1. NFA- to NFA conversion

2. Purpose This presentation presents an example execution of the algorithm which takes as input an NFA with -transitions and produces as output an equivalent NFA without - transitions Algorithm Specification –Input: NFA M 1 with  -transitions –Output: NFA M 2 without -transitions

3. Input NFA M 1 with  transitions a a bb a,b 1 2 34 5

4. Compute all  closures a a bb a,b 1 2 34 5 While this step is not necessary, it does make all future steps simpler. Compute  ({i}) for 0 < i < 6  ({1}) = {1,2,3}  ({2}) = {2}  ({3}) = {3}  ({4}) = {4}  ({5}) = {5,1,2,3} = {1,2,3,5}

5. Process State 1 a a bb a,b 1 2 34 5 Compute  * (1,b) First compute  ({1})  ({1}) = {1,2,3} Now compute  ({1}),b)  (1,b) = {}  (2,b) = {4}  (3,b) = {3}  ({1}),b) = {3,4} Finally compute  (  ({1}),a)) =  ({3,4})  ({3}) = {3}  ({4}) = {4}  (  ({1}),b)) = {3,4}  * (1,b) = {3,4} Compute  * (1,a) First compute  ({1})  ({1}) = {1,2,3} Now compute  ({1}),a)  (1,a) = {}  (2,a) = {2}  (3,a) = {4}  ({1}),a) = {2,4} Finally compute  (  ({1}),a)) =  ({2,4})  ({2}) = {2}  ({4}) = {4}  (  ({1}),a)) = {2,4}  * (1,a) = {2,4}

6. Process State 2 a a bb a,b 1 2 34 5 Compute  * (2,b) First compute  ({2})  ({2}) = {2} Now compute  ({2}),b)  (2,b) = {4}  ({2}),b) = {4} Finally compute  (  ({2}),b)) =  ({4})  ({4}) = {4}  (  ({2}),b)) = {4}  * (2,b) = {4} Compute  * (2,a) First compute  ({2})  ({2}) = {2} Now compute  ({2}),a)  (2,a) = {2}  ({2}),a) = {2} Finally compute  (  ({1}),a)) =  ({2})  ({2}) = {2}  (  ({2}),a)) = {2}  * (2,a) = {2}

7. Process State 3 a a bb a,b 1 2 34 5 Compute  * (3,b) First compute  ({3})  ({3}) = {3} Now compute  ({3}),b)  (3,b) = {3}  ({3}),b) = {3} Finally compute  (  ({3}),b)) =  ({3})  ({3}) = {3}  (  ({3}),b)) = {3}  * (3,b) = {3} Compute  * (3,a) First compute  ({3})  ({3}) = {3} Now compute  ({3}),a)  (3,a) = {4}  ({3}),a) = {4} Finally compute  (  ({3}),a)) =  ({4})  ({4}) = {4}  (  ({3}),a)) = {4}  * (3,a) = {4}

8. Process State 4 a a bb a,b 1 2 34 5 Compute  * (4,b) First compute  ({4})  ({4}) = {4} Now compute  ({4}),b)  (4,b) = {4}  ({4}),b) = {5} Finally compute  (  ({4}),b)) =  ({5})  ({5}) = {1,2,3,5}  (  ({4}),b)) = {1,2,3,5}  * (4,b) = {1,2,3,5} Compute  * (4,a) First compute  ({4})  ({4}) = {4} Now compute  ({4}),a)  (4,a) = {5}  ({4}),a) = {5} Finally compute  (  ({4}),a)) =  ({5})  ({5}) = {1,2,3,5}  (  ({4}),a)) = {1,2,3,5}  * (4,a) = {1,2,3,5}

9. Process State 5 a a bb a,b 1 2 34 5 Compute  * (5,b) First compute  ({5})  ({5}) = {1,2,3,5} Now compute  ({5}),b)  (1,b) = {}  (2,b) = {4}  (3,b) = {3}  (5,b) = {}  ({5}),b) = {3,4} Finally compute  (  ({5}),b)) =  ({3,4})  ({3}) = {3}  ({4}) = {4}  (  ({5}),b)) = {3,4}  * (5,b) = {3,4} Compute  * (5,a) First compute  ({5})  ({5}) = {1,2,3,5} Now compute  ({5}),a)  (1,a) = {}  (2,a) = {2}  (3,a) = {4}  (5,a) = {}  ({5}),a) = {2,4} Finally compute  (  ({5}),a)) =  ({2,4})  ({2}) = {2}  ({4}) = {4}  (  ({5}),a)) = {2,4}  * (5,a) = {2,4}

10. Showing All Transitions a a bb a,b 1 2 34 5 a a bb 1 2 34 5 a b b a The initial state of the new NFA is the same as the initial state of the original NFA- .

11. Accepting States a a bb a,b 1 2 34 5 a a bb 1 2 34 5 a b b a First make all original accepting states accepting states: In this case, this means state 5 should be an accepting state Next check to see if  ({1}) includes an accepting state In this case, no. Thus, we do not make the initial state an accepting state