Lecture Notes in Computer Science 3472

s1 s2 s3
s1 s3 s3
a/0
a/0
s1 s2 s3
s3 s1 s1
s1 s2 s3
s1 s3 s3
b/0
b/0
s1
s4
s2 s3
s4 s4
a/0
s4
s2
s2 s3
s4 s4
s1 s2 s3 s4
s1 s2 s3 s4
a/1 b/0
s1
s1
s2
s1
s3
s4
3 State Verification 81
b/1
b/1 a/0
b/1
s1 s4
s2 s2
s1 s2 s4
s4 s2 s4
a/1
b/1
s1 s2 s4
s4 s2 s4
Fig. 3.6. The pruned UIO tree for the machine in Figure 3.3. Each node in the tree
is labeled by the corresponding path vector. Path vectors are depicted with two lines,
where the first is the initial vector, and the second is the current vector. For instance, it
can be read from the figure that the input sequence ab gives output 01 for initial states
s2 and s3. In both cases, it takes the machine to state s4. The node that is reached
from the root by following the path labeled (a/0)(b/1) has a homogeneous path vector
and is thus terminal. Therefore, it is a leaf in the tree.
not only keep track of the states for which a certain input-output sequence is
consistent, but also the states the machine is taken to by the sequence. In each
internal node, the UIO tree branches for every input-output label such that there
is a state in the current vector of the node that has the label on one of its
outgoing edges. The splitting tree, on the other hand, branches only on one
input symbol per interior node, and splits the states in the node label according
to their behavior on this input. This means that there are many different possible
splitting trees, while the UIO tree is unique for each machine. The splitting tree
is used to find adaptive distinguishing sequences. If such a sequence exists, there
is always one of at most quadratic depth. Thus we never have to build very deep
splitting trees. In fact it is enough to construct trees of total size O(n 2 ); see
Chapter 2. Since UIO trees are used to find UIO sequences, which may have
superpolynomial length, we might have to build exponentially large trees.
Inference Rules. In many cases, it is possible to infer UIO sequences for one
state, given a UIO sequence for some other state, as discussed in Section 3.3.