Properties of Regular Languages

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Presentation transcript:

Properties of Regular Languages Prof. Busch - LSU

For regular languages and we will prove that: Union: Are regular Languages Concatenation: Star: Reversal: Complement: Intersection: Prof. Busch - LSU

We say: Regular languages are closed under Union: Concatenation: Star: Reversal: Complement: Intersection: Prof. Busch - LSU

A useful transformation: use one accept state NFA 2 accept states Equivalent NFA 1 accept state Prof. Busch - LSU

In General NFA Equivalent NFA Single accepting state Prof. Busch - LSU

NFA without accepting state Extreme case NFA without accepting state Add an accepting state without transitions Prof. Busch - LSU

Take two languages Regular language Regular language NFA NFA Single accepting state Regular language NFA Single accepting state NFA Prof. Busch - LSU

Example Prof. Busch - LSU

Union NFA for Prof. Busch - LSU

Example NFA for Prof. Busch - LSU

Concatenation NFA for Prof. Busch - LSU

Example NFA for Prof. Busch - LSU

Star Operation NFA for Prof. Busch - LSU

Example NFA for Prof. Busch - LSU

Reverse NFA for 1. Reverse all transitions 2. Make initial state accepting state and vice versa Prof. Busch - LSU

Example Prof. Busch - LSU

Complement 1. Take the DFA that accepts 2. Make accepting states non-final, and vice-versa Prof. Busch - LSU

Example Prof. Busch - LSU

Intersection regular We show regular regular Prof. Busch - LSU

DeMorgan’s Law: regular Prof. Busch - LSU

Example regular regular regular Prof. Busch - LSU

Another Proof for Intersection Closure Machine Machine DFA for DFA for Construct a new DFA that accepts simulates in parallel and Prof. Busch - LSU

States in State in State in Prof. Busch - LSU

DFA DFA transition transition DFA New transition Prof. Busch - LSU

DFA DFA initial state initial state DFA New initial state Prof. Busch - LSU

Both constituents must be accepting states DFA DFA accept state accept states DFA New accept states Both constituents must be accepting states Prof. Busch - LSU

Example: Prof. Busch - LSU

Automaton for intersection Prof. Busch - LSU

simulates in parallel and accepts string if and only if: accepts string and accepts string Prof. Busch - LSU