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Regular Expressions

Regular Expressions. Regular Expressions. Regular expressions describe regular languages Example: describes the language. Given regular expressions and. Are regular expressions. Recursive Definition. Primitive regular expressions:. Not a regular expression:.

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Regular Expressions

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  1. Regular Expressions COMP 335

  2. Regular Expressions • Regular expressions • describe regular languages • Example: • describes the language COMP 335

  3. Given regular expressions and Are regular expressions Recursive Definition Primitive regular expressions: COMP 335

  4. Not a regular expression: Examples A regular expression: COMP 335

  5. Languages of Regular Expressions • : language of regular expression • Example: COMP 335

  6. Definition • For primitive regular expressions : COMP 335

  7. Definition (continued) • For regular expressions and COMP 335

  8. Example • Regular expression: COMP 335

  9. Example • Regular expression COMP 335

  10. Example • Regular expression COMP 335

  11. = {all strings with at least two consecutive 0} Example • Regular expression COMP 335

  12. = { all strings without two consecutive 0 } Example • Regular expression COMP 335

  13. Equivalent Regular Expressions • Definition: • Regular expressions and • are equivalent if COMP 335

  14. and are equivalent Reg. expressions Example = { all strings without two consecutive 0 } COMP 335

  15. Regular ExpressionsandRegular Languages COMP 335

  16. Theorem Languages Generated by Regular Expressions Regular Languages COMP 335

  17. 1. For any regular expression the language is regular Theorem - Part 1 Languages Generated by Regular Expressions Regular Languages COMP 335

  18. 2. For any regular language , there is a regular expression with Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages COMP 335

  19. 1. For any regular expression the language is regular Proof by induction on the size of Proof - Part 1 COMP 335

  20. NFAs regular languages Induction Basis • Primitive Regular Expressions: COMP 335

  21. Inductive Hypothesis • Assume for regular expressions and • that and are regular languages COMP 335

  22. Inductive Step • We will prove: are regular Languages. COMP 335

  23. By definition of regular expressions: COMP 335

  24. We also know: Regular languages are closed under: Union Concatenation Star By inductive hypothesis we know: and are regular languages COMP 335

  25. Therefore: Are regular languages COMP 335

  26. And trivially: is a regular language COMP 335

  27. Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression COMP 335

  28. Since is regular, take an • NFA that accepts it Single final state COMP 335

  29. From , construct an equivalent • Generalized Transition Graph in which • transition labels are regular expressions Example: COMP 335

  30. Another Example: COMP 335

  31. Reducing the states: COMP 335

  32. Resulting Regular Expression: COMP 335

  33. In General • Removing states: COMP 335

  34. The final transition graph: The resulting regular expression: COMP 335

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