Lecture Notes in Computer Science 3472

2 State Identification 39
PDS either, because it “merges” s1 and s2. Indeed, let k be the number of a’s
applied before the first b. Ifk is even and the automaton was initially in s1 or s2,
it will still be in s1 or s2 after applying a k , and no information is gained (notice
that, since the sequence is preset, we are not allowed to look at the output at
that point). Then, when b is applied, 0 will be output and M4 will move to s1. At
this point, there is no longer hope of knowing whether the machine was initially
in s1 or s2. Ifk is odd and the automaton was initially in s3 or s4 then, after
applying a k , we still end up at s1 or s2 and the previous argument again applies.
Thus, we can conclude that M4 has no PDS.
However, M4 has an ADS. One such ADS is shown in Fig. 2.5. It tells us to
proceed as follows, when executing the test. Initially, apply a. Ifthemachine
outputs 0 then apply b, otherwise, apply a second a, followed by b. The rationale
is as follows. If the machine outputs 0 after the initial a, then we can deduce
that it was initially at state s1 or s2 and it has moved to s3 or s4, respectively.
We can then distinguish s3 and s4 simply by applying b. If, on the other hand,
the machine output 1 after the initial a, then we can deduce that it was initially
at s3 or s4 and has now moved to s1 or s2. In this case, we apply sequence ab to
distinguish the two latter states.
s1
0
b
a
0 1
1
s2
Fig. 2.5. One possible ADS for machine M4.
History and results There is a lot of literature on the state identification problem
and the related homing sequence and state verification problems (Chapters 1
and 3). The earliest work in this field is the seminal 1956 paper by Edward F.
Moore [Moo56]. Then in the 60’s, many papers on these problems followed this
work. These papers were mainly motivated by automata theory and switching
circuits. An overview of the major results of these earlier works is in [Koh78]
by Zvi Kohavi. Later on, works were also motivated by communication protocol
design. Before the work of Lee and Yannakakis [LY94], the proposed algorithms
for solving these problems take exponential time and there was no algorithm for
computing adaptive distinguishing sequences with polynomial length. The algorithms
proposed in these works had exponential time worst-case complexity and
s3
a
0
0
b
1
s4