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Properties of Regular Languages

Properties of Regular Languages. Union:. Concatenation:. Are regular Languages. For regular languages and we will prove that:. Star:. Reversal:. Complement:. Intersection:. Union:. Concatenation:. We say: Regular languages are closed under. Star:. Reversal:.

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Properties of Regular Languages

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  1. Properties of Regular Languages Costas Busch - RPI

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

  3. Union: Concatenation: We say: Regular languages are closed under Star: Reversal: Complement: Intersection: Costas Busch - RPI

  4. A useful transformation: use one accept state NFA 2 accept states Equivalent NFA 1 accept state Costas Busch - RPI

  5. In General NFA Equivalent NFA Single accepting state Costas Busch - RPI

  6. Add an accepting state without transitions Extreme case NFA without accepting state Costas Busch - RPI

  7. Take two languages Regular language Regular language NFA NFA Single accepting state Single accepting state Costas Busch - RPI

  8. Example Costas Busch - RPI

  9. Union • NFA for Costas Busch - RPI

  10. Example NFA for Costas Busch - RPI

  11. Concatenation • NFA for Costas Busch - RPI

  12. Example • NFA for Costas Busch - RPI

  13. Star Operation • NFA for Costas Busch - RPI

  14. Example • NFA for Costas Busch - RPI

  15. Reverse NFA for 1. Reverse all transitions 2. Make initial state accepting state and vice versa Costas Busch - RPI

  16. Example Costas Busch - RPI

  17. Complement 1. Take the DFAthat accepts 2. Make accepting states non-final, and vice-versa Costas Busch - RPI

  18. Example Costas Busch - RPI

  19. Intersection regular We show regular regular Costas Busch - RPI

  20. regular regular regular regular regular DeMorgan’s Law: Costas Busch - RPI

  21. Example regular regular regular Costas Busch - RPI

  22. Another Proof for Intersection Closure Machine Machine DFA for DFA for Construct a new DFA that accepts simulates in parallel and Costas Busch - RPI

  23. States in State in State in Costas Busch - RPI

  24. DFA DFA transition transition DFA New transition Costas Busch - RPI

  25. DFA DFA initial state initial state DFA New initial state Costas Busch - RPI

  26. DFA DFA accept state accept states DFA New accept states Both constituents must be accepting states Costas Busch - RPI

  27. Example: Costas Busch - RPI

  28. Automaton for intersection Costas Busch - RPI

  29. simulates in parallel and accepts string if and only if: accepts string and accepts string Costas Busch - RPI

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