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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 12: Context-Free Grammars

CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 12: Context-Free Grammars. Chapter 12: Context-Free Grammar I. Theory of Automata  II. Theory of Formal Languages III. Theory of Turing Machines …. Chapter 12: Context-Free Grammars. programming languages

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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 12: Context-Free Grammars

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  1. CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 12: Context-Free Grammars Chapter 12: Context-Free Grammar I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines … Zaguia/Stojmenovic

  2. Chapter 12: Context-Free Grammars • programming languages • compiling a program: an operation that generates an equivalent program in machine or assembler language. • 2 phases: • parsing  • translation to machine language Zaguia/Stojmenovic

  3. Chapter 12: Context-Free Grammars Example: AE (Arithmetic Expressions) • Rule 1: Any number is in AE • Rule 2: If x and y are in AE, then so are: (x) – (x) (x+y) (x–y) (x*y) A different way for defining the set AE is to use a set of substitutions rules similar to the grammatical rules: Zaguia/Stojmenovic

  4. Chapter 12: Context-Free Grammars Substitution rules that define the AE’s: S  AE AE  (AE + AE) AE  (AE–AE) AE  (AE*AE) AE  (AE) AE –(AE) AE  NUMBER NUMBERS?? NUMBER  FIRST-DIGIT FIRST-DIGIT  FIRST-DIGIT OTHER-DIGIT FIRST-DIGIT  1 2 3 4 5 6 7 8 9 OTHER-DIGIT  0 1 2 3 4 5 6 7 8 9 Zaguia/Stojmenovic

  5. Chapter 12: Context-Free Grammars S  AE  (AE*AE)  ((AE+AE)*AE)  ((AE+AE)*(AE+AE)) …  ((3+4)*(6+7)) AE: S  AE AE  (AE + AE) AE  (AE–AE) AE  (AE*AE) AE  (AE) AE –(AE) AE  NUMBER How to generate the number 1066? NUMBER  FIRST-DIGIT  FIRST-DIGIT OTHER-DIGIT  FIRST-DIGIT OTHER-DIGIT OTHER-DIGIT  FIRST-DIGIT OTHER-DIGIT OTHER-DIGIT OTHER-DIGIT  1 0 6 6 NUMBERS NUMBER  FIRST-DIGIT FIRST-DIGIT  FIRST-DIGIT OTHER-DIGIT FIRST-DIGIT  1 2 3 4 5 6 7 8 9 OTHER-DIGIT  0 1 2 3 4 5 6 7 8 9 Zaguia/Stojmenovic

  6. Chapter 12: Context-Free Grammars Definition: A context free grammar (CFG) is: • an alphabet S of letters, called terminals. • a set of symbols, called nonterminals or variables. One symbol S is called the start symbol. • a finite set of productions of the form: A  a where A is a nonterminal and a is a finite sequence (word) of nonterminals and terminals. Zaguia/Stojmenovic

  7. EA: S  AE AE  (AE + AE) AE  (AE–AE) AE  (AE*AE) AE  (AE) AE –(AE) AE  NUMBER Chapter 12: Context-Free Grammars • Examples: Terminals: (, ), +, -, *, numbers Nonterminals: S, AE NUMBERS NUMBER  FIRST-DIGIT FIRST-DIGIT  FIRST-DIGIT OTHER-DIGIT FIRST-DIGIT  1 2 3 4 5 6 7 8 9 OTHER-DIGIT  0 1 2 3 4 5 6 7 8 9 Terminals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Nonterminals: S, FIRST-DIGIT, OTHER-DIGIT Zaguia/Stojmenovic

  8. Chapter 12: Context-Free Grammars Definition: A sequence of applications of productions starting with the start symbol and ending in a sequence of terminals is called a derivation. Definition:The language generated by a CFG is the set of all sequences of terminals produced by derivations. We also say language defined by, language derived from, or language produced by the CFG. Definition: A language generated by a CFG is called a context-free language. Zaguia/Stojmenovic

  9. Chapter 12: Context-Free Grammars Examples: S  aS S Λ S  aS  aaS  aaaS  aaaaS  aaaaaS  aaaaaaS  aaaaaaΛ = aaaaaa Generated Language: {Λ, a, aa, aaa, …} = language(a*). Zaguia/Stojmenovic

  10. Chapter 12: Context-Free Grammars S  SS S  a S Λ S  SS  SSS  SaS  SaSS  ΛaSS  ΛaaS  ΛaaΛ=aa (An infinite number of derivations for the word aa.) Generated Language: {Λ, a, aa, aaa, …} = language(a*). Zaguia/Stojmenovic

