CSE322 PROPERTIES OF REGULAR LANGUAGES

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Presentation transcript:

CSE322 PROPERTIES OF REGULAR LANGUAGES Lecture #12

Closure properties of Regular set Set Union Concatenation Closure(iteration) Transpose Set intersection Complementation

Properties of Regular Languages

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

Union: 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 Homework 2

Complement Homework 2

Intersection regular We show regular regular

DeMorgan’s Law: regular

Example regular regular regular

Construct a new FA that accepts 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