Non-Deterministic Finite Automata

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Non-Deterministic Finite Automata
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Nondeterministic Finite Automaton (NFA)
Alphabet
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Alphabet
Two choices
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Alphabet
Two choices
No transition
No transition
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First Choice
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First Choice
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First Choice
All input is consumed
accept
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Second Choice
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Second Choice
Input cannot be consumed
Automaton Halts
reject
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An NFA accepts a string if there is a
computation of the NFA that accepts the string
i.e., all the input string is processed and the
automaton is in an accepting state
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is accepted by the NFA
accept
reject
because this computation accepts
this computation is ignored
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Rejection example
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First Choice
reject
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Second Choice
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Second Choice
reject
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Another Rejection example
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First Choice
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First Choice
Input cannot be consumed
reject
Automaton halts
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Second Choice
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Second Choice
Input cannot be consumed
Automaton halts
reject
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An NFA rejects a string if there is no
computation of the NFA that accepts the string.
For each computation

• All the input is consumed and the
• automaton is in a non final state

OR

• The input cannot be consumed

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is rejected by the NFA
reject
reject
All possible computations lead to rejection
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is rejected by the NFA
reject
reject
All possible computations lead to rejection
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Language accepted
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Lambda Transitions
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(No Transcript)
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(No Transcript)
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input tape head does not move
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all input is consumed
accept
String is accepted
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Rejection Example
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(No Transcript)
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(read head doesnt move)
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Input cannot be consumed
Automaton halts
reject
String is rejected
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Language accepted
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Another NFA Example
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(No Transcript)
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(No Transcript)
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accept
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Another String
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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accept
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Language accepted
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Another NFA Example
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Language accepted
(redundant state)
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Remarks

• The symbol never appears on the
• input tape

• Simple automata

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

NFA
DFA
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Formal Definition of NFAs

Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Accepting states
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Transition Function
resulting states with following one transition
with symbol
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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Extended Transition Function

Same with but applied on strings
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(No Transcript)
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(No Transcript)
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Special case
for any state
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In general
there is a walk from to with label
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The Language of an NFA

• The language accepted by is
• where
• and there is some

(accepting state)
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NFAs accept the Regular Languages

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Equivalence of Machines

• Definition
• Machine is equivalent to machine
• if

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Example of equivalent machines

NFA
DFA
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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
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Proof
we only need to show
Languages accepted by NFAs
Regular Languages
AND
Languages accepted by NFAs
Regular Languages
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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
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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
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Conversion NFA to DFA

NFA
DFA
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NFA
DFA
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empty set

NFA
DFA
trap state
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NFA
union
DFA
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NFA
union
DFA
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NFA
DFA
trap state
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END OF CONSTRUCTION

NFA
DFA
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General Conversion Procedure

• Input an NFA
• Output an equivalent DFA
• with

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• The NFA has states
• The DFA has states from the power set

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Conversion Procedure Steps

• 1. Initial state of NFA
• Initial state of DFA

step
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Example

NFA
DFA
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step

• 2. For every DFAs state
• compute in the NFA
• add transition to DFA

Union
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Example

NFA
DFA
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• 3. Repeat Step 2 for every state in DFA and
  symbols in alphabet until no more states can be
  added in the DFA

step
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Example

NFA
DFA
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step

• 4. For any DFA state
• if some is accepting state in NFA
• Then,
• is accepting state in DFA

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Example

NFA
DFA
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Lemma

If we convert NFA to DFA then the two
automata are equivalent
Proof
We only need to show
AND
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First we show
We only need to prove
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NFA
Consider
symbols
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symbol
denotes a possible sub-path like
symbol
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We will show that if
NFA
then
DFA
state label
state label
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More generally, we will show that if in
(arbitrary string)
NFA
then
DFA
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Proof by induction on
Induction Basis
NFA
DFA
is true by construction of
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Induction hypothesis
Suppose that the following hold
NFA
DFA
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Induction Step
Then this is true by construction of
NFA
DFA
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Therefore if
NFA
then
DFA
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We have shown
With a similar proof we can show
Therefore
END OF LEMMA PROOF