Lecture Notes in Computer Science 3472

60 Moez Krichen
Definition 2.19. A splitting tree associated with a Mealy machine M =
(I , O, S,δ,λ) is a rooted tree T such that:
• Each node of T is labeled by a subset of S and the root of T is labeled with
S,
• The children’s labels of an internal node of T are disjoint and the label of
an internal node of T is the union of its children’s labels,
• With each internal node of T is associated an input sequence,
• With each internal node u of T , with associated set-label Su and input stringlabel
xu, is associated a mapping fu : Su → S such that fu(s) =δ(s, xu) for
each s in Su,
• Each edge of T is labeled with an output symbol from O.
Let π(T ) denotes the collection of sets of states formed by the labels of the leaves
of the splitting tree T (it is easy to see that π(T ) is a partition of S). T is a
complete splitting tree if π(T ) is the discrete partition of S. ⊓⊔
u4
{s1}
0
0
u8
{s5}
u2
{s1, s3, s5}
ba
1
0
u5
{s3, s5}
aaba
u1
{s1, s2, s3, s4, s5, s6}
a
1
u9
{s3}
0
u6
{s6}
u3
{s2, s4, s6}
aba
u10
Fig. 2.14. A complete splitting tree of machine M6.
1
0
1
u7
{s2, s4}
bba
u11
{s2} {s4}
Example. A complete splitting tree for machine M6 (Fig. 2.8) is shown in Fig. 2.14.
The mappings fu associated with this splitting tree are given in Table 2.4.
The way for computing such a complete splitting tree is given by Algorithm 7.
The Algorithm uses the following notation:
• ST0 is the tree with a single node whose root is labeled with S.
• πd is the discrete partition of S.
1