Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

11.6.1 State Transition Diagram
for PTL Interpretation
Consider a state transition graph where the nodes denote the temporal states
and the arc denotes the transition from one state to another through passage of
time. For instance, the state transition graph of fig. 11.9 describes the
transition of temporal states from s1 to s2 and s3, from s2 to s3 and s4, from s3 to
s2 and s4 to s4 itself. Further, each state si corresponds to a temporal value of
the propositions p and q. For brevity of representation, we use the positive or
negative literals like {p, q} or {¬p, q} instead of {p= true, q= true} or {p=
false, q= true} respectively.
Now, suppose we want to evaluate the truth value of the formula
X =Ā(p) ∨ Ā(q)
in each state.
In state s1, X is true as its next states s2 and s3 both satisfy X. X is also
true in s2, as its next state s3 satisfies A. X is also found to be true in s3 as its
next state s2 supporting X. X is not true in s4 as itself and its net state, which
too is s4, does not support A.
p
¬q
S1
S2
S4
¬p
q
¬p
¬q
Fig. 11.9: A state transition graph representing PTL interpretation.
We now formally prove a few identities using the concept of the state
transition graphs.
p
q
S3