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Context-Free Languages

Context-Free Languages. Context-Free Languages. Regular Languages. Context-Free Languages. Context-Free Grammars. Pushdown Automata. stack. automaton. Context-Free Grammars. Grammars. Grammars express languages Example: the English language grammar.

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Context-Free Languages

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  1. Context-Free Languages Costas Busch - RPI

  2. Context-Free Languages Regular Languages Costas Busch - RPI

  3. Context-Free Languages Context-Free Grammars Pushdown Automata stack automaton Costas Busch - RPI

  4. Context-Free Grammars Costas Busch - RPI

  5. Grammars • Grammars express languages • Example: the English language grammar Costas Busch - RPI

  6. Costas Busch - RPI

  7. Derivation of string “the dog walks”: Costas Busch - RPI

  8. Derivation of string “a cat runs”: Costas Busch - RPI

  9. Language of the grammar: L = { “a cat runs”, “a cat sleeps”, “the cat runs”, “the cat sleeps”, “a dog runs”, “a dog sleeps”, “the dog runs”, “the dog sleeps” } Costas Busch - RPI

  10. Productions Sequence of Terminals (symbols) Variables Sequence of Variables Costas Busch - RPI

  11. Another Example Sequence of terminals and variables Grammar: Variable The right side may be Costas Busch - RPI

  12. Grammar: • Derivation of string : Costas Busch - RPI

  13. Grammar: • Derivation of string : Costas Busch - RPI

  14. Grammar: Other derivations: Costas Busch - RPI

  15. Grammar: Language of the grammar: Costas Busch - RPI

  16. A Convenient Notation • We write: • Instead of: for zero or more derivation steps Costas Busch - RPI

  17. In general we write: If: in zero or more derivation steps Trivially: Costas Busch - RPI

  18. Example Grammar Possible Derivations Costas Busch - RPI

  19. Another convenient notation: Costas Busch - RPI

  20. Formal Definitions Grammar: Set of variables Set of terminal symbols Start variable Set of productions Costas Busch - RPI

  21. Context-Free Grammar: All productions in are of the form Variable String of variables and terminals Costas Busch - RPI

  22. Example of Context-Free Grammar productions start variable variables terminals Costas Busch - RPI

  23. Language of a Grammar: • For a grammar with start variable String of terminals or Costas Busch - RPI

  24. Example: context-free grammar : Since, there is derivation for any Costas Busch - RPI

  25. A language is context-free • if there is a context-free grammar • with Context-Free Language: Costas Busch - RPI

  26. Example: is a context-free language since context-free grammar : generates Costas Busch - RPI

  27. Another Example Context-free grammar : Example derivations: Palindromes of even length Costas Busch - RPI

  28. Another Example Context-free grammar : Example derivations: Describes matched parentheses: () ((( ))) (( )) Costas Busch - RPI

  29. Derivation OrderandDerivation Trees Costas Busch - RPI

  30. Derivation Order Consider the following example grammar with 5 productions: Costas Busch - RPI

  31. Leftmost derivation order of string : At each step, we substitute the leftmost variable Costas Busch - RPI

  32. Rightmost derivation order of string : At each step, we substitute the rightmost variable Costas Busch - RPI

  33. Leftmost derivation of : Rightmost derivation of : Costas Busch - RPI

  34. Derivation Trees Consider the same example grammar: And a derivation of : Costas Busch - RPI

  35. yield Costas Busch - RPI

  36. yield Costas Busch - RPI

  37. yield Costas Busch - RPI

  38. yield Costas Busch - RPI

  39. Derivation Tree (parse tree) yield Costas Busch - RPI

  40. Sometimes, derivation order doesn’t matter Leftmost derivation: Rightmost derivation: Give same derivation tree Costas Busch - RPI

  41. Ambiguity Costas Busch - RPI

  42. Grammar for mathematical expressions Example strings: Denotes any number Costas Busch - RPI

  43. A leftmost derivation for Costas Busch - RPI

  44. Another leftmost derivation for Costas Busch - RPI

  45. Two derivation trees for Costas Busch - RPI

  46. take Costas Busch - RPI

  47. Good Tree Bad Tree Compute expression result using the tree Costas Busch - RPI

  48. Two different derivation trees may cause problems in applications which use the derivation trees: • Evaluating expressions • In general, in compilers • for programming languages Costas Busch - RPI

  49. Ambiguous Grammar: A context-free grammar is ambiguous if there is a string which has: two different derivation trees or two leftmost derivations (Two different derivation trees give two different leftmost derivations and vice-versa) Costas Busch - RPI

  50. Example: this grammar is ambiguous since string has two derivation trees Costas Busch - RPI

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