Lecture Notes in Computer Science 3472

• An input sequence x is a valid input sequence for C if:
2 State Identification 45
∀ s, s ′ ∈ C : s �= s ′ ⇒ λ(s, x ) �= λ(s ′ , x )orδ(s, x ) �= δ(s ′ , x ).
• a (resp., x ) is a valid input (resp., valid input sequence) for a collection of
sets of states τ if a (resp., x )isvalidforeachmemberofτ.
⊓⊔
Let us illustrate this notion of validity on machine M5 (Fig. 2.6). It has been
already shown that ab is a PDS of this machine. When executing this input
sequence on M5, the current uncertainty evolves as follows
{{s1, s2, s3}} a
−→{{s1, s2}0, {s3}1} b
−→{{s3}01, {s1}00, {s2}11}.
First, it is easy to check that both a and ab are valid for {s1, s2, s3} and that b
is valid for {s1, s2}. More precisely, b is valid for {{s1, s2}, {s3}}, but since {s3}
is singleton then that it is equivalent to say that it is valid for {s1, s2}.
The second thing to notice is that the total number of states contained in
each of the three uncertainties of the example above equals 3 the number of
states of the machine. In particular, the last uncertainty is made of as many
singletons as the number of states of the machine.
Proposition 2.6, below, argues that these observations remain true in the
general case. The proposition uses the following notation:
• Super(M) is the set of multisets of non-empty blocks of states such that
W = {B1, B2, ··· , Bl} is in Super(M ) if and only if � Bi ∈W |Bi| = |S|.
• Singleton(M) is the subset of Super(M ) the members of which are sets of
singletons.
• For a given multiset of blocks τ and a given input symbol a, µ(τ,a) is
obtained by partitioning each member C of τ w.r.t λ(·, a), then applying
δ(·, a) on each block of the obtained partition.
and
For instance, if S = {s1, s2} then:
and
Super(M )=[{{s1}, {s2}}; {{s1}, {s1}}; {{s2}, {s2}}; {{s1, s2}}].
Singleton(M )=[{{s1}, {s2}}; {{s1}, {s1}}; {{s2}, {s2}}] 1 .
Moreover, for machine M5 we have:
µ({{s1, s2, s3}}, a) ={{s1, s2}0, {s3}1}
µ({{s1, s2}0, {s3}1}, b) ={{s3}01, {s1}00, {s2}11}.
1
Here, we use the notation [C1; C2] instead of the classical notation {C1, C2} for
making things more readable.