CS 267 – Spring 2011 – Homework Assignment 1
Due Thursday, April 14th
Do not discuss the problems with anyone other than the instructor.
1.
Consider the following two transition systems:
M
1
= (
AP
1
,S
1
,R
1
,I
1
,L
1
) with the set of states
S
1
=
{
0
,
1
,
2
,
3
}
, the initial set of states
I
1
=
{
0
}
, the transition relation
R
1
=
{
(0
,
1)
,
(1
,
2)
,
(2
,
3)
,
(2
,
1)
,
(3
,
3)
}
, the set of atomic
propositions
AP
1
=
{
p,q
}
and the labeling function
L
1
:
S
1
→
2
AP
where
L
1
(0) =
∅
,
L
1
(1) =
{
p
}
,
L
1
(2) =
{
p,q
}
, and
L
1
(3) =
{
q
}
.
M
2
= (
AP
2
,S
2
,R
2
,I
2
,L
2
) with the set of states
S
2
=
{
0
,
1
,
2
,
3
}
, the initial set of states
I
2
=
{
0
}
, the transition relation
R
2
=
{
(0
,
1)
,
(1
,
2)
,
(2
,
3)
,
(1
,
1)
,
(3
,
2)
}
, the set of atomic
propositions
AP
2
=
{
p,q
}
and the labeling function
L
2
:
S
→
2
AP
where
L
2
(0) =
{
p
}
,
L
2
(1) =
{
p,q
}
,
L
2
(2) =
{
p,q
}
, and
L
2
(3) =
{
q
}
.
(a)
For each of the CTL formulas show the states which satisfy them:
EG
p,
AG
p,
EG
q,
AG
q,
EF
p,
AF
p, p
EU
q, p
AU
q,
EG(EF
q
)
,
AG(EF
q
)
,
AG(AF
q
)
,
EG(AF
q
)
.
For the ACTL properties above that fail for the given transition systems, give a counter-
example. For the ECTL properties above that hold for the given transition systems, give a