Video Converter
Audio Converter
Image Converter
PDF Converter
File Compressor
regular expressions res

Regular Expressions (REs) Regular Expressions (REs) p.1/37 - PowerPoint PPT Presentation

Aug 16, 2022 β€’271 likes β€’1.52k views

Regular Expressions (REs) Regular Expressions (REs) p.1/37 Expressions In arithmetic: Regular Expressions (REs) p.2/37 Expressions In arithmetic: 1. expressions are constructed from numbers and variables using arithmetic

  1. βœ‚ οΏ½ ✁ οΏ½ βœ‚ οΏ½ ✁ οΏ½ βœ„ βœ† βœ„ ☎ ✁ βœ‚ οΏ½ βœ„ βœ„ βœ‚ οΏ½ βœ‚ οΏ½ βœ‚ οΏ½ οΏ½ ☎ οΏ½ βœ‚ Precedence relation Denoting the rule describing the order of ops in the evaluation of arithmetic and regular expressions we have: ✁ βœ‚βœ Arithmetic expressions: Regular expressions: ✁ ✁� Note: denotes expressions enclosed in parentheses Regular Expressions (REs) – p.7/37

  2. οΏ½ οΏ½ οΏ½ Formal definitions Consider an alphabet. A regular expression (RE) over is defi ned recursivelyby the rules: Regular Expressions (REs) – p.8/37

  3. οΏ½ οΏ½ οΏ½ Formal definitions Consider an alphabet. A regular expression (RE) over is defi ned recursivelyby the rules: Recursion basis: Regular Expressions (REs) – p.8/37

  4. βœ‚ βœ„ οΏ½ βœ‚ οΏ½ οΏ½ οΏ½ ✁ ☎ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ Formal definitions Consider an alphabet. A regular expression (RE) over is defi ned recursivelyby the rules: Recursion basis: 1. For any , is a RE describing the language Regular Expressions (REs) – p.8/37

  5. οΏ½ ✝ οΏ½ ✁ ☎ βœ‚ βœ„ οΏ½ βœ„ οΏ½ οΏ½ βœ‚ ✝ ✁ οΏ½ βœ‚ ☎ ✁ οΏ½ ✝ οΏ½ βœ‚ οΏ½ βœ‚ Formal definitions Consider an alphabet. A regular expression (RE) over is defi ned recursivelyby the rules: Recursion basis: 1. For any , is a RE describing the language 2. is a RE describing the language Regular Expressions (REs) – p.8/37

  6. βœ‚ βœ„ ✝ οΏ½ βœ„ ✁ ✝ ✝ βœ‚ οΏ½ ✁ ☎ βœ‚ οΏ½ οΏ½ οΏ½ ☎ οΏ½ βœ„ βœ‚ ✁ οΏ½ οΏ½ οΏ½ βœ‚ οΏ½ ☎ οΏ½ οΏ½ οΏ½ βœ‚ Formal definitions Consider an alphabet. A regular expression (RE) over is defi ned recursivelyby the rules: Recursion basis: 1. For any , is a RE describing the language 2. is a RE describing the language 3. is a RE describing the empty language . Regular Expressions (REs) – p.8/37

  7. οΏ½ ☎ ☎ βœ‚ οΏ½ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ βœ„ οΏ½ Formal definitions, continutation Recursion body: If and are REs evaluating to the οΏ½ ✁� languages and then: Regular Expressions (REs) – p.9/37

  8. οΏ½ οΏ½ βœ‚ βœ‚ βœ‚ οΏ½ ✁ ✁ οΏ½ οΏ½ οΏ½ βœ‚ ☎ βœ‚ οΏ½ οΏ½ βœ„ ✁ ☎ οΏ½ οΏ½ οΏ½ βœ„ βœ‚ ✁ βœ‚ οΏ½ βœ„ οΏ½ βœ„ Formal definitions, continutation Recursion body: If and are REs evaluating to the οΏ½ ✁� languages and then: 1. is a RE evaluating to Regular Expressions (REs) – p.9/37

  9. οΏ½ οΏ½ βœ‚ ✁ βœ„ οΏ½ οΏ½ ✁ βœ‚ ✁ βœ„ οΏ½ οΏ½ βœ‚ βœ‚ οΏ½ οΏ½ οΏ½ ✁ βœ‚ οΏ½ βœ‚ βœ‚ ✁ βœ„ οΏ½ βœ‚ βœ‚ οΏ½ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ ✁ βœ„ βœ„ οΏ½ οΏ½ βœ‚ ☎ βœ‚ βœ‚ οΏ½ οΏ½ βœ‚ Formal definitions, continutation Recursion body: If and are REs evaluating to the οΏ½ ✁� languages and then: 1. is a RE evaluating to 2. is a RE evaluating to Regular Expressions (REs) – p.9/37

  10. βœ‚ οΏ½ βœ„ ✁ βœ† οΏ½ οΏ½ βœ‚ βœ‚ οΏ½ βœ„ βœ‚ βœ‚ οΏ½ ✁ οΏ½ οΏ½ βœ„ ✁ βœ‚ βœ‚ βœ„ οΏ½ ✁ βœ‚ ✁ οΏ½ οΏ½ βœ„ ☎ οΏ½ βœ‚ οΏ½ βœ„ οΏ½ βœ„ ✁ οΏ½ βœ‚ ✁ οΏ½ οΏ½ βœ‚ βœ‚ ☎ οΏ½ οΏ½ ☎ βœ‚ οΏ½ οΏ½ βœ„ ✁ ☎ οΏ½ οΏ½ οΏ½ βœ„ βœ‚ βœ„ βœ‚ οΏ½ ☎ ✝ οΏ½ οΏ½ ✁ ✁ οΏ½ οΏ½ βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ βœ„ βœ‚ ✁ ✁ οΏ½ οΏ½ βœ„ βœ„ βœ‚ βœ‚ οΏ½ ✁ οΏ½ Formal definitions, continutation Recursion body: If and are REs evaluating to the οΏ½ ✁� languages and then: 1. is a RE evaluating to 2. is a RE evaluating to 3. is a RE evaluating to ✁ ✁� where and . Regular Expressions (REs) – p.9/37

  11. Note A defi nition of the form expressed by rule (4) above is called an inductive defi nition Regular Expressions (REs) – p.10/37

  12. οΏ½ οΏ½ ✁ Potential confusions Do not confuse the REs and ; Regular Expressions (REs) – p.11/37

  13. οΏ½ οΏ½ ✁ ✝ Potential confusions Do not confuse the REs and ; 1. represents the language containing just the empty string Regular Expressions (REs) – p.11/37

