
X/Y F2xy
F3x
Y/X F4xy
F5y
XOR XY F6xyxy
OR XY F7xy
universal NOR X Y F8(xy)
Equivalence X Y F9xyxy
F10y
F11xy
F12x
F13xy
universal NAND X Y F14(xy)
F15 1
13
????? ?????? ???????????? ????????? ???????
????????.
A
FA B AND
F
B
A
FAB OR
F
B
FA Inverter
A
A
FA Buffer
A
A
A
F(A B)AB NAND
F
B
A
F(AB)A B NOR
F
B
14
A
F
B
A
F
B
15
????? ????? ??????
Semantics of NOR?

16
????NOR/NAND ????? ??????
NAND (A,B,C) (ABC)
NOR (A,B,C) (ABC)
????XOR ????? ??????
17
?????? ????? -Universal Systems

• ????? ??? ??????? ???????? ????? ?????? ?"? ????
  ?????? ??/? ????? ??????. ??? ?? ??????? ????????
  ????? ?????? ?"? ????? ??????????
• NOT, AND, OR , ,
• ????? ?????????? ???? ???? (Universal) ?? ????
  ???? ????? ?????? ?????? ?? ????????? ??????? ??
  ??????? ????????.
• ???? ?. NOT, OR ??? ????
• ?. NOT, AND??? ????
• ????? (???? ?)
• ????? ?? F ??????. ????? ???? ?????
• ?. F (G) ????? ?- NOT.
• ?. F GQ ????? ?- OR.
• ?. F GQ F ((GQ)) (GQ)
• ????? ?OR ?NOT ????!
• ??? ??? ?????????? ????? ??????????

18
NOR and NAND - Universal Systems

• ?????? ? - NOT, AND ??? ???? ????? ?? ???? ????
  ?? AND ?-NOT ?"?
• NAND ????

X (X X) NAND(X,X)
A B ((A B)) )NAND)A,B))
NAND(NAND(A,B),NAND(A,B))
?????? ? - NOT, OR ??? ???? ????? ?? ???? ????
?? OR ?-NOT ?"? NOR ????
X (X X) NOR(X,X)
A B ((A B)) )NOR)A,B))
NOR(NOR(A,B),NOR(A,B))
Equivalence
A
A
F
F
B
B
19
??????? ???????? ?? ???? ??? ????? ????? ?????.
CAB
ABC F
1
0
1
0
1
0
1
Delay Time
0
1
1
Spikes ???? ????? ?"? ????? ????? ??/? ?????
??????.
CA
(C A) (C B) AB C
CB