Lecture Notes in Computer Science 3472

fu1 =(s1, s2), (s3, s5), (s4, s5), (s5, s6), (s6, s1)
fu2 = (s1, s2), (s2, s3), (s5, s1)
fu3 = (s2, s5), (s4, s1), (s6, s2)
fu5 = (s3, s1), (s5, s2)
fu7 = (s2, s2), (s4, s1)
2 State Identification 61
Table 2.4. The mappings fu of the splitting tree of machine M6 (Fig. 2.14).
• u(B) is the leaf of ST with set-label B.
• u ·set and u ·string are the set and string-labels of the node u, respectively.
• append(ST, u, B ′ , a) isST towhichweappendanewnodev from u such
that the set-label of v is B ′ and the edge from u to v is labeled by input
symbol a.
• λ(s ∈ B ′ , a)= λ(s, a) for any arbitrary s in B ′ .
• label(v, w) is the label of the edge of ST which is from v to w.
The algorithm proceeds as follows. It starts with the one node splitting tree.
At each iteration, it considers a block B from R the set of blocks of π with largest
cardinality. B is the set-label of one of the leaves of the current splitting tree.
Then, it looks for the shortest input sequence that may refine B. Threewaysare
then possible for finding such an input sequence. If B has an a-valid input symbol
(“case 1”) then the searched input sequence is the found a-valid input symbol. If
no such leaf exists then we look for a leaf which has a b-valid input symbol (w.r.t
to π the partition of states induced by the current splitting tree). If such a leaf
is found (“case 2”), then we identify the node of the current splitting tree whose
set-label contains δ(B, a), where a is the b-valid input symbol. If σ is the input
string-label associated with this node then it is clear that B can be refined by
executing aσ. Now, if the set-label of the considered leaf has neither an a-valid
nor a b-valid input symbol then we check whether there exists a sequence of
c-valid inputs for B (w.r.t π) which makes the states of B (the set-label to be
refined) move to another set-label C which has just been refined by the execution
of an input sequence τ . If such a path σ exists (“case 3”) then it is clear that
στ can refine the set-label B.
If none of the three ways works, we conclude that the considered machine
has no ADS. In the third case, for obtaining the shortest input sequence which
refines B we shall look for the shortest path σ which goes from B to some other
possible set-label C . This can be done by performing a reachability analysis on
Gπ[R] the subgraph of Gπ (the implication graph corresponding to the partition
π) induced by R to find a path from B to some possible block C .
For each of the three cases, the splitting tree is updated as follows: we assign
the found input sequence to the input string-label of the considered leaf then
new leaves are attached to this (old) leaf. The number of attached leaves equals
the number of subsets to which B is refined.
Example. We apply Algorithm 7 to M6 to obtain the complete splitting tree
shown in Fig. 2.14.