  14. οΏ½ οΏ½ ✁ ✝ οΏ½ Potential confusions Do not confuse the REs and ; 1. represents the language containing just the empty string 2. is the RE that represents the the empty language Regular Expressions (REs) – p.11/37

  15. βœ„ οΏ½ οΏ½ ☎ οΏ½ οΏ½ ☎ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ οΏ½ οΏ½ ✝ ✁ οΏ½ Potential confusions Do not confuse the REs and ; 1. represents the language containing just the empty string 2. is the RE that represents the the empty language Note the distinction between and . is an expression and is the set of strings specifi ed by Regular Expressions (REs) – p.11/37

  16. οΏ½ βœ† βœ„ οΏ½ βœ† Example REs 1. , Regular Expressions (REs) – p.12/37

  17. βœ„ οΏ½ ✁ ☎ βœ‚ βœ† οΏ½ οΏ½ βœ† οΏ½ ✁ βœ„ βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ οΏ½ Example REs 1. , contains exactly a single 1 Regular Expressions (REs) – p.12/37

  18. βœ‚ οΏ½ βœ‚ βœ„ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ„ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ† βœ† οΏ½ βœ„ βœ† οΏ½ βœ† Example REs 1. , contains exactly a single 1 2. , Regular Expressions (REs) – p.12/37

  19. ✁ βœ† βœ‚ ✁ βœ‚ βœ† βœ„ βœ‚ βœ‚ οΏ½ βœ„ οΏ½ βœ‚ βœ† βœ„ βœ‚ βœ† οΏ½ ✁ ☎ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† οΏ½ βœ„ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ ☎ ✁ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 Regular Expressions (REs) – p.12/37

  20. βœ‚ βœ† βœ† βœ„ οΏ½ βœ† οΏ½ βœ„ οΏ½ βœ‚ βœ„ βœ„ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ† βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ† βœ„ οΏ½ βœ† βœ‚ βœ„ οΏ½ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , Regular Expressions (REs) – p.12/37

  21. ☎ οΏ½ βœ‚ βœ† βœ‚ οΏ½ ✁ οΏ½ ✁ οΏ½ ✁ βœ‚ οΏ½ βœ‚ βœ† οΏ½ βœ† βœ„ βœ‚ βœ† ✁ βœ„ οΏ½ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ‚ βœ† βœ„ βœ‚ ☎ βœ‚ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† ☎ οΏ½ ✁ οΏ½ ✁ οΏ½ βœ„ βœ‚ οΏ½ βœ‚ βœ† βœ„ βœ‚ βœ† οΏ½ βœ„ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring Regular Expressions (REs) – p.12/37

  22. ✁ βœ‚ βœ‚ ☎ ✁ οΏ½ οΏ½ οΏ½ βœ‚ βœ‚ βœ„ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ† βœ‚ βœ‚ ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ„ οΏ½ οΏ½ οΏ½ βœ† βœ‚ οΏ½ βœ„ βœ† βœ„ οΏ½ ☎ οΏ½ βœ† βœ„ οΏ½ βœ† βœ† βœ„ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ ✁ βœ† βœ„ βœ‚ οΏ½ βœ„ βœ‚ βœ† βœ‚ βœ† οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ ✁ οΏ½ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , Regular Expressions (REs) – p.12/37

  23. οΏ½ βœ‚ οΏ½ βœ† βœ‚ οΏ½ βœ„ βœ‚ βœ† βœ„ βœ„ οΏ½ βœ† βœ† βœ‚ ☎ βœ‚ ✁ οΏ½ βœ‚ ✁ βœ„ βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ βœ‚ βœ‚ οΏ½ βœ† οΏ½ βœ‚ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ ✁ ☎ βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ ✁ βœ„ βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ οΏ½ οΏ½ ✁ οΏ½ ☎ βœ‚ βœ† βœ‚ βœ„ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ‚ βœ„ βœ† βœ‚ ✁ βœ‚ βœ† Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length Regular Expressions (REs) – p.12/37

  24. βœ† οΏ½ βœ‚ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ‚ βœ„ ✁ οΏ½ βœ‚ βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ‚ ☎ οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ βœ„ ☎ βœ† βœ‚ βœ‚ οΏ½ ✁ ✁ βœ‚ οΏ½ βœ„ οΏ½ οΏ½ ✁ οΏ½ οΏ½ βœ‚ βœ† βœ† ✁ οΏ½ ✁ ☎ βœ‚ βœ† οΏ½ βœ„ οΏ½ βœ‚ οΏ½ βœ„ βœ† βœ† οΏ½ βœ„ βœ† οΏ½ οΏ½ βœ‚ οΏ½ βœ† ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ‚ βœ„ βœ‚ βœ‚ οΏ½ βœ„ βœ‚ βœ† βœ‚ βœ„ βœ† βœ‚ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , Regular Expressions (REs) – p.12/37

  25. οΏ½ ☎ βœ‚ βœ„ οΏ½ οΏ½ βœ‚ οΏ½ ✁ οΏ½ ✁ βœ‚ βœ‚ βœ† βœ‚ βœ„ βœ‚ οΏ½ βœ† βœ‚ οΏ½ βœ„ βœ† βœ‚ οΏ½ βœ‚ ✁ βœ‚ βœ‚ οΏ½ βœ† βœ‚ βœ„ οΏ½ ✁ οΏ½ ☎ βœ‚ βœ† βœ‚ βœ‚ βœ‚ οΏ½ βœ„ οΏ½ βœ† βœ‚ βœ‚ βœ† βœ„ βœ‚ οΏ½ βœ‚ οΏ½ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ„ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ ✁ βœ† οΏ½ ☎ οΏ½ βœ† βœ‚ ☎ βœ‚ ✁ οΏ½ ✁ οΏ½ ✁ βœ‚ βœ„ βœ† βœ‚ βœ„ βœ† βœ‚ οΏ½ βœ„ οΏ½ βœ† βœ‚ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three Regular Expressions (REs) – p.12/37

