I cannot seem to create a DFA for this language

Question

{w∈{a,b}∗|w has baba as a substring} 

I am confused with this. How can I capture the following input for example?

aaababa
abbbaba
ababbba
ababaaaa

It seems there must always be baba in either middle or start or end. I must decide whether to put it always in start or middile or end?

Thanks

Answer 1

By the Myhill-Nerode theorem, the states are:

• Already matched 'baba' (accepting state)
• Immediately after 'bab', and haven't matched 'baba'
• Immediately after 'ba', and haven't matched 'baba'
• Immediately after 'b', and none of the above
• None of the above (start state)

It's essentially the same DFA created by the Knuth-Morris-Pratt search algorithm.