Properties of Regular Languages Prof Busch LSU 1
Properties of Regular Languages Prof. Busch - LSU 1
For regular languages we will prove that: and Union: Concatenation: Are regular Languages Star: Reversal: Complement: Intersection: Prof. Busch - LSU 2
We say: Regular languages are closed under Union: Concatenation: Star: Reversal: Complement: Intersection: Prof. Busch - LSU 3
A useful transformation: use one accept state NFA 2 accept states Equivalent NFA 1 accept state Prof. Busch - LSU 4
NFA In General Equivalent NFA Single accepting state Prof. Busch - LSU 5
Extreme case NFA without accepting state Add an accepting state without transitions Prof. Busch - LSU 6
Take two languages Regular language NFA Single accepting state Prof. Busch - LSU 7
Example Prof. Busch - LSU 8
Union NFA for Prof. Busch - LSU 9
Example NFA for Prof. Busch - LSU 10
Concatenation NFA for Prof. Busch - LSU 11
Example NFA for Prof. Busch - LSU 12
Star Operation NFA for Prof. Busch - LSU 13
Example NFA for Prof. Busch - LSU 14
Reverse NFA for 1. Reverse all transitions 2. Make initial state accepting state and vice versa Prof. Busch - LSU 15
Example Prof. Busch - LSU 16
Complement 1. Take the DFA that accepts 2. Make accepting states non-final, and vice-versa Prof. Busch - LSU 17
Example Prof. Busch - LSU 18
Intersection regular We show regular Prof. Busch - LSU 19
De. Morgan’s Law: regular regular Prof. Busch - LSU 20
Example regular Prof. Busch - LSU regular 21
Another Proof for Intersection Closure Machine DFA for Construct a new DFA that accepts simulates in parallel Prof. Busch - LSU and 22
States in State in Prof. Busch - LSU 23
DFA transition DFA New transition Prof. Busch - LSU 24
DFA initial state DFA New initial state Prof. Busch - LSU 25
DFA accept states DFA New accept states Both constituents must be accepting states Prof. Busch - LSU 26
Example: Prof. Busch - LSU 27
Automaton for intersection Prof. Busch - LSU 28
simulates in parallel accepts string and if and only if: accepts string and accepts string Prof. Busch - LSU 29
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