Lecture Notes in Computer Science 3472

2 State Identification 57
The following example allows to better understand the preceding proof.
Example. We apply Algorithm 6 on machine M6 (Fig. 2.9). Initially, we start
with the one block partition
π = {{s1, s2, s3, s4, s5, s6}}.
Only a is valid for {s1, s2, s3, s4, s5, s6} (b merges s2 and s6). It refines π to
π = {{s1, s3, s5}, {s2, s4, s6}},
where {s1, s3, s5} corresponds to output 0 and {s2, s4, s6} to 1. Now, {s1, s3, s5}
canberefinedto{s1} and {s3, s5} by use of input symbol b since under b which
is valid for {s1, s3, s5} s1 stays at the same block whereas s3 and s5 move to
the block {s2, s4, s6} of the old partition. Thus, due to the execution of b the
partition π is refined to
π = {{s1}, {s3, s5}, {s2, s4, s6}}.
In a similar way, a refines {s2, s4, s6} to {s2} and {s4, s6} since a is valid for
{s2, s4, s6} and under as2 moves to the block {s1} whereas the states s4 and s6
move to the block {s3, s5} of the old partition. Thus, π is refined to
π = {{s1}, {s3, s5}, {s2}, {s4, s6}}.
Now, b becomes valid for {s3, s5} and it can refine it into {s3} and {s5}. Partition
π becomes
π = {{s1}, {s3}, {s5}, {s2}, {s4, s6}}.
Finally, {s4, s6} can be refined either by a or by b into {s4} and {s6}. Thus,we
end with the discrete partition of the set of states of machine M6 and consequently
we verify that it has an ADS.
A summary of the different steps of the execution of Algorithm 6 on machine
M6 is given in Table 2.3.
i B valid partition π
input of B
0 {s1, s2, s3, s4, s5, s6} a {s1, s3, s5}, {s2, s4, s6} {s1, s3, s5}, {s2, s4, s6}
1 {s1, s3, s5} b {s1}, {s3, s5} {s1}, {s3, s5}, {s2, s4, s6}
2 {s2, s4, s6} a {s2, s4}, {s6} {s1}, {s3, s5}, {s2, s4}, {s6}
3 {s3, s5} a {s3}, {s5} {s1}, {s3}, {s5}, {s2, s4}, {s6}
4 {s2, s4} a {s2}, {s4} {s1}, {s3}, {s5}, {s2}, {s4}, {s6}
Table 2.3. Different steps of the execution of Algorithm 6 on machine M6.
Now from the preceding calculations, we show how a new ADS (Fig. 2.13)
for machine M6 can be constructed. The nodes of this new ADS are numbered
in order to explain how it is computed.