Lecture Notes in Computer Science 3472

56 Moez Krichen
B of π, a is a-valid for B if it is valid for B and there are two states of B which
produce different outputs on a. The input a is b-valid for B with respect to π
if it is valid for B and all the states of B produce the same output on a and
move to the same block of π. Finally, a is c-valid for B w.r.t π if it is valid for
B and it is neither a-valid nor b-valid for B.
Moreover, we define the implication graph of a given machine correspondingtoapartitionπof
the set of states of this machine as the directed graph Gπ
the nodes of which are the blocks of π and such that B a/b
−→B ′ is an arc of Gπ if
a is c-valid for B and each state in B produces b and moves to a state in B ′ on
a.
Finally, we introduce the notion of closed experiments. An experiment T is
said to be closed if the current set of each of its leaves is contained in the initial
set of some (possibly different) leaf.
Proof. (“if” direction) We assume that Algorithm 6 terminates with the discrete
partition and we construct an ADS T for M .First,Tis initialized to the one
node tree. At each step, every node of T is assigned some initial and current
sets. Due to the initialization step, the root of T is assigned S the whole set of
states as both initial and current sets.
• If T is not closed then we choose a leaf u of T the current set of which
intersects the initial sets of more than one leaf. We identify v the lowest
common ancestor (in T ) of all such leaves. The tree T is then updated by
applying on the current set of u the input sequence spelt by the path from
the root of T to v.
• It is when T is closed that the partitioning resulted in by Algorithm 6 is
going to be helpful. In that case, we choose a leaf u of T such that I (u) is
of maximal cardinality in π(T ).
It is not difficult to see that there exists a block B in π(T ) such that
C (u) =B (since T is closed and I (u) is of maximal cardinality). On the implication
graph Gπ(T ), we identify the blocks B1, B2, ··· , Bk of π(T )which
are reachable from B. It is not difficult to see that B1, B2, ··· , Bk are of the
same cardinality as B.
Now, since Algorithm 6 terminates with the discrete partition we deduce
that there exists a step of the execution of the algorithm which splits for the
first time some block Bi. Letabe the valid input symbol used during this
step and τ a possible path in Gπ which is from B to Bi. The string τa is the
input sequence we are going to apply on B the current set of u and update
correspondingly the tree T .
The input a can only be either a-valid or b-valid for Bi and not c-valid
(w.r.t π(T )). It can not be so because all the other blocks Bj �= Bi (the only
possible successors of Bi on c-valid inputs) are not split yet at that step. If
a is a-valid for Bi then the new obtained tree T has necessarily more leaves
than the old one (i.e., π(T ) has been refined). Otherwise, if a is b-valid for
Bi then it is easy to see that the updated tree T is not closed.
Thus, after a finite number of iterations an ADS for the considered machine
is obtained. ⊓⊔