  26. βœ‚ οΏ½ οΏ½ ☎ βœ‚ ✁ βœ„ οΏ½ οΏ½ οΏ½ οΏ½ ✁ ✁ βœ‚ ☎ ✁ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ‚ οΏ½ βœ‚ βœ‚ οΏ½ βœ‚ βœ„ βœ‚ βœ† βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ βœ‚ βœ‚ οΏ½ βœ† ✁ οΏ½ ✁ ☎ βœ† βœ‚ βœ‚ βœ‚ οΏ½ βœ„ βœ† βœ„ βœ† βœ‚ οΏ½ ✁ βœ‚ οΏ½ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ„ βœ† βœ† οΏ½ οΏ½ βœ„ βœ„ βœ† οΏ½ βœ„ βœ† οΏ½ βœ‚ βœ„ βœ‚ οΏ½ βœ„ οΏ½ οΏ½ βœ† βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ ✁ ✁ βœ‚ ☎ βœ‚ βœ† βœ‚ βœ„ βœ† βœ‚ οΏ½ βœ„ βœ„ βœ† οΏ½ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , Regular Expressions (REs) – p.12/37

  27. οΏ½ οΏ½ ☎ βœ† βœ‚ βœ‚ βœ‚ οΏ½ βœ„ ✁ βœ† βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ βœ‚ βœ„ βœ„ βœ‚ οΏ½ ☎ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ ✁ ✁ βœ‚ βœ‚ οΏ½ οΏ½ βœ‚ βœ„ οΏ½ βœ„ ✁ οΏ½ ✁ ☎ βœ† βœ‚ βœ‚ οΏ½ βœ‚ οΏ½ βœ„ οΏ½ βœ† βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ οΏ½ βœ‚ βœ„ βœ† βœ„ ✁ βœ‚ βœ‚ βœ„ βœ† βœ‚ οΏ½ βœ‚ βœ„ οΏ½ ✁ οΏ½ ✁ ☎ βœ† βœ„ οΏ½ βœ„ βœ† οΏ½ οΏ½ βœ„ βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ βœ† βœ„ οΏ½ βœ‚ οΏ½ βœ† βœ‚ οΏ½ βœ„ ✁ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† οΏ½ οΏ½ βœ‚ βœ„ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ βœ† βœ‚ βœ„ βœ† βœ‚ βœ„ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , Regular Expressions (REs) – p.12/37

  28. βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ βœ‚ βœ‚ οΏ½ οΏ½ ✁ οΏ½ ✁ ☎ βœ† βœ‚ βœ‚ βœ‚ οΏ½ βœ„ βœ‚ βœ† βœ‚ βœ† βœ‚ οΏ½ οΏ½ βœ‚ ✁ βœ„ οΏ½ οΏ½ βœ‚ βœ† οΏ½ βœ„ βœ‚ βœ„ οΏ½ ✁ βœ„ οΏ½ οΏ½ βœ„ οΏ½ βœ„ οΏ½ ✁ βœ„ οΏ½ βœ‚ βœ„ βœ„ οΏ½ οΏ½ ✁ οΏ½ ✁ ☎ βœ† βœ‚ βœ‚ βœ‚ βœ‚ οΏ½ βœ„ βœ„ ✁ ✁ ✁ οΏ½ βœ„ ✁ βœ† βœ‚ βœ„ βœ† βœ‚ οΏ½ βœ‚ ✁ οΏ½ ✁ οΏ½ ☎ βœ† βœ‚ βœ† οΏ½ βœ„ βœ† οΏ½ οΏ½ βœ„ βœ„ βœ† οΏ½ βœ„ βœ† οΏ½ βœ‚ βœ„ οΏ½ βœ‚ ✁ ☎ βœ‚ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ‚ οΏ½ βœ„ βœ‚ βœ† βœ„ βœ‚ οΏ½ οΏ½ βœ† βœ‚ βœ† βœ‚ βœ„ οΏ½ ✁ οΏ½ ✁ ☎ βœ‚ βœ† ☎ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , 7. , Regular Expressions (REs) – p.12/37

  29. οΏ½ βœ‚ βœ† βœ„ βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ βœ† ☎ ✁ οΏ½ ✁ ✁ οΏ½ βœ‚ βœ‚ βœ‚ ☎ ✁ βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ† ✁ βœ‚ βœ„ ✁ οΏ½ βœ† βœ‚ βœ‚ οΏ½ οΏ½ βœ† βœ‚ ✁ βœ‚ οΏ½ οΏ½ βœ‚ βœ† οΏ½ βœ„ οΏ½ βœ‚ βœ† βœ„ ✁ οΏ½ ✁ βœ„ βœ„ βœ„ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ βœ‚ βœ„ οΏ½ βœ„ βœ„ ✁ οΏ½ βœ„ βœ‚ ☎ ✁ οΏ½ βœ† βœ‚ βœ„ βœ† βœ‚ ✁ βœ‚ βœ† βœ„ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ βœ‚ βœ† βœ„ βœ‚ βœ† οΏ½ ✁ ☎ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ ☎ ✁ βœ‚ ✁ βœ‚ οΏ½ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ ✁ βœ„ οΏ½ οΏ½ οΏ½ βœ„ ✁ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ οΏ½ ✁ οΏ½ ✁ βœ‚ ☎ βœ‚ βœ† οΏ½ βœ† βœ„ βœ‚ βœ† βœ‚ βœ„ οΏ½ βœ‚ οΏ½ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , 7. , starts and ends with the same symbol Regular Expressions (REs) – p.12/37

  30. οΏ½ οΏ½ βœ„ οΏ½ βœ‚ βœ‚ ✝ οΏ½ βœ„ ✁ βœ„ οΏ½ ✁ βœ„ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ ✁ ✁ ☎ βœ‚ οΏ½ ✁ οΏ½ βœ‚ οΏ½ βœ‚ βœ‚ βœ‚ βœ† ☎ βœ† βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ βœ† βœ‚ ✁ βœ‚ βœ„ οΏ½ βœ‚ βœ† οΏ½ ✁ βœ„ βœ‚ βœ† ✁ βœ„ οΏ½ ✁ βœ„ βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ οΏ½ οΏ½ βœ† βœ„ βœ„ οΏ½ βœ„ βœ‚ οΏ½ οΏ½ βœ‚ οΏ½ βœ„ ✁ βœ„ βœ‚ βœ† βœ„ ✁ οΏ½ ✁ ✁ ☎ βœ† ✁ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ οΏ½ βœ† οΏ½ βœ‚ βœ‚ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ‚ βœ‚ βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ οΏ½ βœ„ οΏ½ βœ‚ βœ† βœ„ βœ‚ βœ† βœ† βœ„ βœ„ βœ† βœ† βœ‚ βœ† βœ‚ οΏ½ οΏ½ βœ‚ βœ‚ βœ† οΏ½ βœ„ βœ† βœ„ οΏ½ οΏ½ βœ‚ βœ‚ βœ„ οΏ½ οΏ½ βœ„ βœ„ οΏ½ βœ‚ βœ† οΏ½ οΏ½ βœ‚ βœ„ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ‚ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , 7. , starts and ends with the same symbol 8. , Regular Expressions (REs) – p.12/37

