Lecture Notes in Computer Science 3472

21 Finite State Machines 613
• λ(s, a) =λ(t, a) for each input symbol a ∈ I ,and
• δ(s, a) andδ(t, a) are in the same block.
It is straight-forward to check that s and t are in the same block if and only if
they are equivalent.
The straight-forward solution to finding this coarsest partition is to start
with an initial partition of S, where two states s and t are in the same block if
and only if λ(s, a) =λ(t, a) for each input symbol a ∈ I . Thereafter, this initial
partition is repeatedly refined by the following method. Take a block Bi of the
current partition. Examine δ(s, a) foreachs∈Biand a ∈ I . Partition Bi so
that s and t are in the same block if and only if δ(s, a) andδ(t, a) areinthe
same block of the current partition. The process is is iterated until no further
refinements are possible. This process yields a quadratic algorithm, since each
refinement requires linear time, and there can be at most |S| refinements. A
more efficient algorithm has been developed by Hopcroft ([Hop71] and [AHU74,
Sec. 4.13]).
21.3 Initial and Current State Uncertainty
Chapters 1 – 4 use some basic definitions of the conclusions that can be made
from observations of a machine’s response to input strings.
The initial state uncertainty describes what we know about the initial state
after applying an input string and observing the resulting output string.
Definition 21.2. The initial state uncertainty after applying input sequence
x ∈ I ∗ is a partition π(x ) def
= {B1, B2,...,Br } of S, such that two states s, t are
in the same block Bi if and only if λ(s, x )=λ(t, x ).
In other words, each block in the initial state uncertainty π(x ) is the set of states
from which a particular output sequence is produced in response to the input
sequence x .
The current state uncertainty describes what we know about the current
(final) state after having applied an input string and observed the resulting
output string.
Definition 21.3. The current state uncertainty after applying input sequence
x ∈ I ∗ is the set of subsets σ(x ) def
= {δ(Bi, x ):Bi ∈ π(x )} of S.
In other words, each block in the current state uncertainty σ(x )isthesetofstates
that the machine can end up in after generating a particular output sequence
in response to the input sequence x . While the initial state uncertainty is a
partition, the current state uncertainty does not need to be a partition, but just
a set of nonempty blocks
21.4 Distinguishing Experiments
Chapters 1 – 4 present algorithms that investigate the structure and current
state of a Mealy machine by performing experiments, i.e., applying a sequence