Theory of Computation Nondeterministic Finite Automaton NFA An

Theory of Computation

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. 2

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. 3

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

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

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

Converting an FA to an equivalent NFA n If a language is accepted by an FA, then there exists a TG accepting that language(Kleene’s theorem). n 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. 7

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

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

Note n n 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. 10

NFA to FA 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. 11

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

Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13

Alphabet = Two choices 14

Alphabet = Two choices No transition 15

First Choice 16

First Choice 17

First Choice 18

First Choice All input is consumed “accept” 19

Second Choice 20

Second Choice 21

Second Choice No transition: the automaton hangs 22

Second Choice Input cannot be consumed “reject” 23

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 24

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

Rejection example 26

First Choice 27

First Choice “reject” 28

Second Choice 29

Second Choice 30

Second Choice “reject” 31

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

Rejection example 33

First Choice 34

First Choice No transition: the automaton hangs 35

First Choice Input cannot be consumed “reject” 36

Second Choice 37

Second Choice 38

Second Choice No transition: the automaton hangs 39

Second Choice Input cannot be consumed “reject” 40

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

Language accepted: 42

Lambda Transitions 43

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(read head does not move) 46

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all input is consumed “accept” String is accepted 48

Rejection Example 49

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(read head doesn’t move) 51

No transition: the automaton hangs 52

Input cannot be consumed “reject” String is rejected 53

Language accepted: 54

Another NFA Example 55

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“accept” 59

Another String 60

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“accept” 67

Language accepted 68

Another NFA Example 69

Language accepted (redundant state) 70

Remarks: • The symbol never appears on the input state • Simple automata: 71

• NFAs are interesting because we can express languages easier than DFAs NFA DFA 72

Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e. Transition function Initial state Accepting states 73

Transition Function 74

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Extended Transition Function 78

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We will prove: Languages accepted by NFAs Regular Languages accepted by DFAs 81

NFA Convert NFA to DFA 82

Convert NFA to DFA NFA DFA 83

Convert NFA to DFA NFA n DFA 84

Convert NFA to DFA NFA n DFA 85

Convert NFA to DFA NFA n DFA 86

Convert NFA to DFA NFA n DFA 87

Convert NFA to DFA NFA n DFA 88

Thank You! 89