Is my transition function correct (String matching with finite automata)

Question

I am learning String matching with finite automata from CLRS . I am solving some exercise problems .For the exercise problem 32.3-1 ,

Construct the string-matching automaton for the pattern P = aabab and illustrate its operation on the text string T = aaababaabaababaab.

the following is my transition function ,

states   a   b
 0       1   0
 1       2   0
 2       2   3
 3       4   3
 4       4   5
 5       ?   ?

Is my transition function correct ? And how do i fill the last row ? Any help

Answer 1

I assume you are creating a Finite Automata which accepts a string containing the pattern aabab .

There are two mistakes in your finite automata,

on state 3 and state 4,

For state 3 , if the input is b , you have to go back to state 0 . For example the pattern aabb will force you back to state 0 . Here you have to start all over again from state 0 .

For state 4 , if the input is a , you have to go back to state 2 because you have the pattern aa . For example the pattern aabaa will force you back to state 2 .

The corrected Finite Automaton is given below,

states   a   b
 0       1   0
 1       2   0
 2       2   3
 3       4   0
 4       2   5
 5       5   5

Here 5 is your Accepting state. You will reach this state only when you have found the required pattern in a string. Once a pattern is found no matter what the string remains in the accepting state. Hence for both inputs a and b on state 5 remains on 5 itself.

The transition function is that of an fa accepting a string with sub string ' aabab '. If you are going back to state 1 for a and 0 for b , then the transition function accepts strings ending with the sub string ' aabab '. Given that only state 5 is the accepting state.