Non Deterministic Automata

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Non Deterministic Automata
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Nondeterministic Finite Accepter (NFA)
Alphabet
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Nondeterministic Finite Accepter (NFA)
Alphabet
Two choices
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Nondeterministic Finite Accepter (NFA)
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
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First Choice
accept
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Second Choice
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Second Choice
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Second Choice
No transition the automaton hangs
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Second Choice
reject
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Observation
An NFA accepts a string if there is a computation
of the NFA that accepts the string
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Example
is accepted by the NFA
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Lambda Transitions
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accept
String is accepted
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Language accepted
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Another NFA Example
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accept
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Another String
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accept
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Language accepted
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Another NFA Example
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Language accepted
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Formal Definition of NFAs

Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Final states
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Transition Function
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Extended Transition Function

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Formally
It holds
if and only if
there is a walk from to with label
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The Language of an NFA

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Formally

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

(final state)
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Equivalence of NFAs and DFAs

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Equivalence of Machines

• For DFAs or NFAs
• Machine is equivalent to machine
• if

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Example
NFA

DFA
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• Since
• machines and are equivalent

NFA
DFA
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Equivalence of NFAs and DFAs
Question NFAs DFAs ?
Same power? Accept the same languages?
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Equivalence of NFAs and DFAs
Question NFAs DFAs ?
YES!
Same power? Accept the same languages?
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We will prove
Languages accepted by NFAs
Languages accepted by DFAs
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We will prove
Languages accepted by NFAs
Languages accepted by DFAs
NFAs and DFAs have the same computation power
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
Proof
Every DFA is also an NFA
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
Proof
Every DFA is also an NFA
A language accepted by a DFA is also accepted by
an NFA
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
Any NFA can be converted to an equivalent DFA
Proof
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
Any NFA can be converted to an equivalent DFA
Proof
A language accepted by an NFA is also accepted by
a DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA

NFA
DFA
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NFA to DFA Remarks

• We are given an NFA
• We want to convert it
• to an equivalent DFA
• With

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

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Procedure NFA to DFA

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

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Example

NFA
DFA
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Procedure NFA to DFA

• 2. For every DFAs state
• Compute in the NFA
• Add transition

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Exampe

NFA
DFA
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Procedure NFA to DFA

• Repeat Step 2 for all letters in alphabet,
• until
• no more transitions can be added.

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Example

NFA
DFA
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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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Example

NFA
DFA
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Theorem
Take NFA

Apply procedure to obtain DFA
Then and are equivalent

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