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1 Non Deterministic Automata

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2 Alphabet = Nondeterministic Finite Accepter (NFA)

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3 Two choices Alphabet = Nondeterministic Finite Accepter (NFA)

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4 No transition Two choices No transition Alphabet = Nondeterministic Finite Accepter (NFA)

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5 First Choice

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8 “accept” First Choice

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9 Second Choice

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10 Second Choice

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11 Second Choice No transition: the automaton hangs

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12 Second Choice “reject”

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13 Observation An NFA accepts a string if there is a computation of the NFA that accepts the string

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14 Example is accepted by the NFA:

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15 Lambda Transitions

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18 (read head doesn’t move)

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20 “accept” String is accepted

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21 Language accepted:

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22 Another NFA Example

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26 “accept”

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27 Another String

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34 “accept”

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35 Language accepted

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36 Another NFA Example

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37 Language accepted

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38 Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Final states

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39 Transition Function

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43 Extended Transition Function

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46 Formally It holds if and only if there is a walk from to with label

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47 The Language of an NFA

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52 Formally The language accepted by NFA is: where and there is some (final state)

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54 Equivalence of NFAs and DFAs

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55 Equivalence of Machines For DFAs or NFAs: Machine is equivalent to machine if

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56 Example NFA DFA

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57 Since machines and are equivalent DFA NFA

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58 Equivalence of NFAs and DFAs Question: NFAs = DFAs ? Same power? Accept the same languages?

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59 Equivalence of NFAs and DFAs Question: NFAs = DFAs ? Same power? Accept the same languages? YES!

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60 We will prove: Languages accepted by NFAs Languages accepted by DFAs

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61 We will prove: Languages accepted by NFAs Languages accepted by DFAs NFAs and DFAs have the same computation power

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62 Languages accepted by NFAs Languages accepted by DFAs Step 1

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63 Languages accepted by NFAs Languages accepted by DFAs Step 1 Proof: Every DFA is also an NFA

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64 Languages accepted by NFAs Languages accepted by DFAs Step 1 Proof: Every DFA is also an NFA A language accepted by a DFA is also accepted by an NFA

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65 Languages accepted by NFAs Languages accepted by DFAs Step 2

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66 Languages accepted by NFAs Languages accepted by DFAs Step 2 Proof: Any NFA can be converted to an equivalent DFA

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67 Languages accepted by NFAs Languages accepted by DFAs Step 2 Proof: Any NFA can be converted to an equivalent DFA A language accepted by an NFA is also accepted by a DFA

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68 NFA to DFA NFA DFA

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69 NFA to DFA NFA DFA

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70 NFA to DFA NFA DFA

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71 NFA to DFA NFA DFA

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72 NFA to DFA NFA DFA

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73 NFA to DFA NFA DFA

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74 NFA to DFA NFA DFA

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75 NFA to DFA: Remarks We are given an NFA We want to convert it to an equivalent DFA With

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76 If the NFA has states the DFA has states in the powerset

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77 Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA:

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78 Example NFA DFA

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79 Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add transition

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80 Exampe NFA DFA

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81 Procedure NFA to DFA Repeat Step 2 for all letters in alphabet, until no more transitions can be added.

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82 Example NFA DFA

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83 Procedure NFA to DFA 3. For any DFA state If some is a final state in the NFA Then, is a final state in the DFA

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84 Example NFA DFA

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85 Theorem Take NFA Apply procedure to obtain DFA Then and are equivalent :

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86 Finally We have proven Languages accepted by NFAs Languages accepted by DFAs

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87 Languages accepted by NFAs Languages accepted by DFAs We have proven Regular Languages

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88 Languages accepted by NFAs Languages accepted by DFAs We have proven Regular Languages

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