Theory of Computation

1. Theory of Computation

2. Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite states with one initial and some final states 2) Finite set of input letters (Σ) from which input strings are formed 3) Finite set of transitions, showing where to move if a letter is input at certain state, there may be more than one transition for certain letters and there may not be any transition for certain letters.

3. Observations The string is supposed to be accepted, if there exists at least one successful path, otherwise rejected. NFA can be considered to be an intermediate structure between FA and TG.

4. Example The above NFA accepts the language consisting of a and ab. 2+ a a 1- 3 a b 4 5+

5. Example The above NFA accepts the language of strings, defined over Σ = {a, b}, containing aa. a, b a,b a a 1- 3+ 2

6. Non-determinism This refers to situations in which the next state of a computation is not uniquely determined by the current state. In these situations we know only a range of possibilities

7. Converting an FA to an equivalent NFA If a language is accepted by an FA, then there exists a TG accepting that language(Kleene’s theorem). Since, an NFA is also a TG, there exists an NFA accepting the language accepted by the given FA. In this case these FA and NFA are said to be equivalent to each others.

8. Example Consider the following FA corresponding to (a+b)*b The above FA may be equivalent to the following NFA a b b - + a a, b b - +

9. Example Consider the following FA The above FA may be equivalent to the following NFA a,b b a a - + 1 b a,b a, b a a - + 1

10. Note Every FA can be considered to be an NFA, but the converse may not true. Every NFA can be considered to be a TG, but the converse may not true.

11. An NFA can be considered to be a TG, so a RE corresponding to the given NFA can be determined Again using Kleene’s theorem, an FA can be built corresponding to that RE. NFA to FA

12. Example 2 b a 1- 4+ b a 3 a 1- 2 b 4+ a b a a, b 3 b a, b 

13. Nondeterministic Finite Automaton (NFA) Cont… Alphabet =

14. Alphabet = Two choices

15. Alphabet = Two choices No transition No transition

16. First Choice

17. First Choice

18. First Choice

19. First Choice All input is consumed “accept”

20. Second Choice

21. Second Choice

22. Second Choice No transition: the automaton hangs

23. Second Choice Input cannot be consumed “reject”

24. An NFA accepts a string: if there is a computation of the NFA that accepts the string i.e., all the input is consumed and the automaton is in an accepting state

25. Example is accepted by the NFA: “accept” “reject” because this computation accepts

26. Rejection example

27. First Choice

28. First Choice “reject”

29. Second Choice

30. Second Choice

31. Second Choice “reject”

32. Example is rejected by the NFA: “reject” “reject” All possible computations lead to rejection

33. Rejection example

34. First Choice

35. First Choice No transition: the automaton hangs

36. First Choice Input cannot be consumed “reject”

37. Second Choice

38. Second Choice

39. Second Choice No transition: the automaton hangs

40. Second Choice Input cannot be consumed “reject”

41. is rejected by the NFA: “reject” “reject” All possible computations lead to rejection

42. Language accepted:

43. Lambda Transitions

46. (read head does not move)

48. all input is consumed “accept” String is accepted

49. Rejection Example