Lecture Notes in Computer Science 3472

1 Homing and Synchronizing Sequences 9
versions of the homing and synchronizing sequences problems: first an algorithm
to compute homing sequences for minimized Mealy machines (Section 1.3.1),
then an algorithm to compute synchronizing sequences (Section 1.3.2). Section
1.3.3 unifies these algorithms into one for computing homing sequences
for general (not necessarily minimized) machines – this algorithm can be used
both to compute homing and synchronizing sequences. The two algorithms for
computing homing sequences are then modified to compute adaptive homing
sequences in Section 1.3.4. Finally, Section 1.3.5 gives exponential algorithms to
compute minimal length homing and synchronizing sequences.
Section 1.4 turns from algorithms to complexity. First, Section 1.4.1 shows
that it is NP-hard to find the shortest homing or synchronizing sequence. Second,
Section 1.4.2 shows that it is PSPACE-complete to determine if a machine has
a homing or synchronizing sequence, if it is known that the initial state is in a
particular subset Q ⊆ S. In both cases it means that polynomial algorithms for
the problems are unlikely to exist.
Section 1.5 gives an overview of research in the area and mentions some related
areas, and Section 1.6 summarizes the most important ideas in the chapter.
Flogsta Ekeby
H˚aga Eriksberg K˚abo
Fig. 1.3. The subway map of Uppsala. The five stations are connected by two one-way
lines: white and grey.
Exercise 1.1. The schematic map of the subway in Uppsala looks as in Figure 1.3.
You do not know at which station you are, and there are no signs or other
characteristics that reveal the current station, but you have to get to Flogsta
by moving from station to station, in each step taking either the white or the
grey line. What type of sequence does this correspond to? Find a sequence if one
exists.
(Hint: use that if you are in Flogsta or H˚aga, the white line takes you to Eriksberg
for sure.)
1.2 Initial and Current State Uncertainty
Consider a Mealy machine to which we apply some input string and receive an
output string. Even if the input string was not homing, we may still draw some