MELJUN CORTES Automata Lecture Nondeterministic Finite Automata 2

8/21/2019 MELJUN CORTES Automata Lecture Nondeterministic Finite Automata 2

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Theory of Computation (With Automata Theory)

* Property of STI

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Nondeterministic Finite Automata

Nondeterministic Finite

Automata

Nondeterminism

Introduction

• The main characteristic of a

DFA is that for each input

symbol, the automaton can

move to only one state from

any given current state.

• Consider a DFA that accepts

strings over the alphabet =

{0, 1} that starts with a 0 and

ends with a 1.

0 1q 0

0 1

q 2q 1

0

q 3

0, 1

1


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• Consider now the following

finite automaton:

This machine also accepts

strings that start with a 0 and

end with 1.

Observe that if the machine is

at state q1 and a 1 arrives, the

machine goes to state q2 and

stays at q1 at the same time.

This machine is not a DFA

because there is a situation in

which an input symbol causes

the automaton to move to

more than one state.

0 1q 0

0, 1

q 2q 1


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• The given graph is the state

diagram for a

nondeterm in is t ic f in i te

automaton (NFA).

• Because an NFA can move to

more than one state given a

single input symbol, an NFA issaid to have “choices.”

Take the given NFA as an

example. If the NFA iscurrently in state q1 and an

input symbol 1 arrives, the

NFA has the option of staying

in state q1 or going to state q2.


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In this case, the NFA will

consider both possibilities.

The NFA will make a “copy” of

itself. One copy continues the

computation at state q1 while

the other continues at state q2.

Effectively, the NFA is said to

be in two states, q1 and q2.

This is in contrast with a DFA

since DFAs can only be in one

state at any given time.


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• Notice also that in an NFA, it ispossible for a state not to have

a transition edge for one or

more input symbols.

In the given example, takenote that for state q0, there is a

transition edge for input

symbol 0 but none for 1.

If a copy of the NFA is at stateq0 and an input symbol 1

arrives, this copy stops

processing because it has

nowhere to go. This copy of

the NFA simply “dies.”

Similarly, if a copy of the NFA

is at q2 and a 1 or a 0 arrives,

this copy also stops

processing and dies.


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• Here is an example to illustrate

how an NFA processesstrings:

Given the following NFA:

Determine if the string 01011

will be accepted.

1. Initially, the NFA will be at

state q0. When the first

input symbol (0) arrives,

the NFA goes to state q1as shown below:

0 1q 0

0, 1

q 2q 1

q 0

q 1

0

input state

start


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2. Assume now that the

second input symbol (1)arrives. The NFA has two

options, either it stays at q1or goes to q2. The

machine will try both

possibilities at the sametime. This is shown below:

The machine made

another copy of itself. One

copy to continue the

processing at q1 and the

other at q2.

q 0

q 1

0

input state

(start)

q 1

1

q 2


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3. Assume now that the third

input symbol (0) arrives.

One copy of the NFA (the

one at q1) stays at q1. The

other copy (the one at q2)

dies because it has

nowhere to go.

At this point in time, there

will only be one remaining

copy of the NFA (the one

that is still at q1).

q 0

q 1

0

input state

(start)

q 1

1

q 2

q 1

(stopped)

0


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4. Assume now that the

fourth input symbol (1)arrives. There are again

two options, either stay at

q1 or go to q2. The NFA

will again consider both

possibilities.

q 0

q 1

0

input state

(start)

q 1

1

q 2

q 1

(stopped)

0

q 1 q 2

1


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5. Assume now that the fifth

and last input symbol (1)arrives. One copy of the

NFA (the one at q1) has

the option of staying at q1or going to q2. As usual,

the NFA will consider both.The other copy (the one at

q2) stops processing.

q 0

q 1

0

input state

(start)

q 1

1

q 2

q 1

(stopped)

0

q 1 q 2

1

q 1 q 2

(stopped)

1


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There are two copies of theNFA. One copy is currently at

state q1 and the other at state

q2. Since one of the copies is

at a final state, the NFA

accepts the string 01011.

If there is no copy of the NFA

that is in a final state, then the

string will be rejected.

• Another example:



will be accepted.

1 0, 1q 0

0, 1

q 1 q 3q 2

0, 1


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the NFA has two options,

either to stay at q0

or go to

q1. The machine will try

both possibilities.

There will now be two

copies of the NFA. One

copy is at state q0 while the

other is at state q1.

q 0

1

input state

(start)

q 1q 0


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2. When the second input

symbol (1) arrives, thecopy of the NFA which is at

state q0 has two options—

stay at q0 or go to q1. The

machine will try both

possibilities. The copywhich is at state q1 will go

to q2.

There will now be three

copies of the NFA. One is

at state q0, the second one

is at state q1, and the third

is at state q2.

q 0

1

input state

(start)

1

q 0 q 1 q 2

q 1q 0


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3. When the third input

symbol (0) arrives, the

copy of the NFA which is at

state q0 stays at q0, the

copy which is at state q1goes to q2, and the copy

which is at q2 goes to state

q3.

q 0

1

input state

(start)

1

0

q 2q 0 q 3

q 0 q 1 q 2

q 1q 0


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4. If the fourth input symbol

(1) arrives, the copy of theNFA which is at state q0has two options. Stay at q0or go to q1. The machine

will try both possibilities.

The copy which is at stateq2 will go to q3. The copy

which is at state q3 stops

computing because it has

nowhere to go.

q 0

1

input state

(start)

1

0

1

q 1q 0 q 3

q 2q 0 q 3 (stopped)

q 0 q 1 q 2

q 1q 0


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5. When the fifth input symbol

(1) arrives, the copy of theNFA which is at state q0has two options—stay at q0or go to q1. The machine

will try both possibilities.

