Lecture Notes in Computer Science 3472

1 Homing and Synchronizing Sequences 33
in Section 1.3.1, which works by concatenating many separating sequences. Synchronizing
sequences do not always exist, but the cubic time algorithm of Section
1.3.2 computes one if it exists, or reports that none exists, by concatenating
many merging sequences. Synchronizing sequences have at most cubic length, but
it is an open problem to determine if this can be improved to quadratic. Combining
the methods of these two algorithms, we get the algorithm of Section 1.3.3
for computing homing sequences for general (non-minimized) machines.
It is practically important to compute as short sequences as possible. Unfortunately,
the problems of finding the shortest possible homing or synchronizing
sequences are NP-complete, so it is unlikely that no polynomial algorithm exists.
This was proved in Section 1.4.1, and Section 1.3.5 gave exponential algorithms
for both problems. Section 1.4.2 shows that only a small relaxation of the problem
statement gives a PSPACE-complete problem.