  31. οΏ½ βœ„ βœ‚ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ βœ„ οΏ½ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ βœ‚ ☎ ✁ οΏ½ οΏ½ βœ„ ✝ βœ‚ βœ‚ οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ βœ† βœ„ ✁ ✁ βœ‚ βœ‚ βœ‚ βœ‚ βœ† ☎ ✁ οΏ½ οΏ½ βœ„ οΏ½ βœ„ ✁ βœ„ βœ‚ βœ† βœ„ ✁ οΏ½ ✁ βœ‚ βœ† ☎ ✁ οΏ½ ✁ οΏ½ βœ‚ ✝ βœ‚ ✁ οΏ½ βœ‚ οΏ½ βœ„ βœ„ βœ‚ βœ† οΏ½ βœ‚ βœ† οΏ½ ✁ βœ‚ οΏ½ βœ† βœ„ ✁ οΏ½ ✁ βœ„ βœ„ βœ„ οΏ½ βœ„ ✁ οΏ½ βœ† βœ† βœ„ βœ‚ βœ† ✁ βœ„ οΏ½ βœ‚ βœ„ βœ„ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ† βœ‚ βœ‚ βœ„ οΏ½ οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ βœ„ οΏ½ βœ† βœ† βœ„ οΏ½ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ‚ βœ‚ οΏ½ βœ‚ οΏ½ βœ„ ☎ βœ‚ βœ‚ οΏ½ βœ‚ βœ‚ βœ† ✁ βœ† βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ† ☎ ✁ οΏ½ οΏ½ βœ† βœ‚ οΏ½ οΏ½ βœ„ βœ‚ βœ† οΏ½ βœ„ οΏ½ βœ† ✁ οΏ½ οΏ½ βœ„ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ οΏ½ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , 7. , starts and ends with the same symbol 8. , Regular Expressions (REs) – p.12/37

  32. ☎ βœ‚ βœ„ ☎ βœ„ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ ☎ ✁ ✁ οΏ½ βœ„ βœ„ οΏ½ βœ„ βœ‚ βœ‚ οΏ½ οΏ½ βœ„ βœ† βœ‚ βœ‚ βœ‚ βœ† ✁ βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ† οΏ½ βœ† ✁ οΏ½ ✁ βœ„ οΏ½ οΏ½ βœ‚ ✁ βœ‚ οΏ½ βœ‚ ✝ βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ„ βœ‚ βœ‚ οΏ½ ✁ οΏ½ ✁ ✝ βœ‚ βœ„ βœ† ✁ βœ„ οΏ½ βœ„ οΏ½ ✁ βœ„ βœ„ βœ‚ βœ† βœ„ ✁ οΏ½ ✁ βœ„ βœ† οΏ½ ✁ οΏ½ βœ‚ βœ† οΏ½ ✁ βœ„ βœ‚ βœ† βœ„ βœ‚ οΏ½ οΏ½ βœ‚ βœ„ βœ‚ βœ† βœ„ βœ„ οΏ½ βœ‚ βœ† βœ„ βœ† βœ‚ βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ οΏ½ βœ‚ βœ‚ βœ† ✁ βœ† βœ† οΏ½ βœ† βœ„ οΏ½ βœ† βœ‚ βœ„ οΏ½ οΏ½ βœ„ βœ‚ οΏ½ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ ✝ βœ‚ οΏ½ βœ„ βœ‚ οΏ½ οΏ½ βœ‚ βœ‚ βœ‚ βœ† βœ‚ βœ„ οΏ½ βœ‚ οΏ½ βœ‚ βœ† ☎ ✁ οΏ½ ✁ οΏ½ βœ† βœ‚ βœ‚ βœ„ οΏ½ οΏ½ βœ„ βœ‚ βœ† ✝ βœ„ οΏ½ βœ‚ βœ† οΏ½ βœ„ οΏ½ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ ✁ οΏ½ οΏ½ Example REs 1. , contains exactly a single 1 2. , contains at least one 1 3. , contains as a substring 4. , is a string of even length 5. , the length of is a multiple of three 6. , 7. , starts and ends with the same symbol 8. , 9. , Regular Expressions (REs) – p.12/37

  33. Regular Expressions (REs) – p.12/37 βœ‚ οΏ½ ✁ βœ„ βœ„ οΏ½ οΏ½ βœ‚ βœ† οΏ½ ✁ βœ„ βœ† βœ„ βœ„ ✁ οΏ½ ✁ βœ„ βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ βœ‚ ✁ βœ† οΏ½ ☎ βœ„ ✁ βœ„ βœ„ οΏ½ οΏ½ βœ„ ✁ οΏ½ βœ„ βœ‚ ✁ βœ‚ οΏ½ βœ„ βœ„ οΏ½ βœ„ βœ‚ οΏ½ βœ‚ βœ† οΏ½ ✁ βœ„ οΏ½ ✁ βœ‚ βœ„ ✁ ✝ βœ‚ βœ„ οΏ½ οΏ½ οΏ½ ✁ ✝ βœ‚ οΏ½ ✁ οΏ½ ✝ βœ‚ βœ‚ ☎ ✁ ✝ βœ„ βœ„ βœ„ οΏ½ βœ„ βœ‚ βœ„ βœ‚ ✝ βœ† βœ‚ βœ„ βœ† βœ„ οΏ½ οΏ½ οΏ½ ✁ ✝ βœ‚ βœ„ βœ‚ ✝ ☎ ✁ οΏ½ βœ„ βœ† ✁ βœ„ βœ† βœ‚ οΏ½ οΏ½ ✁ οΏ½ οΏ½ ✁ βœ‚ βœ† οΏ½ οΏ½ βœ„ βœ‚ βœ† βœ„ βœ† βœ„ οΏ½ βœ† οΏ½ οΏ½ οΏ½ οΏ½ βœ„ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ βœ‚ βœ‚ οΏ½ βœ„ οΏ½ βœ‚ ✁ ☎ βœ‚ βœ† βœ„ βœ‚ βœ† βœ‚ οΏ½ οΏ½ βœ‚ βœ† βœ† βœ„ βœ‚ βœ† βœ‚ ☎ ✁ οΏ½ ✁ οΏ½ ✁ βœ‚ βœ‚ οΏ½ οΏ½ οΏ½ βœ† ✁ βœ† οΏ½ βœ‚ βœ‚ βœ‚ οΏ½ βœ† οΏ½ βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ‚ βœ† ✁ ✁ βœ„ βœ‚ οΏ½ βœ„ οΏ½ βœ‚ οΏ½ οΏ½ ✁ βœ‚ βœ‚ ☎ βœ† οΏ½ βœ„ βœ„ οΏ½ βœ‚ βœ‚ βœ‚ βœ† ☎ starts and is a multiple of three as a substring contains exactly a single 1 contains at least one 1 is a string of even length contains the length of οΏ½ β˜Žβœ„ ends with the same symbol Example REs , , , , , , , , , 1. 2. 3. 4. 5. 6. 7. 8. 9.

