Theory of Computation Nondeterministic Finite Automaton NFA An

Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite

Observations The string is supposed to be accepted, if there exists at least one

Example a 1 - a 2+ 3 a 4 b 5+ The above NFA

Example a, b 1 - a, b a 2 a 3+ The above NFA

Non-determinism This refers to situations in which the next state of a computation is

Converting an FA to an equivalent NFA n If a language is accepted by

Example n Consider the following FA corresponding to (a+b)*b a b b n +

Example n Consider the following FA b a n a, b b a 1

Note n n Every FA can be considered to be an NFA, but the

NFA to FA An NFA can be considered to be a TG, so a

Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13

First Choice All input is consumed “accept” 19

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

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

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

First Choice No transition: the automaton hangs 35

First Choice Input cannot be consumed “reject” 36

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

all input is consumed “accept” String is accepted 48

Input cannot be consumed “reject” String is rejected 53

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

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

Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e.

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

Slides: 89

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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