Boolean Functions 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping...
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Boolean Functions 11 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping truth table {0,1} {0,1} : n f
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Transcript of Boolean Functions 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping...
Boolean Functions
1 1 1 1
0000111
0011001
0101010
0000011
1x 2x 3x f
mappingtruth table{0,1}{0,1}: nf
Representation of Boolean Functions
Disjunctive Normal Form (canonical):
1 1 1 1
0000111
0011001
0101010
0001010
Example:
OR of AND terms (not unique):
321321321321 ),,( xxxxxxxxxxxxf
3231321 ),,( xxxxxxxf
1x 2x 3x f
Representation of Boolean Functions
Conjunctive Normal Form (canonical):
Example:
AND of OR terms (not unique):
)(
)()(
)()(),,(
321
321321
321321321
xxx
xxxxxx
xxxxxxxxxf
)()(),,( 213321 xxxxxxf 1 1 1 1
0000111
0011001
0101010
0001010
1x 2x 3x f
Representation of Boolean Functions
Example:
XOR of AND terms:
unique?
1 1 1 1
0000111
0011001
0101010
0001010
1x 2x 3x f
)1 as represent literals; positive(only xx
3213231321 ),,( xxxxxxxxxxf
representation is unique.
Dependence on variables is explicit.
functions,2 m2
XOR of AND terms:
Representation of Boolean Functions
For m variables, express a boolean function as the sum of some combination of the product terms:m2
3211312121 ||,,,|,,|1 xxxxxxxxxxxx nnn
sums,distinct 2 m2
Logic Circuits
Logic GateBuilding Block:
1x2x
dx
di
ix
,,1allfor
{0,1}
{0,1}{0,1}: dg
),,( 1 dxxg
Logic Circuits
Logic GateBuilding Block:
1x2x
dx
),,( 1 dxxg
feed-forward device
Logic Circuits
“AND” gate
0001
Common Gate:
1x
2x
g0011
0101
1x 2x g
Logic Circuits
“OR” gate
0011
0101
0111
Common Gate:
1x
2x
g
1x 2x g
Logic Circuits
“XOR” gate
0011
0101
0110
Common Gate:
1x
2x
g
1x 2x g
Logic Circuits
“NOT” gate
1
0
Common Gate:
0
1
1x g
1x g
),,( 11 mxxf a
),,( 12 mxxf a
),,( 1 mn xxf a
inputs outputs
Logic Circuits
1x
2x
mx
mi
ix
,,1allfor
{0,1}
nj
mjf
,,1allfor
{0,1}{0,1}:
network oflogic gates
),,( 1 mn xxf a
),,( 11 mxxf a
),,( 12 mxxf a),,( 1 mxxf a
Logic Circuits
inputs outputs
1x
2x
mx
network oflogic gates gate
mi
ix
,,1allfor
{0,1}
nj
mjf
,,1allfor
{0,1}{0,1}:
Example
Logic Circuits
x
y
x
y
z
z
c
s
Logic Circuits
0
1
0
1
1
1
0
1
1
0
1
Example
Logic Circuits
• Size: number of gates.• Depth: longest path from an input to an output.
Measures
x
y
x
y
z
z
c
s
size = 5, depth = 3
Logic Circuits
Why study logic circuits? It all seems simple enough....
1
0110100
Construct XOR from AND/OR/NOT gates.
1 2 3
1 1 1
0000111
0011001
0101010
x x x ),,( 321 xxxXOR
3 variables(draw circuit)
Logic Circuits
Construct XOR from AND/OR/NOT gates.
4 variables(draw circuit)
1
1111111
1 2 3
0 1 1
1000111
0100101
0010110
x x x ),,( 321 xxxXOR , 4x4
1
0001011
x
......
......
x
y),( yxXOR
x
y),( yxOR
x
y),( yxAND )(xNOTx
1} {0,, yx
Example: Logic Gates
Models of Computation
Models of Computation
Example: Linear Threshold Gates
1x
2x)sgn(
10
n
iii xww
nx
1w
2w
nw0w
Models of Computation
Example: Comparators and Balancers
x
y
min(x, y)
max(x, y)
x
y
2
yx
2
yx
Models of Computation
Example: Switching Circuits
a b
c
ed
S D
dcbacedeabf
Switching Circuits
x1
x2
x3
x4
x5
S D
(Shannon, 1938) DSf
)( 41 xandx)( 52 xandx
)( 531 xandxandx)( 432 xandxandx
w
Example: Encoding FSMs
nox
noxF
to fromn transitioa causes
1),,(
Systems with states analyzed.2010
Finite State Machine
x
),( 21 ooo
input
current state
next state),( 21 nnn
A
B
D
F
Logic Circuits
Construct XOR from AND/OR/NOT gates.
construction:
lower bound:
XOR of n variables
)1(5.2 n gates
12 n gates
For n > 4, optimal size is unknown.
≤ optimal size ≤ )1(5.2 n12 n
Linear Threshold Gates
1x
2x
nx
1w
2w
nw
0w... ),,( 1 nxxf x
ni
ix
,,1allfor
{0,1}
0if1
0if0
2210
2210
nn
nn
xwxwwxw
xwxwwxwf
w
ni
iw
,,0allfor
R
Linear Threshold Gates
Useful Model?
