Lecture Notes in Computer Science 3472

1 Homing and Synchronizing Sequences 13
The Algorithm. Algorithm 1 computes a homing sequence for a minimized
machine as follows. It first finds a separating sequence for some two states of the
machine. By the definition of separating sequence, the two states give different
outputs, hence they now belong to different blocks of the resulting current state
uncertainty. Next iteration finds two new states that belong to the same block of
the current state uncertainty and applies a separating sequence for them. Again,
the two states end up in different blocks of the new current state uncertainty. This
process is repeated until the current state uncertainty contains only singleton
blocks, at which point we have a homing sequence.
Algorithm 1 Computing a homing sequence for a minimized machine.
1 function Homing-For-Minimized(Minimized Mealy machine M)
2 x ← ε
3 while there is a block B ∈ σ(x )with|B| > 1
4 find two different states s, t ∈ B
5 let y be a separating sequence for s and t
6 x ← xy
7 return x
Step 5 of the algorithm can always be performed because the machine is
minimized. Since y is separating, the block B splits into at least two new blocks
(one containing s and one containing t). Thus, Lemma 1.10 holds with strict
inequality between any two iterations, i.e., the quantity �
B∈σ(x )
|B| −|σ(x )|
strictly decreases in each iteration. When the algorithm starts, it is n − 1and
when it terminates it is 0. Hence the algorithm terminates after concatenating
at most n − 1 separating sequences.
The algorithm is due to Ginsburg [Gin58] who relies on the adaptive version
of Moore [Moo56] which we will see more of in Section 1.3.4.
Length of Separating Sequences. We now show that any two states in a
minimized machine have a separating sequence of length at most n − 1. Since
we only need to concatenate n − 1 separating sequences, this shows that the
computed homing sequence has length at most (n − 1) 2 . The argument will also
help understanding how to compute separating sequences.
Define a sequence ρ0,ρ1,... of partitions, so that two states are in the same
class of ρi if and only if they do not have any separating sequence of length i.
Thus, ρi is the partition induced by the relation s ≡i t def
⇔ “λ(s, x )=λ(t, x )for
all x ∈ I ∗ of length at most i”. In particular, ρ0 = {S}, andρi+1 is a refinement
of ρi. These partitions are also used in algorithms for machine minimization; cf.
Section 21. The following lemma is important.
Lemma 1.12 ([Moo56]). If ρi+1 = ρi for some i, then the rest of the sequence
of partitions is constant, i.e., ρj = ρi for all j > i.