  34. ✁ ✁ βœ‚ βœ„ ✁ ✁ βœ„ οΏ½ βœ‚ βœ„ ✁ ☎ οΏ½ More example 10. , ; Note: concatenating to any set yields Regular Expressions (REs) – p.13/37

  35. ✁ οΏ½ οΏ½ οΏ½ οΏ½ ☎ βœ„ ✁ οΏ½ βœ„ οΏ½ βœ„ ✁ ✁ ✁ βœ„ βœ„ βœ‚ ✁ βœ„ οΏ½ ☎ ✁ ✁ βœ‚ οΏ½ More example 10. , ; Note: concatenating to any set yields 11. , ; by defi nition, is in the star operation applied on any language; if the language is empty becomes the one element Regular Expressions (REs) – p.13/37

  36. οΏ½ Identities If is a RE then the following identities take place: Regular Expressions (REs) – p.14/37

  37. ✁ οΏ½ οΏ½ ☎ οΏ½ βœ„ οΏ½ οΏ½ ✁ οΏ½ ☎ οΏ½ βœ„ Identities If is a RE then the following identities take place: ; adding the empty language to any other language will not change that language Regular Expressions (REs) – p.14/37

  38. οΏ½ οΏ½ βœ„ οΏ½ ☎ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ ☎ ☎ οΏ½ οΏ½ βœ„ οΏ½ ☎ ✁ ✁ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ Identities If is a RE then the following identities take place: ; adding the empty language to any other language will not change that language ; concatenating any string with the empty string does not change that string Regular Expressions (REs) – p.14/37

  39. βœ„ οΏ½ οΏ½ ☎ οΏ½ ✁ οΏ½ οΏ½ βœ„ ✁ οΏ½ οΏ½ ☎ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ✁ οΏ½ ☎ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ ✁ οΏ½ οΏ½ Note . Example: if then ; Regular Expressions (REs) – p.15/37

  40. ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ οΏ½ ✁ ☎ οΏ½ οΏ½ βœ„ οΏ½ ☎ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ οΏ½ οΏ½ οΏ½ ✁ οΏ½ ✁ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ✁ οΏ½ ☎ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ οΏ½ οΏ½ ☎ βœ„ οΏ½ οΏ½ ✁ οΏ½ ☎ βœ„ Note . Example: if then ; . Example: if then but Regular Expressions (REs) – p.15/37

  41. οΏ½ Application REs are useful tools for the design of compilers Regular Expressions (REs) – p.16/37

  42. οΏ½ οΏ½ Application REs are useful tools for the design of compilers Language lexicon is described by REs. Regular Expressions (REs) – p.16/37

  43. ✟ ✝ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ† βœ‚ οΏ½ οΏ½ ☎ ✁ οΏ½ βœ„ βœ„ βœ„ βœ„ βœ† βœ† ✟ βœ„ βœ„ βœ‚ οΏ½ ✟ βœ„ ✁ ☎ βœ„ βœ† ✝ βœ† βœ‚ οΏ½ οΏ½ οΏ½ βœ† ✁ οΏ½ οΏ½ βœ„ Application REs are useful tools for the design of compilers Language lexicon is described by REs. Example: numerical constants can be described by: where ✁ β˜Žβœ„ βœ‚ β˜Žβœ„ ☎ β˜Žβœ„ ✞ β˜Žβœ„ Regular Expressions (REs) – p.16/37

  44. οΏ½ οΏ½ ✁ οΏ½ ✟ οΏ½ οΏ½ βœ† βœ‚ βœ„ ☎ οΏ½ ✁ οΏ½ βœ„ βœ„ βœ„ ✝ βœ„ βœ„ βœ† ✟ βœ† βœ„ οΏ½ οΏ½ βœ‚ ✟ ✁ ☎ βœ„ βœ† ✝ βœ† βœ‚ οΏ½ οΏ½ οΏ½ βœ† ✁ οΏ½ οΏ½ βœ„ Application REs are useful tools for the design of compilers Language lexicon is described by REs. Example: numerical constants can be described by: where ✁ β˜Žβœ„ βœ‚ β˜Žβœ„ ☎ β˜Žβœ„ ✞ β˜Žβœ„ From the lexicon description by REs one can generate automatically lexical analyzers Regular Expressions (REs) – p.16/37

  45. οΏ½ βœ„ οΏ½ οΏ½ ☎ βœ„ οΏ½ οΏ½ ☎ New terminology A RE that evaluates to the language is further said that specifi es the language Regular Expressions (REs) – p.17/37

  46. οΏ½ Equivalence with FA REs and fi nite automata are equivalentin their descriptive power Regular Expressions (REs) – p.18/37

  47. οΏ½ οΏ½ οΏ½ οΏ½ Equivalence with FA REs and fi nite automata are equivalentin their descriptive power Any RE can be converted into a fi nite automaton, βœβœ„βœ‚ , that recognizes the language specifi ed by Regular Expressions (REs) – p.18/37

  48. οΏ½ οΏ½ ✁ ✁ οΏ½ οΏ½ οΏ½ Equivalence with FA REs and fi nite automata are equivalentin their descriptive power Any RE can be converted into a fi nite automaton, βœβœ„βœ‚ , that recognizes the language specifi ed by Vice-versa, any fi nite automaton recognizing a language can be converted into a RE that οΏ½ ✁� specifi es the language Regular Expressions (REs) – p.18/37

  49. Theorem 1.54 Language is regular iff some RE specifi es it Regular Expressions (REs) – p.19/37

  50. Theorem 1.54 Language is regular iff some RE specifi es it Proof idea: this proof has two parts: Regular Expressions (REs) – p.19/37

  51. οΏ½ Theorem 1.54 Language is regular iff some RE specifi es it Proof idea: this proof has two parts: First part: we show that a language specified by a RE is regular, i.e., there is a finite automaton that recognizes it. Regular Expressions (REs) – p.19/37

