Formale Systeme
Week 6
Equivalences with quantifiers
http://cs.uni-salzburg.at/~anas/teaching/FormaleSysteme/
Equivalences with quantifiers
Bound variables
x
x
P :Q
val
P :Q
val
y
P y for x : Q y for x
y
P y for x : Q y for x
if y is not free in P and Q
Domain splitting
Examples:
Domain splitting
Examples:
x
val
x
x
x
1
x
1:x
2
5 : x2
6x
5
6x
0
5
0
x
x
5:x
2
6x
5
0
Domain splitting
Examples:
x
val
x
k
val
val
k
k
x
x
0
0
0
1
x
1:x
2
k
k
k
5 : x2
6x
5
6x
5
0
n : k 2 10
2
n 1 k n:k
n 1 : k 2 10
0
x
x
10
k k
5:x
2
6x
n : k2
5
10
0
Domain splitting
Examples:
x
val
x
k
val
val
k
k
x
x
0
0
0
1
x
1:x
2
k
k
k
5 : x2
6x
5
6x
5
0
0
x
n : k 2 10
2
n 1 k n:k
n 1 : k 2 10
x
10
k k
Domain splitting
x
x
P
P
Q:R
val
Q:R
val
x
P :R
x
Q:R
x
P :R
x
Q:R
5:x
2
6x
n : k2
5
10
0
Equivalences with quantifiers
One-element rule
x
x
x
x
n:Q
val
Q n for x
n:Q
val
Q n for x
Equivalences with quantifiers
One-element rule
x
x
x
x
n:Q
val
Q n for x
n:Q
val
Q n for x
Example:
3:2 x
x x
1
val
2 3
1
Equivalences with quantifiers
One-element rule
x
x
x
x
n:Q
val
Q n for x
n:Q
val
Q n for x
Example:
3:2 x
x x
1
val
2 3
1
Empty domain
x
x
F :Q
val
T
F :Q
val
F
Equivalences with quantifiers
One-element rule
x
x
x
x
n:Q
val
Q n for x
n:Q
val
Q n for x
Example:
3:2 x
x x
1
val
2 3
1
Empty domain
x
“All Marsians are green”
x
F :Q
val
T
F :Q
val
F
Domain weakening
Intuition: The following are equivalent
x
x
x
x
D:A x
D:A x
and
and
x
x
x D
x D
Ax
Ax
The same can be done to parts of the domain
Domain weakening
Intuition: The following are equivalent
x
x
x
x
D:A x
D:A x
and
and
x
x
x D
x D
Ax
Ax
The same can be done to parts of the domain
Domain weakening
x
P
x
P
Q:R
Q:R
val
val
x
x
P :Q
P :Q
R
R
Domain weakening
Intuition: The following are equivalent
x
x
x
x
D:A x
D:A x
and
and
x
x
x D
x D
Ax
Ax
The same can be done to parts of the domain
Domain weakening
x
P
x
P
Q:R
Q:R
val
val
val
x
x
P :Q
P :Q
R
R
P
Q|
P
Equivalences with quantifiers
De Morgan
x
x
P :Q
P :Q
val
val
x
x
P: Q
P: Q
Equivalences with quantifiers
De Morgan
x
x
P :Q
P :Q
val
val
x
x
P: Q
P: Q
not for all = at least for one not
not exists = for all not
Equivalences with quantifiers
De Morgan
x
x
Hence:
P :Q
P :Q
val
val
x
x
P: Q
P: Q
and
not for all = at least for one not
not exists = for all not
Equivalences with quantifiers
De Morgan
x
x
P :Q
P :Q
val
val
x
x
not for all = at least for one not
P: Q
P: Q
Hence:
not exists = for all not
and
It holds further that:
x
x
x
x
x
x