The copy which is at stateq1 will go to q2. The copy

which is at q3 dies.

q 0

1

input state

(start)

1

0

1

(stopped)

1

q 0 q 1 q 2

q 1q 0 q 3


q 0 q 1 q 2

q 1q 0


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6. When the sixth and final

input symbol (0) arrives,the copy of the NFA which

is at state q0 stays at q0.

The copy at state q1 goes

to q2. The copy at state q2

goes to q3.

q 0

1

input state

(start)

1

0

1

q 0 q 2

(stopped)

1

q 3

q 0 q 1 q 2

q 1q 0 q 3


q 0 q 1 q 2

q 1q 0

0


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The NFA is now at threestates, q0, q2, and q3. Since

one of these states is a final

state (q3), then the NFA

accepts the input string

110110.

• ε -Transitions

One other difference between

an NFA and a DFA is that an

NFA may have ε -transitions.

An ε -transition is shown below:

e

q jq i


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Whenever there is an ε -

transition from state qi to stateq j, this means that once an

NFA goes to state qi, it has the

option of staying at qi or it can

go right away to state q j even

without any additional inputarriving.

Example:



will be accepted.

0e

q 0

0

q 2q 1

1


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the NFA goes to q1. At q1, it

has the option ofimmediately going to q2without any new input

symbol arriving (because

of the ε -transition from q1to q2) . The NFA willconsider both and will be in

two states, q1 and q2.

q 0

q 1

0

input state

(start)

q 2


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2. When the second input

symbol (1) arrives, the

copy of the NFA which is at

state q1 stops computing

since it has nowhere to go.The copy at q2 will go to q0.

The NFA will only be in

one state which is q0.

q 0

q 1

0

input state

(start)

q 0

1

q 2(stopped)


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3. When the third inputsymbol (0) arrives, the

NFA goes to q1. Once at

q1, it has the option of

immediately going to q2

without any new inputsymbol arriving. The NFA

will consider both and will

be in two states, q1 and q2.

q 0

q 1

0

input state

(start)

q 0

1

q 2(stopped)

q 1 q 2

0


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4. When the fourth input

symbol (0) arrives, thecopy of the NFA which is at

state q1 stays at q1. Since

there is an ε -transition from

q1 to q2, it will also

immediately go to q2. Theother copy of the NFA (the

one at state q2) stops

because it has nowhere to

go.

q 0

q 1

0

input state

(start)

q 0

1

q 2(stopped)

q 1 q 2

0

(stopped)

q 1 q 2

0


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5. When the fifth input symbol(1) arrives, the copy of the

NFA which is at state q1stops since it has nowhere

to go. The remaining copy

of the NFA (the one atstate q2) goes to q0.

q 0

q 1

0

input state

(start)

q 0

1

q 2(stopped)

q 1 q 2

0

(stopped)

q 1 q 2

q 0

0

1

(stopped)


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6. When the sixth and last


the NFA goes to q1. At q1,

it has the option ofimmediately going to q2without any new input

symbol arriving (because

of the ε -transition from q1

to q2).

The NFA will then be at

state q1

and state q2

.

Since one of these states

is a final state (q1), the

string 010010 is accepted.


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q 0

q 1

0

input state

(start)

q 0

1

q 2(stopped)

q 1 q 2

0

(stopped)

q 1 q 2

q 0

0

1

q 1 q 2

0

(stopped)


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• To summarize the differences

between an NFA and a DFA:

1. NFAs can move to more

than one state from anygiven current state. They

can therefore be in several

states at the same time.

2. In a given state, NFAs may

not necessarily have a

transition edge for each

symbol in the alphabet.

3. NFAs can move to a state

even without any input

symbol arriving (ε-

transitions).


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Formal Definition:

• The formal definition of a NFA

is similar to that of a DFA.

• The main difference lies in the

transition function.

• The transition function of anNFA must consider the

possibility of state transitions

even if there is no input

symbol (ε -transitions).

• The transition function must

also consider the possibility

that the NFA may go to

several states given the same

input symbol.


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• For example:

The formal definition of this

NFA is a 5-tuple N 1 = {Q, , ,

q0, F } where:

1. Q = {q0, q1, q2}2. = {0, 1}

3. :

4. Start State = q0.

5. F = {q2}

0 1 ε

q0 q1 -- --

q1 q1 q1, q2 --

q2 -- -- --

0 1q 0

0, 1

q 2q 1


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• Another example:

The formal definition of this

NFA is a 5-tuple N 2 = {Q, , ,

q0, F } where:

1. Q = {q0, q1, q2}

2. = {0, 1}

3. :

4. Start State = q0

5. F = {q1}

0 1 ε

q0 q1 -- --

q1 q1 -- q2

q2 -- q0 --

0e

q 0

0

q 2q 1

1


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More exercises:

• NFA N 1:

• NFA N 2:

10 1

0

q 2q 1q 0

e

1q 0

0, 1

q 2q 1

0, 1

1


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More exercises:

• NFA N 3:

• NFA N 4:

1

q 1

e

q 3

q 0

q 4 q 50 1

0

q 2

1 0

10 e

q 0

0, 1

q 2 q 3q 1

0, 1

1


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• In an NFA, a state may have

zero, one, or many outgoing

edges for each input symbol.

DFAs can only have one

outgoing edge for each inputsymbol.

This means that a DFA is a

restricted type of NFA. It can

therefore be said that all DFAs

are NFAs.

However, not all NFAs are

DFAs.

In other words,

nondeterminism is a

generalization of determinism.

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MELJUN CORTES Automata Lecture Nondeterministic Finite Automata 2