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Why do we care about regular languages?

Why do we care about regular languages?. Programs are composed of tokens: Identifier Number Keyword Special symbols Each of these can be defined by regular grammars . Problem: How do we handle multiple symbol operators (e.g., ++ in C, =+ in C, := in Pascal)? ?? -multiple final states?.

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Why do we care about regular languages?

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  1. Why do we care about regular languages? • Programs are composed of tokens: • Identifier • Number • Keyword • Special symbols • Each of these can be defined by regular grammars. • Problem: How do we handle multiple symbol operators (e.g., ++ in C, =+ in C, := in Pascal)? • ?? -multiple final states?

  2. Sample token classes

  3. Scanner for a language turns out to be a giant NFA for grammar.(i.e., have -rules going from start state to the start state of each token-type on previous slide). integer   identifier  keyword  symbol

  4. Grammars • Grammar: generative description of a language • Automaton: analytical description • Example: programming languages are defined by a grammar (BNF),but recognized with an analytical description (the parser of a compiler), • Language theory establishes links between analytical and generative language descriptions.

  5. Grammars • Grammars express languages • Example: the English language

  6. A derivation of “the boy walks”:

  7. A derivation of “a dog runs”:

  8. Language of the grammar: L = { “a boy runs”, “a boy walks”, “the boy runs”, “the boy walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

  9. Notation Variable or Non-terminal Terminal Production rule

  10. Another Example • Grammar: • Derivation of sentence :

  11. Grammar: • Derivation of sentence :

  12. Other derivations:

  13. Language of the grammar

  14. More Notation • Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

  15. Example • Grammar :

  16. More Notation • Sentential Form: • A sentence that contains • variables and terminals • Example: Sentential Forms sentence

  17. We write: • Instead of:

  18. In general we write: • If:

  19. By default:

  20. Example Grammar Derivations

  21. Example Grammar Derivations

  22. Another Grammar Example • Grammar : Derivations:

  23. More Derivations

  24. Language of a Grammar • For a grammar • with start variable : String of terminals

  25. Example • For grammar : Since:

  26. A Convenient Notation

  27. Linear Grammars

  28. Linear Grammars • Grammars with • at most one variable at the right side • of a production • Examples:

  29. A Non-Linear Grammar Grammar :

  30. Another Linear Grammar • Grammar :

  31. Right-Linear Grammars • All productions have form: • Example: or

  32. Left-Linear Grammars • All productions have form: • Example: or

  33. Regular Grammars

  34. Regular Grammars • A regular grammaris any • right-linear or left-linear grammar • Examples:

  35. Observation • Regular grammars generate regular languages • Examples:

  36. Regular Grammars GenerateRegular Languages

  37. Theorem Languages Generated by Regular Grammars Regular Languages

  38. Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

  39. Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

  40. Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

  41. The case of Right-Linear Grammars • Let be a right-linear grammar • We will prove: is regular • Proof idea: We will construct NFA • with

  42. Grammar is right-linear Example:

  43. Construct NFA such that • every state is a grammar variable: special final state

  44. Add edges for each production:

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