Lecture Notes in Computer Science 3472

36 Moez Krichen
b/1
a/0
s1
b/1
a/1
s2
Fig. 2.1. Machine M1.
many versions, depending on which types of experiments are allowed. A first
classification can be made between simple experiments and multiple experiments
[Moo56, Gil61]. In a simple experiment we dispose of a single copy of
the machine whereas in a multiple experiment we dispose of multiple identical
copies of the machine which occupy the same initial (unknown) state. We will
only consider simple experiments in this chapter called distinguishing sequences.
Another classification can be made between preset and adaptive distinguishing
sequences. A preset distinguishing sequence (PDS) is a sequence of inputs
whereas an adaptive distinguishing sequence (ADS) is really a tree where inputs
may be different depending on the outputs (thus, the term “sequence” can be
considered a misnomer, but it is kept for reasons of tradition). Thus, a PDS
models an experiment where the sequence of inputs is determined in advance,
independently of the responses of the machine. On the other hand, an ADS models
an experiment where at each step the experimenter decides which input to
apply next based on the observed output. Clearly, an ADS is a more powerful
experiment than a PDS. Indeed, as we will see, there are cases where a machine
has an ADS but has no PDS. Thus, it makes sense to make a distinction between
the two.
This brings us to another difference between the problem of finding a homing
sequence and the state identification problem. We know that for minimal
machines a homing sequence always exists. Such a sequence is, by definition,
a “preset homing sequence”. Indeed, there is no need to consider adaptive sequences
since they do not add to the power of the experimenter. On the other
hand, a minimal machine may not have a PDS; in fact, it may not have an ADS
either. Thus, contrary to the homing problem, there are machines for which the
state identification problem cannot be solved, even though these machines are
minimal.
As a last difference between the two problems, notice the following. When
a machine is not minimal, we are not sure whether it has a homing sequence