Lecture Notes in Computer Science 3472

24 Sven Sandberg
b
s1
a
s2
b
s3
a
a, b
s1, s2
s1, s2
s2
a
a b
a
s1, s2, s3
s1, s3
b
s1, s2, s3
Fig. 1.6. A machine and its corresponding synchronizing tree. Note that the rightmost
and leftmost nodes of the tree have been made leaves due to rule (3b) of Definition 1.18.
Since the lowest leaf is labeled with a singleton s2, the path leading to it from the root
indicates a shortest synchronizing sequence, aba.
It is possible to replace condition (3b) of Definition 1.18 with a stronger one,
allowing to prune the tree more: stipulate that a node becomes a leaf also if its
label is a superset of some node of smaller depth. This clearly works, because if
P ⊆ Q ⊆ S are the node labels, then δ(P, x ) ⊆ δ(Q, x ) for all sequences x .The
drawback is that it can be more costly to test this condition.
The homing tree is used analogously to compute shortest homing sequences.
Definition 1.20. The homing tree (for a Mealy machine) is a rooted tree
where edges are labeled with input symbols and nodes with current state uncertainties
(i.e., sets of sets of states), satisfying the following conditions.
(1) Each non-leaf has exactly |I | outgoing edges, labeled with different input
symbols.
(2) Each node is labeled with σ(x ), where x is the sequence of input symbols
occurring as edge labels on the path from the root to the node.
(3) A node is a leaf iff:
(a) either each block of its label is a singleton set,
(b) or it has the same label as a node of smaller depth. ⊓⊔
Condition (3b) can be strengthened in a similar way for the homing tree as
for the synchronizing tree. Here we turn a node into a leaf also if each block of
its label is a superset of some block in the label of another node at a smaller
depth.
The homing tree method was introduced by Gill [Gil61] and the synchronizing
tree method has been described by Hennie [Hen64]. Synchronizing sequences are
sometimes used in test generation for circuits without a reset (a reset, here,
would be an input symbol that takes every state to one and the same state,