  52. οΏ½ οΏ½ Theorem 1.54 Language is regular iff some RE specifi es it Proof idea: this proof has two parts: First part: we show that a language specified by a RE is regular, i.e., there is a finite automaton that recognizes it. Second part: we show that if a language is regular then there is a RE that specifies it Regular Expressions (REs) – p.19/37

  53. Lemma 1.55 If a language is specifi ed by a RE, then it is regular. Regular Expressions (REs) – p.20/37

  54. ✁ οΏ½ Lemma 1.55 If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE that evaluate to the language . Regular Expressions (REs) – p.20/37

  55. ✁ οΏ½ οΏ½ οΏ½ Lemma 1.55 If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE that evaluate to the language . 1. We will show how to convert into an NFA that recognizes . Regular Expressions (REs) – p.20/37

  56. οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ Lemma 1.55 If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE that evaluate to the language . 1. We will show how to convert into an NFA that recognizes . 2. Then by corollary 1.40, if an NFA recognizes then is regular Regular Expressions (REs) – p.20/37

  57. οΏ½ οΏ½ Proof Convert into an NFA by the following six-steps proce- dure: Regular Expressions (REs) – p.21/37

  58. βœ„ ✁ βœ‚ ✁ βœ„ βœ‚ βœ‚ ☎ οΏ½ οΏ½ ☎ ✁ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ οΏ½ οΏ½ βœ„ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ Step 1: If , then and the NFA recognizing is in Figure 1 Figure 1: NFA recognizing Regular Expressions (REs) – p.22/37

  59. οΏ½ ✁ βœ‚ ✁ βœ„ βœ‚ βœ‚ ☎ οΏ½ ☎ βœ„ ✁ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ ☎ οΏ½ οΏ½ βœ„ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ Step 1: If , then and the NFA recognizing is in Figure 1 Figure 1: NFA recognizing Note: this is an NFA but not a DFA because it has states with no exiting arrow for each possible input symbol Regular Expressions (REs) – p.22/37

  60. οΏ½ ✁ οΏ½ οΏ½ ✁ οΏ½ ☎ οΏ½ οΏ½ οΏ½ βœ‚ ☎ βœ‚ βœ‚ οΏ½ ✁ ☎ ☎ οΏ½ ✁ οΏ½ βœ„ οΏ½ οΏ½ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ ✁ οΏ½ βœ‚ ✁ οΏ½ ☎ ✁ βœ‚ ✁ οΏ½ οΏ½ ✁ οΏ½ οΏ½ βœ‚ ✁ οΏ½ Formal construction οΏ½ ✁� Formally where: , οΏ½ βœβœ„ , for and Regular Expressions (REs) – p.23/37

  61. ☎ ✁ ✝ ✁ οΏ½ βœ‚ βœ‚ ☎ οΏ½ οΏ½ βœ„ οΏ½ ✁ οΏ½ οΏ½ οΏ½ ☎ οΏ½ οΏ½ βœ„ οΏ½ οΏ½ οΏ½ Step 2: If then and the NFA that recognizes is in Figure 2 Figure 2: The NFA recognizing Regular Expressions (REs) – p.24/37

  62. οΏ½ οΏ½ ☎ ✁ ✁ βœ‚ βœ„ ☎ ✁ ☎ ✁ ✁ οΏ½ οΏ½ ✁ βœ‚ ✁ οΏ½ ✁ ✁ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ ☎ Formal construction οΏ½ ✁� οΏ½ ✁� Formally where: οΏ½ βœβœ„ for any and Regular Expressions (REs) – p.25/37

  63. οΏ½ ☎ βœ„ ☎ βœ‚ ☎ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ οΏ½ ✁ οΏ½ οΏ½ οΏ½ Step 3: If then , and the NFA that recognizes is in Figure 3 Figure 3: he NFA recognizing Regular Expressions (REs) – p.26/37

  64. ✁ ✁ ✁ οΏ½ βœ‚ ✁ ☎ βœ„ ✁ οΏ½ βœ‚ ☎ ✁ οΏ½ ✁ ✁ ☎ οΏ½ οΏ½ οΏ½ ☎ Formal construction οΏ½ ✁� Formally where: οΏ½ βœβœ„ for any and Regular Expressions (REs) – p.27/37

  65. ☎ οΏ½ ✁ ☎ οΏ½ οΏ½ βœ„ οΏ½ βœ‚ οΏ½ βœ„ οΏ½ ☎ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ„ Step 4: If then . οΏ½ ✁� Regular Expressions (REs) – p.28/37

  66. βœ„ οΏ½ βœ„ ✁ ☎ βœ‚ οΏ½ ☎ οΏ½ ☎ οΏ½ οΏ½ βœ„ οΏ½ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ Step 4: If then . οΏ½ ✁� Note: in view with the inductive nature of we may assume that: Regular Expressions (REs) – p.28/37

  67. οΏ½ οΏ½ οΏ½ οΏ½ βœ„ βœ‚ ✁ ☎ ✁ ☎ οΏ½ βœ‚ ✁ βœ„ βœ‚ βœ„ ✁ ✁ βœ„ βœ„ ✁ ☎ οΏ½ βœ‚ βœ„ οΏ½ οΏ½ οΏ½ οΏ½ ✁ οΏ½ βœ‚ βœ‚ οΏ½ οΏ½ οΏ½ ☎ οΏ½ βœ„ οΏ½ ✁ ☎ ✁ βœ„ βœ„ Step 4: If then . οΏ½ ✁� Note: in view with the inductive nature of we may assume that: 1. is an NFA recognizing Regular Expressions (REs) – p.28/37

  68. ✁ οΏ½ βœ‚ βœ„ ✁ βœ‚ βœ„ βœ„ ✁ βœ„ βœ‚ ✁ βœ‚ βœ‚ βœ„ οΏ½ ✁ ✁ βœ‚ βœ„ ☎ βœ‚ ☎ οΏ½ οΏ½ βœ‚ βœ„ βœ‚ βœ„ ✁ βœ‚ βœ„ βœ„ οΏ½ βœ‚ οΏ½ οΏ½ οΏ½ οΏ½ οΏ½ ✁ οΏ½ βœ‚ βœ‚ βœ„ οΏ½ οΏ½ ☎ οΏ½ βœ„ βœ‚ οΏ½ οΏ½ ☎ ✁ ☎ βœ‚ βœ„ οΏ½ οΏ½ ☎ ☎ βœ‚ οΏ½ οΏ½ βœ„ ✁ βœ„ Step 4: If then . οΏ½ ✁� Note: in view with the inductive nature of we may assume that: 1. is an NFA recognizing 2. is an NFA recognizing Regular Expressions (REs) – p.28/37

