Construct a DFA given the following conditions

Question

Construct a DFA over {0,1}* where the string when converted to binary must be divisible by 2 or 3. Also, the number of 1s in the string must not be divisible by

Answer 1

Let the states be (a, b) where a is in {0,1,2,3,4,5} and b is either 0 or 1. a will record the input number mod 6, and b the parity of 1s in the input. The starting state is (0, 0) and the accepting states will be (0, 1), (2, 1), (3, 1), (4, 1) -- that is, it's divisible by 2 or 3 (ie: it's 0, 2, 3 or 4 mod 6), and has an odd number of 1s.

Then transitions are:

• (a, b) -0-> (2a mod 6, b)
• (a, b) -1-> (2a+1 mod 6, 1-b)

This is a picture of the state machine (circle is initial state, double octagon is an accepting state).

The state machine diagram is generated in dot format using this python program:

def state(a, b):
    return 'q%d%d' % (a, b)

print('digraph g {')
print('  node [shape=plaintext];')
print('  %s [shape=circle]' % state(0, 0))
for i in (0, 2, 3, 4):
    print('  %s [shape=doubleoctagon]' % state(i, 1))
for a in range(6):
    for b in range(2):
        for x in range(2):
            s0 = state(a, b)
            s1 = state((2*a+x) % 6, (b+x) % 2)
            print('  %s -> %s [label=%s]' % (s0, s1, str(x)))
print('}')

And then run the output through the dot command. On linux that can be something like this: python fsm.py > g.dot && dot g.dot -Tpng -o g.png (assuming the above code is saved as fsm.py).