Non-Deterministic%20Automata%20%20Regular%20Expressions

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Non-Deterministic AutomataRegular Expressions
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NFA Example
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NFA Example
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Formal Definition of NFAs

Set of states, 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

• For DFAs or NFAs
• Machine is equivalent to
• 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

• Every DFA is also an NFA
• A language accepted by a DFA will be accepted by
  an NFA
• An NFA is as least as powerful as a DFA
• Is an DFA as powerful as an NFA?

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Is a DFA as Powerful as an NFA?

• Answer YES!
• A language accepted by an NFA will be accepted by
  some DFA
• For every NFA there is an equivalent DFA
• that accepts the same language

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NFAs Accept the Regular Languages

• For every NFA there is an equivalent DFA
• The language accepted by a DFA is regular
• The language accepted by an NFA is regular

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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

NFA
DFA
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NFA to DFA Observations

• 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
• Giving union
• Add a transition

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Exampe

NFA
DFA
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Procedure NFA to DFA

• Repeat Step 2 for all letters,
• 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 for the
  NFA
• Then,
• is a final state for 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