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NFAs accept the Regular Languages

NFAs accept the Regular Languages. Equivalence of Machines. Definition: Machine is equivalent to machine if. Example of equivalent machines. NFA. DFA. Theorem:. Languages accepted by NFAs. Regular Languages. Languages accepted by DFAs.

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NFAs accept the Regular Languages

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  1. NFAs accept the Regular Languages Prof. Busch - LSU

  2. Equivalence of Machines Definition: Machine is equivalent to machine if Prof. Busch - LSU

  3. Example of equivalent machines NFA DFA Prof. Busch - LSU

  4. Theorem: Languages accepted by NFAs Regular Languages Languages accepted by DFAs NFAs and DFAs have the same computation power, accept the same set of languages Prof. Busch - LSU

  5. Proof: we only need to show Languages accepted by NFAs Regular Languages AND Languages accepted by NFAs Regular Languages Prof. Busch - LSU

  6. Proof-Step 1 Languages accepted by NFAs Regular Languages Every DFA is trivially an NFA Any language accepted by a DFA is also accepted by an NFA Prof. Busch - LSU

  7. Proof-Step 2 Languages accepted by NFAs Regular Languages Any NFA can be converted to an equivalent DFA Any language accepted by an NFA is also accepted by a DFA Prof. Busch - LSU

  8. Conversion NFA to DFA NFA DFA Prof. Busch - LSU

  9. NFA DFA Prof. Busch - LSU

  10. empty set NFA DFA trap state Prof. Busch - LSU

  11. NFA union DFA Prof. Busch - LSU

  12. NFA union DFA Prof. Busch - LSU

  13. NFA DFA trap state Prof. Busch - LSU

  14. END OF CONSTRUCTION NFA DFA Prof. Busch - LSU

  15. General Conversion Procedure Input: an NFA Output: an equivalent DFA with Prof. Busch - LSU

  16. The NFA has states The DFA has states from the power set Prof. Busch - LSU

  17. Conversion Procedure Steps 1. Initial state of NFA: Initial state of DFA: step Prof. Busch - LSU

  18. Example NFA DFA Prof. Busch - LSU

  19. step 2. For every DFA’s state compute in the NFA add transition to DFA Union Prof. Busch - LSU

  20. Example NFA DFA Prof. Busch - LSU

  21. 3. Repeat Step 2 for every state in DFA and symbols in alphabet until no more states can be added in the DFA step Prof. Busch - LSU

  22. Example NFA DFA Prof. Busch - LSU

  23. step 4. For any DFA state if some is accepting state in NFA Then, is accepting state in DFA Prof. Busch - LSU

  24. Example NFA DFA Prof. Busch - LSU

  25. Lemma: If we convert NFA to DFA then the two automata are equivalent: Proof: We only need to show: AND Prof. Busch - LSU

  26. First we show: We only need to prove: Prof. Busch - LSU

  27. NFA Consider symbols Prof. Busch - LSU

  28. symbol denotes a possible sub-path like symbol Prof. Busch - LSU

  29. We will show that if NFA then DFA state label state label Prof. Busch - LSU

  30. More generally, we will show that if in : (arbitrary string) NFA then DFA Prof. Busch - LSU

  31. Proof by induction on Induction Basis: NFA DFA is true by construction of Prof. Busch - LSU

  32. Induction hypothesis: Suppose that the following hold NFA DFA Prof. Busch - LSU

  33. Induction Step: Then this is true by construction of NFA DFA Prof. Busch - LSU

  34. Therefore if NFA then DFA Prof. Busch - LSU

  35. We have shown: With a similar proof we can show: Therefore: END OF LEMMA PROOF Prof. Busch - LSU

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