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Single Final State for NFAs and DFAs

Learn how any NFA or DFA can be converted to an equivalent NFA with a single final state. Explore examples and properties of regular languages, including union, concatenation, and star operations. Discover the connection between regular expressions and regular languages.

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Single Final State for NFAs and DFAs

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  1. Single Final State for NFAs and DFAs

  2. Observation • Any Finite Automaton (NFA or DFA) • can be converted to an equivalent NFA • with a single final state

  3. Equivalent NFA Example NFA

  4. Equivalent NFA Single final state In General NFA

  5. Add a final state Without transitions Extreme Case NFA without final state

  6. Some Properties of Regular Languages

  7. Union: Concatenation: Are regular Languages Star: Properties For regular languages and we will prove that:

  8. Union: Concatenation: Star: We Say: Regular languages are closed under

  9. Regular language Regular language NFA NFA Single final state Single final state

  10. Example

  11. Union • NFA for

  12. Example NFA for

  13. Concatenation • NFA for

  14. Example • NFA for

  15. Star Operation • NFA for

  16. Example • NFA for

  17. Regular Expressions

  18. Regular Expressions • Regular expressions • describe regular languages • Example: • describes the language

  19. Given regular expressions and Are regular expressions Recursive Definition Primitive regular expressions:

  20. Not a regular expression: Examples A regular expression:

  21. Languages of Regular Expressions • : language of regular expression • Example

  22. Definition • For primitive regular expressions:

  23. Definition (continued) • For regular expressions and

  24. Example • Regular expression:

  25. Example • Regular expression

  26. Example • Regular expression

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

  28. = { all strings without two consecutive 0 } Example • Regular expression

  29. Equivalent Regular Expressions • Definition: • Regular expressions and • are equivalent if

  30. and are equivalent regular expr. Example = { all strings without two consecutive 0 }

  31. Regular ExpressionsandRegular Languages

  32. Theorem Languages Generated by Regular Expressions Regular Languages

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

  34. 2. For any regular language there is a regular expression with Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages

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

  36. NFAs regular languages Induction Basis • Primitive Regular Expressions:

  37. Inductive Hypothesis • Assume • for regular expressions and • that • and are regular languages

  38. Inductive Step • We will prove: Are regular Languages

  39. By definition of regular expressions:

  40. We also know: Regular languages are closed under union concatenation star By inductive hypothesis we know: and are regular languages

  41. Therefore: Are regular languages

  42. And trivially: is a regular language

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

  44. Since is regular take the • NFA that accepts it Single final state

  45. From construct the equivalent • Generalized Transition Graph • transition labels • are regular expressions Example:

  46. Another Example:

  47. Reducing the states:

  48. Resulting Regular Expression:

  49. In General • Removing states:

  50. The final transition graph: The resulting regular expression:

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