Question: Find a finite automaton that accepts bit strings whose last five bits include a 1.
Answer: This is the set of strings that do not end with 00000 and are not the empty string, 0, 00, 000, or 0000. Here is the transition table with a start state of A.
input
0 1
A B a [we have seen nothing so far]
B C a [we have seen 0 so far]
C D a [we have seen 00 so far]
D E a [we have seen 000 so far]
E f a [we have seen 0000 so far]
a b a [we just saw a 1]
b c a [we saw a 1 one bit ago but not since]
c d a [we saw a 1 two bits ago but not since]
d e a [we saw a 1 three bits ago but not since]
e f a [we saw a 1 four bits ago but not since]
f f a [we have seen five zeroes in the last four characters]
The accepting states are a, b, c, d, and e.