Lecture Notes in Computer Science 3472

42 Moez Krichen
∀ s, s ′ ∈ S : s �= s ′ ⇒ λ(s, x ) �= λ(s ′ , x ).
i.e., for any two distinct initial states in S, M produces distinct output sequences
after executing x . ⊓⊔
So, in practice, if we want to check if x is a PDS of M then we need to
compute the list of responses (output sequences) M may produce on x for all
its possible initial states. If an output sequence is observed more than once (the
same output sequence for distinct initial states) then x is not an PDS of M .
Equivalently, we deduce that x is a PDS of M if and only if all the responses
are distinct from each other.
a/1
s3
b/1
a/0
s1
b/1
Fig. 2.6. Machine M5.
Example. Table 2.1 gives the responses of the machine M5 (Fig. 2.6) on ab. The
obtained responses ({01, 00, 11}) are distinct from each other, so ab is a PDS of
M5.
s λ(s, ab)
s1 01
s2 00
11
s3
Table 2.1. Possible responses of machine M5 for the input sequence ab.
Remark 2.2 (A PDS is a homing sequence). Since we are dealing with deterministic
machines, when the initial state of the machine is known, it is easy to
b/0
s2
a/0