NFAs accept the Regular Languages

Equivalence of Machines Definition: Machine is equivalent to machine if

Example of equivalent machines NFA FA

same computation power We will prove: Languages accepted by NFAs Regular Languages Languages accepted by FAs NFAs and FAs have the same computation power

We will show: Languages accepted by NFAs Regular Languages Languages accepted by NFAs Regular Languages

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

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

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

Convert NFA to FA NFA FA

NFA to FA: Remarks We are given an NFA We want to convert it to an equivalent FA With

If the NFA has states the FA has states in the powerset

Procedure NFA to FA 1. Initial state of NFA: Initial state of FA:

Example NFA FA

Procedure NFA to FA 2. For every FA’s state Compute in the NFA Add transition to FA

Exampe NFA FA

Procedure NFA to FA Repeat Step 2 for all letters in alphabet, until no more transitions can be added.

Example NFA FA

Procedure NFA to FA 3. For any FA state If is accepting state in NFA Then, is accepting state in FA

Example NFA FA

Theorem Take NFA Apply procedure to obtain FA Then and are equivalent :

Proof AND

First we show: Take arbitrary: We will prove:

denotes

We will show that if then

More generally, we will show that if in : (arbitrary string) then

Proof by induction on Induction Basis: Is true by construction of

Induction hypothesis:

Induction Step:

Induction Step:

Therefore if then

We have shown: We also need to show: (proof is similar)

Single Accepting State for NFAs

Any NFA can be converted to an equivalent NFA with a single accepting state

Example NFA Equivalent NFA

In General NFA Equivalent NFA Single accepting state

Extreme Case NFA without accepting state Add an accepting state without transitions

Properties of Regular Languages

For regular languages and we will prove that: Union: Are regular Languages Concatenation: Star: Reversal: Complement: Intersection:

We say: Regular languages are closed under Union: Concatenation: Star: Reversal: Complement: Intersection:

Regular language Regular language NFA NFA Single aceepting state Single accepting state NFA

Example

Union NFA for

Example NFA for

Concatenation NFA for

Example NFA for

Star Operation NFA for

Example NFA for

Reverse NFA for 1. Reverse all transitions 2. Make initial state accepting state and vice versa

Example

Complement 1. Take the FA that accepts 2. Make final states non-final, and vice-versa

Example

Intersection regular We show regular regular

DeMorgan’s Law: regular

Example regular regular regular

Another Proof for Intersection Closure Machine Machine FA for FA for Construct a new FA that accepts simulates in parallel and

States in State in State in

FA FA transition transition FA transition

FA FA initial state initial state FA Initial state

FA FA accept state accept states FA accept states Both constituents must be accepting states

Example:

Automaton for intersection

simulates in parallel and accepts string if and only if accepts string and accepts string