Deterministic Finite Automata pattern

Question

I'm trying to solve this problem using Deterministic Finite Automata :

inputs:     {a,b}
conditions: 
a. must have exactly 2 a  
b. have more than 2 b

so a correct input should be like this abbba or bbbaa or babab

now my question is, "is there a pattern to solve this things?"

Answer 1

Yes there is a pattern. You can take each statement and deduct pre-states from them. Then you take the cross-product of those pre-states, which will comprise the final states. In this example:

a. will yield states: 0a, 1a, 2a, 2+a (you've seen 0 a, 1 a, 2 as or more than 2 as) b. will yield states: 0b, 1b, 2b, 2+b (you've seen 0 b, 1 b, 2 bs or more than 2 bs)

The cross product of these states result in 4x4=16 states. You'll start from {0a,0b} states. The inputs can be 3 types: a, b or something else. From that you should be able to go. Do you need more help?

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Answer 2

Always draw such things first.

Feel free to give states any meanings. What you need here is states like: q2: (1 b, 2 a's). Draw states like this, until the accept state and connect them with lines. The accept state is qx: 2 a's 3 b's.

After reaching the accept state, if input is "b" that line goes to itself, the accept state. If the input is "a", draw a new state, that will get into an endless loop and goes into itself no matter what the input is.

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