Incorrect Answer.
We determine the meaning of the CTL formula AG (p 
A[q U r]) by
- breaking it up into significant subformulas,
- determining the meaning of these subformulas, and by
- determining the meaning of the entire formula by composing the meanings
of subformulas in the manner suggested by the logic operators
that are present.
To wit,
- the meaning of A[q U r] at any state s is that 'for all computation paths
that begin in s, the predicate q is true until the path reaches a state in
which r is true (and such a state has to be reached)'.
- The meaning of p A[q U r] at
any state s is that 'if p holds in state s, then A[q U r] has to hold at
state s as well'.
- The meaning of AG(p A[q U r]) at
any state s is that 'p A[q U r] holds
at any state t that is reachable from state s'.
The English specification "If p is true in every reachable state, then there is a path along which q is continuously true, until r becomes true" is a far
cry from the meaning of the formula above.
The premise "p is true in every reachable state" can be modeled by
AG(p). The conclusion "there is a path along which q is continuously true, until r becomes true" may be expressed by
E[q u r]. Thus, the English language specification is modeled by the
CTL formula
AG(p) E[q u r]. Note that this has
a completely different meaning than
AG(p A[q U r]).
For example, consider a state s in a model where each state t, reachable
from s, satisfies p. Then for former formula insists on a computation
path beginning in s that satisfies "q U r". The latter formula, however,
insists on A[q U r] to hold at all reachable states t, for p holds for
all of them, thereby "triggering" the implication!
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