Homing and synchronizing sequences

Abstract

We considered two fundamental and closely related testing problems for Mealy machines. In both cases, we look for a sequence of input symbols to apply to the machine so that the final state becomes known. A synchronizing sequence takes the machine to one and the same state no matter what the initial state was. A homing sequence produces output, so that one can learn the final state by looking at this output. These problems can be completely solved in polynomial time. Homing sequences always exist if the machine is minimized. They have at most quadratic length and can be computed in cubic time, using the algorithm 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. © Springer-Verlag Berlin Heidelberg 2005.