Lecture Notes in Computer Science 3472

2 State Identification 55
of the considered input symbol: two states of this block will be in the same
subblock if they produce the same output symbol and move to states which are
in the same block of this partition. When no more refinement is possible, the
algorithm terminates.
In this algorithm, Partition(B,π ′ ) denotes the partition of B grouping states
which produce the same output and move to the same block of π ′ under all
possible inputs.
Algorithm 6 Checking whether an machine M has an ADS([LY94]).
• Initialization: π := {S}
• Iterations: while ∃ B ∈π, a ∈I valid input for B and s1, s2 ∈B such that λ(s1, a) �=
λ(s2, a) orδ(s1, a) andδ(s2, a) not in the same block from π
(1) π ′ := π
(2) π := π/B (we omit B from π)
(3) π := π ∪ Partition(B,π ′ )
Depending on whether the last obtained partition is the discrete one or not,
the conclusion about the existence of an ADS for the considered machine is given
by Theorem 2.17.
Theorem 2.17. ([LY94]) A given machine M has an ADS if and only if Algorithm
6 applied on it ends with the discrete partition of the set of states of
M.
We sketch a proof for this theorem. The proof is split into two parts. We
start with the “only if” direction.
Proof. (“only if” direction) We do it by contradiction: we assume that M has an
ADS T and that the partition π with which Algorithm 6 terminates is not the
discrete one. We consider a block B in π with maximal cardinality (i.e., |B| � 2).
We assume that the initial state of the machine is in B and we try to identify
it by executing T . By induction on the length of the experiment, it is easy to
prove that for the current node u of T there exists a block of states Bu such
that
|Bu| = |B| and Bu ⊆ C (u).
In particular, this remains true when the distinguishing experiment T terminates.
Thus, there exists a leaf of T the current set of which contains more than
one state. So, we come to a contradiction with the fact that T is an ADS (by
Proposition 2.15). ⊓⊔
For proving the “if” direction, we need to introduce some extra definitions.
Given a partition π of the set of states of the considered machine, we distinguish
between three types of valid inputs: a-valid, b-valid and c-valid. For some block