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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 FA
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same computation power
We will prove: Languages accepted by NFAs Regular Languages Languages accepted by FAs NFAs and FAs have the same computation power
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We will show: Languages accepted by NFAs Regular Languages Languages accepted by NFAs Regular Languages
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Proof-Step 1 Languages accepted by NFAs Regular Languages Proof: Every FA is trivially an NFA Any language accepted by a FA is also accepted by an NFA
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Proof-Step 2 Languages accepted by NFAs Regular Languages Proof: Any NFA can be converted to an equivalent FA Any language accepted by an NFA is also accepted by a FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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Convert NFA to FA NFA FA
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NFA to FA: Remarks We are given an NFA We want to convert it
to an equivalent FA With
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If the NFA has states the FA has states in the powerset
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Procedure NFA to FA 1. Initial state of NFA: Initial state of FA:
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Example NFA FA
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Procedure NFA to FA 2. For every FA’s state Compute in the NFA
Add transition to FA
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Exampe NFA FA
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Procedure NFA to FA Repeat Step 2 for all letters in alphabet, until
no more transitions can be added.
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Example NFA FA
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Procedure NFA to FA 3. For any FA state If is accepting state in NFA
Then, is accepting state in FA
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Example NFA FA
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Theorem Take NFA Apply procedure to obtain FA
Then and are equivalent :
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Proof AND
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First we show: Take arbitrary: We will prove:
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denotes
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We will show that if then
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More generally, we will show that if in :
(arbitrary string) then
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Proof by induction on Induction Basis: Is true by construction of
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Induction hypothesis:
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Induction Step:
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Induction Step:
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Therefore if then
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We have shown: We also need to show: (proof is similar)
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Single Accepting State for NFAs
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Any NFA can be converted
to an equivalent NFA with a single accepting state
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Example NFA Equivalent NFA
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In General NFA Equivalent NFA Single accepting state
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Extreme Case NFA without accepting state Add an accepting state
without transitions
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Properties of Regular Languages
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For regular languages and
we will prove that: Union: Are regular Languages Concatenation: Star: Reversal: Complement: Intersection:
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We say: Regular languages are closed under
Union: Concatenation: Star: Reversal: Complement: Intersection:
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Regular language Regular language NFA NFA Single aceepting state
Single accepting state NFA
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Example
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Union NFA for
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Example NFA for
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Concatenation NFA for
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Example NFA for
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Star Operation NFA for
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Example NFA for
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Reverse NFA for 1. Reverse all transitions
2. Make initial state accepting state and vice versa
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Example
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Complement 1. Take the FA that accepts 2. Make final states non-final,
and vice-versa
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Example
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Intersection regular We show regular regular
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DeMorgan’s Law: regular
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Example regular regular regular
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Another Proof for Intersection Closure
Machine Machine FA for FA for Construct a new FA that accepts simulates in parallel and
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States in State in State in
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FA FA transition transition FA transition
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FA FA initial state initial state FA Initial state
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FA FA accept state accept states FA accept states Both constituents must be accepting states
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Example:
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Automaton for intersection
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simulates in parallel and
accepts string if and only if accepts string and accepts string
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