CSE 541  Logic in Computer Science
Solutions for Selected Exercises on
Temporal Logic
Exercise 3.4.9
A CTL formula
EFp
is true for a state if
p
is true for that state
already, wheras
EX EFp
need not be true if
p
is true for the
present state.
A formula
AGp
is true for a state
s
if, and only if,
p
is true
for the present state
s
and all states reachable from
s
, wheras
AX AGp
is true for
s
and if, and only if, it is true for all states
reachable from
s
.
A formula
E
[
pUq
] is true for a state if
q
is true for that state
already. The formula
p
∧
EX E
[
pUq
], on the other hand, requires
that (i)
q
be true in a future state, not including the present state,
and (ii)
p
be true in all preceding states, including the present
state.
Exercise 3.4.10
a.
EF φ
and
EG φ
are not equivalent.
Let
φ
be
p
and
M
be a transition system with
States:
S
=
{
s
0
, s
1
}
Transitions:
s
0
→
s
1
,
s
1
→
s
1
Labels:
L
(
s
0
) =
{
p
}
,
L
(
s
1
) =
∅
We have
M
, s
0

=
EF p
but
M
, s
0
6
=
EG p
.
b.
EF φ
∨
EF ψ
and
EF
(
φ
∨
ψ
) are equivalent.
c.
AF φ
∨
AF ψ
and
AF
(
φ
∨
ψ
) are not equivalent.
Take
φ
=
p
and
ψ
=
q
and let
M
be a transition system with
States:
S
=
{
s
0
, s
1
, s
2
}
Transitions:
s
0
→
s
1
,
s
0
→
s
2
,
s
1
→
s
1
,
s
2
→
s
2
Labels:
L
(
s
0
) =
∅
,
L
(
s
1
) =
{
p
}
L
(
s
2
) =
{
q
}
Then
M
, s
0

=
AF
(
p
∨
q
) but
M
, s
0
6
=
AF p
∨
AF q
.