Lecture Notes in Computer Science 3472

and
{s2, s4, s6} a/1
−→{s1, s3, s5}.
2 State Identification 59
According to the execution of Algorithm 6, b allows to refine {s1, s3, s5} since
it makes its states move to distinct blocks of π(T ). Thus, T can be extended
by executing the input sequence ab from u2. Asaresult,nodesu4 and u5
are added to T and node u2 is labeled by a and node u4 by b. The suitable
current and initial sets are assigned to each new node.
• Step 3: the so far obtained T is not closed since the current set {s1, s4, s6} of
u5 intersects both the initial set of u3 and u5 (the current leaves of T ). The
lowest common ancestor of these two nodes is u1. So, the input sequence to
be used is a. Thus at this step, we add to Tu6 and u7 and to each of them
we assign the corresponding current and initial sets.
• Step 4: T is not closed since the current set {s1, s5} of node u7 intersects both
the initial set of nodes u7 and u9. The lowest common ancestor of u7 and u9
is u5. So, the input sequence to be used is aaba (spelt by the path from u1
to u5). T is consequently formed by nodes u1 to u12. Due to this step the
two extra states s1 (node u11) ands3 (node u12) have become identifiable.
• Step 5: T is not closed since the current set {s1, s3, s5} of node u3 is the
union of the initial sets of the nodes u6, u11 and u12. The lowest common
ancestor of these nodes is u5. Soasinstep4,weapplyaaba on the current
set of u3. Consequently, we append to T nodes from u13 to u17.
• Step 6: T is not closed since the current set {s1, s5} of node u17 is the union
of the initial sets of the nodes u6 and u11. The lowest common ancestor of
these nodes is still u5. Soasinstep4,weapplyaaba on the current set of
u3. At this step, nodes from u18 to u22 are appended to T .
It is easy to see that the obtained T due to step 6 is an ADS for machine M6.
Remark 2.18. There is a similarity between Algorithm 6 and the classical algorithm
for FSM minimization. The only difference between them is that Algorithm
6 uses only valid inputs, whereas the minimization algorithm uses all
input symbols.
2.4.4 Computing a Polynomial ADS
In the proof of Theorem 2.17, we gave a first method for constructing an ADS
(when Algorithm 6 terminates with the discrete partition of some given machine).
This method may give exponential ADSs. For example, the ADS of machine
M6 resulted in by this method (Fig. 2.13) is not optimal (it is longer than
the one shown in Fig. 2.9).
In [LY94], Lee and Yannakakis propose methods for computing ADSs with
polynomial size and in polynomial time. For computing “optimal” ADSs, they
introduce an intermediary structure: the splitting tree. The latter is defined as
follows.