  69. ✁ βœ„ ☎ οΏ½ βœ‚ ✁ οΏ½ οΏ½ βœ„ ☎ βœ‚ οΏ½ βœ‚ βœ„ ✁ βœ„ βœ„ βœ‚ οΏ½ ✁ βœ‚ οΏ½ βœ„ βœ‚ βœ‚ βœ‚ βœ‚ βœ„ βœ„ οΏ½ βœ„ βœ‚ ✁ βœ„ βœ‚ βœ„ βœ‚ ✁ βœ„ βœ‚ βœ„ οΏ½ βœ„ οΏ½ ☎ οΏ½ οΏ½ ☎ ☎ βœ‚ οΏ½ ✁ οΏ½ οΏ½ οΏ½ οΏ½ βœ‚ ✁ βœ‚ οΏ½ βœ„ ✁ οΏ½ οΏ½ ☎ ✁ ☎ οΏ½ βœ„ οΏ½ ✁ οΏ½ ☎ βœ‚ οΏ½ οΏ½ οΏ½ Step 4: If then . οΏ½ ✁� Note: in view with the inductive nature of we may assume that: 1. is an NFA recognizing 2. is an NFA recognizing The NFA recognizing is given in Figure 4 Regular Expressions (REs) – p.28/37

  70. ✝ βœ‚ οΏ½ οΏ½ βœ„ οΏ½ ☎ βœ‚ βœ„ βœ„ βœ„ βœ„ βœ„ βœ„ οΏ½ ✝ βœ‚ ✁ βœ‚ βœ‚ βœ„ βœ„ βœ„ βœ‚ βœ‚ βœ„ βœ„ βœ„ βœ‚ ✁ βœ‚ ✁ ✁ NFA recognizing βœ‚βœ†β˜Ž Figure 4: Construction of to recognize Regular Expressions (REs) – p.29/37

  71. βœ‚ ✁ ☎ βœ‚ ☎ βœ„ βœ„ οΏ½ ✁ οΏ½ ☎ ✁ βœ‚ βœ„ βœ„ ✁ ☎ οΏ½ ✁ Construction procedure 1. : That is, the states of are all states on and with the addition of a new state Regular Expressions (REs) – p.30/37

Recommend

Objectives You should be able to ... Regular Languages Use the syntax of regular expressions

Objectives You should be able to ... Regular Languages Use the syntax of regular expressions

Objectives Regular Expressions Syntax of Regular Expressions Objectives Regular Expressions Syntax of Regular Expressions Objectives You should be able to ... Regular Languages Use the syntax of regular expressions to model a given set

447 views β€’ 3 slides
Regexp Lecture 26: Regular Expressions Regular Expressions Regular expressions are a small

Regexp Lecture 26: Regular Expressions Regular Expressions Regular expressions are a small

Regexp Lecture 26: Regular Expressions Regular Expressions Regular expressions are a small programming language over strings Regex or regexp are not unique to Python They let us to succinctly and compactly represent classes of strings In this

741 views β€’ 29 slides
Regular Expressions Regular Expressions and Automata and Automata Berlin Chen 2003 References:

Regular Expressions Regular Expressions and Automata and Automata Berlin Chen 2003 References:

Regular Expressions Regular Expressions and Automata and Automata Berlin Chen 2003 References: 1. Speech and Language Processing, chapter 2 1 Introduction Regular Expressions (REs) Finite-State Automata (FSAs) Formal

630 views β€’ 52 slides
C++0x Regular Expressions Simon Andreas Frimann Lund Datalogisk Institut Kbenhavns

C++0x Regular Expressions Simon Andreas Frimann Lund Datalogisk Institut Kbenhavns

Regular Expressions C++0x Sources C++0x Regular Expressions Simon Andreas Frimann Lund Datalogisk Institut Kbenhavns Universitet Maj 16, 2008 Regular Expressions C++0x Sources Regular Expressions Regular Expression, regex or

550 views β€’ 22 slides
Regular Expressions = Regular Languages Mark Greenstreet, CpSc 421, Term 1, 2008/09 17

Regular Expressions = Regular Languages Mark Greenstreet, CpSc 421, Term 1, 2008/09 17

Regular Expressions = Regular Languages Mark Greenstreet, CpSc 421, Term 1, 2008/09 17 September 2008 p.1/18 Lecture Outline Regular Expressions Regular Expresssions Equivalence of Regular Expressions and Finite Automata 17

633 views β€’ 24 slides
Theory of Computer Science C3. Regular Languages: Regular Expressions, Pumping Lemma Malte

Theory of Computer Science C3. Regular Languages: Regular Expressions, Pumping Lemma Malte

Theory of Computer Science C3. Regular Languages: Regular Expressions, Pumping Lemma Malte Helmert University of Basel March 30, 2016 Regular Expressions Pumping Lemma Minimal Automata (skimmed) Summary Regular Expressions Regular

649 views β€’ 49 slides
Regular Expressions A regular expression describes a language using three operations. Regular

Regular Expressions A regular expression describes a language using three operations. Regular

Regular Expressions A regular expression describes a language using three operations. Regular Expressions A regular expression (RE) describes a language. It uses the three regular operations. These are called union/or , concatenation and star .

455 views β€’ 17 slides
Chapter 7 Expressions and Statements Expressions Arithmetic Expressions Conditional

Chapter 7 Expressions and Statements Expressions Arithmetic Expressions Conditional

Chapter 7 Expressions and Statements Expressions Arithmetic Expressions Conditional Expressions Relational Expressions Logical Expressions Operator Overloading Assignment Statement Control Structure Selection

747 views β€’ 51 slides
Regular Expressions Definitions Equivalence to Finite Automata 1 REs: Introduction

Regular Expressions Definitions Equivalence to Finite Automata 1 REs: Introduction

Regular Expressions Definitions Equivalence to Finite Automata 1 REs: Introduction Regular expressions are an algebraic way to describe languages. They describe exactly the regular languages. If E is a regular expression, then

840 views β€’ 26 slides
Regular Expressions in Python L435/L555 Dept. of Linguistics, Indiana University Fall 2016 1 /

Regular Expressions in Python L435/L555 Dept. of Linguistics, Indiana University Fall 2016 1 /

Regular Expressions in Python RE Module Basics of REs RE functions Regular Expressions in Python L435/L555 Dept. of Linguistics, Indiana University Fall 2016 1 / 9 Regular expression module Regular Expressions in Python RE Module

205 views β€’ 9 slides
CSc 337 LECTURE 26: REGULAR EXPRESSIONS AND SECURITY Regular expressions in JavaScript var str =

CSc 337 LECTURE 26: REGULAR EXPRESSIONS AND SECURITY Regular expressions in JavaScript var str =

CSc 337 LECTURE 26: REGULAR EXPRESSIONS AND SECURITY Regular expressions in JavaScript var str = "The rain in SPAIN stays mainly in the plain"; var res = str.match(/ain/g); - The match function takes a regular expression as a

412 views β€’ 19 slides
Kleene Algebras: The Algebra of Regular Expressions Adam Braude University of Puget Sound May

Kleene Algebras: The Algebra of Regular Expressions Adam Braude University of Puget Sound May

Kleene Algebras: The Algebra of Regular Expressions Kleene Algebras: The Algebra of Regular Expressions Adam Braude University of Puget Sound May 5, 2020 Kleene Algebras: The Algebra of Regular Expressions Regular Expressions Motivation

559 views β€’ 17 slides
CS/COE 1520 pitt.edu/~ach54/cs1520 Regular expressions Regular expressions Formally:

CS/COE 1520 pitt.edu/~ach54/cs1520 Regular expressions Regular expressions Formally:

CS/COE 1520 pitt.edu/~ach54/cs1520 Regular expressions Regular expressions Formally: Expressions that can be generated by regular languages, or that can be produced by a finite automaton Practically speaking: Patterns

233 views β€’ 19 slides
Regular Expressions in .NET Regular Expressions in .NET By: Nasser Alshammari College of

Regular Expressions in .NET Regular Expressions in .NET By: Nasser Alshammari College of

Regular Expressions in .NET Regular Expressions in .NET By: Nasser Alshammari College of Science, Department of Computer Science Old Dominion University Outline Outline Regular Expressions? Why do we need them? RE language RE

353 views β€’ 19 slides
Regular Expressions for Linguists: A Life Skill . Michael Yoshitaka Erlewine mitcho@mitcho.com

Regular Expressions for Linguists: A Life Skill . Michael Yoshitaka Erlewine mitcho@mitcho.com

. Regular Expressions for Linguists: A Life Skill . Michael Yoshitaka Erlewine mitcho@mitcho.com Hackl Lab Turkshop March 2013 . . Regular Expressions What are regular expressions? Regular Expressions (aka regex es or regexp s) are a

382 views β€’ 15 slides
Regular Expressions Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of

Regular Expressions Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of

Objectives Regular Expressions Syntax of Regular Expressions Conversion to Right Linear Grammar Regular Expressions Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of Computer Science Objectives Regular Expressions

315 views β€’ 19 slides
Expressions, and Arithmetic Chapters 4 1 2 Arithmetic Operator Precedence Just as in

Expressions, and Arithmetic Chapters 4 1 2 Arithmetic Operator Precedence Just as in

9/9/2012 For Next Time Read Chapter 4 Expressions, and Arithmetic Chapters 4 1 2 Arithmetic Operator Precedence Just as in normal mathematics: Operator Operation * and / are applied before + and + Addition Parentheses

156 views β€’ 3 slides
Unit 2 Integer Operations 2.2 Summary of Unit 1 memorize powers of 2 starts with 8-F if

Unit 2 Integer Operations 2.2 Summary of Unit 1 memorize powers of 2 starts with 8-F if

2.1 Unit 2 Integer Operations 2.2 Summary of Unit 1 memorize powers of 2 starts with 8-F if negative base-2 Unsigned 0x 9 3 = +147 Base 16 1 0 0 1 0 0 1 1 = +147 Unsigned Base 2 128 64 32 16 8 4 2 1 1 0 0 1 0 0

849 views β€’ 61 slides
Skills & Outcomes You should know and be able to apply the following skills with

Skills & Outcomes You should know and be able to apply the following skills with

2.1 2.2 Skills & Outcomes You should know and be able to apply the following skills with confidence Unit 2 Perform addition & subtraction in unsigned & 2's complement system Integer Operations Determine if overflow has

465 views β€’ 14 slides
ARITHMETIC OPERATIONS WE HAVE THE DATA, WHAT NOW? Operations in C Bit-wise boolean operations

ARITHMETIC OPERATIONS WE HAVE THE DATA, WHAT NOW? Operations in C Bit-wise boolean operations

ARITHMETIC OPERATIONS WE HAVE THE DATA, WHAT NOW? Operations in C Bit-wise boolean operations Logical operation Arithmetic operations 2 BOOLEAN ALGEBRA Algebraic representation of logic Encode True as 1 and False

597 views β€’ 31 slides
Constructing the Integers Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech

Constructing the Integers Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech

Introduction Equivalence Classes Arithmetic Operations Properties Constructing the Integers Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Constructing the Integers Introduction

1.45k views β€’ 126 slides
Operators Lecture 3 COP 3014 Fall 2018 January 15, 2019 Operators Special built-in symbols

Operators Lecture 3 COP 3014 Fall 2018 January 15, 2019 Operators Special built-in symbols

Operators Lecture 3 COP 3014 Fall 2018 January 15, 2019 Operators Special built-in symbols that have functionality, and work on operands operand an input to an operator Arity - how many operands an operator takes unary

402 views β€’ 15 slides
Systems with Generic Operations Previous section: designed systems in which data Topic 15

Systems with Generic Operations Previous section: designed systems in which data Topic 15

Systems with Generic Operations Previous section: designed systems in which data Topic 15 objects can be represented in more than one way Key idea: link the code that specifies the data Generic Arithmetic Operations operations to the

379 views β€’ 6 slides
Arithmetic of Extension Fields of Small Characteristics Recent Developments Abhijit Das

Arithmetic of Extension Fields of Small Characteristics Recent Developments Abhijit Das

Arithmetic of Extension Fields of Small Characteristics Recent Developments Abhijit Das Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Indo-US Workshop Indian Statistical Institute, Calcutta January

187 views β€’ 17 slides

More recommend

Explore More Topics

Stay informed with curated content and fresh updates.

animals pets art culture automotive transportation business finance computer internet construction architecture education-career electronics communication

Logo Sambuz

Unleash a World of Digital Possibilitiesβ€”Browse, Share, and Explore Content Without Boundaries

Useful Links

About Us Privacy Services FAQ's Contact

Newsletter

Stay Informed & Inspiredβ€”Subscribe for the Latest Updates, Tips, and Exclusive Content Directly to Your Inbox.

Mail Us

mail@sambuz.com

2026 SAMBUZ, All right reserved.