  11. Chapter 12: Context-Free Grammars In general: variables: Upper case letters terminals: Lower case letters The empty word: is it a nonterminal? L... no a terminal? LaaL = aa not exactly, because it is erased. N L N can simply be deleted. Zaguia/Stojmenovic

  12. Chapter 12: Context-Free Grammars • S  aS S  bS S  a S  b S  aS  abS  abbS  abba • S  X S  Y X ΛY  aY Y bY Y  a Y  b • S  aS S  bS S  a S  b S Λ S  aS  abS  abbS  abbaS  abba • S  aS S  bS S Λ Zaguia/Stojmenovic

  13. Chapter 12: Context-Free Grammars S  XaaX X  aX X  bX X Λ S  XaaX  aXaaX  abXaaX  abXaabX  abaab How many derivations are possible for the word baabaab? S  XY X  aX X  bX X  a Y  Ya Y Yb Y  a S  XY X  aX | bX | a Y  Ya | Yb | a Abbreviation: Zaguia/Stojmenovic

  14. Chapter 12: Context-Free Grammars S  SS | ES | SE | Λ | DSD E  aa | bb D  ab | ba EVEN-EVEN=language([aa+bb+(ab + ba)(aa+bb)*(ab+ba)]*) S  aSb | Λ S aSb  aaSbb  aaaSbbb  aaaaSbbbb  aaaaaSbbbbb  aaaaabbbbb S  aSa | bSb | Λ ? Zaguia/Stojmenovic

  15. S S S S A A A A A A A A b A A A A b A A A A b A A A A b A b a A a b A b a A a a a Chapter 12: Context-Free Grammars S  AA A  AAA | bA | Ab | a Parse Trees for the word bbaaaab bbaaaab Zaguia/Stojmenovic

  16. Chapter 12: Context-Free Grammars Parse trees are also called syntax trees, generation trees, production trees, or derivation trees. Remark: In a parse tree every internal nodes is labelled with a variable (nonterminal) and every leaf is labelled with a terminal. Zaguia/Stojmenovic

  17. S S S S * S S + + 5 S S * S S 3 3 4 4 5 Chapter 12: Context-Free Grammars Example: S  (S+S) | (S*S) | NUMBER NUMBER  … S (S+S)  (S+(S*S))  …  (3+(4*5)) S (S*S)  ((S+S)*S)  …  ((3+4)*5) Zaguia/Stojmenovic

  18. S S S S 3 + S S + 20 3 + * 5 4 * S S 3 S 4 5 S S * S S + 5 S S S 5 * 7 5 * 3 4 + 4 3 Chapter 12: Context-Free Grammars    23    35 Zaguia/Stojmenovic

  19. S S S S + S + * S S * 3 + 3 * 4 5 3 5 4 5 4 Chapter 12: Context-Free Grammars Lukasiewicz (Prefix) Notation +3*45 Zaguia/Stojmenovic

  20. S S S * S S + 5 S S * * 3 4 + 5 + 5 4 3 4 3 Chapter 12: Context-Free Grammars *+345 Zaguia/Stojmenovic

  21. S * + 6 5 * + + 2 1 4 3 Chapter 12: Context-Free Grammars * + * + 1 2 + 3 4 5 6 + 1 2  * + * 3 + 3 4 5 6 + 3 4  * + * 3 7 5 6 * 3 7  * + 21 5 6 + 21 5  * 26 6 * 26 6  156 Zaguia/Stojmenovic

  22. S B A a b S a a S a a S b Chapter 12: Context-Free Grammars Example:S  AB A  a B  b S AB  aB  ab S AB  Ab  ab • A CFG is ambiguous if there is at least one word in the language that has at least two derivation trees. It is called unambiguous otherwise. • Example: S  aSa S  bSb S  a S  b S L S aSa  aaSaa  aabaa Zaguia/Stojmenovic

  23. S S S S S a a S S a a S S a S a a S S a a S S a S a a a a a Chapter 12: Context-Free Grammars Language(a+) Example:S  aS | Sa | a Example:S  aS | a Zaguia/Stojmenovic

  24. S aa bX aXX aabX bab bb abX aXab aXb aabab aabb abab abb aabab abab aabb abb Chapter 12: Context-Free Grammars Total Language Trees S  aa | bX | aXX X  ab | b Zaguia/Stojmenovic

  25. aSb bS a S aaSbb abSb aab baSb bbS ba  aaaSbbb aabSbb aaabb aaaSbbb aaaSbbb bba . . . . . . . . Chapter 12: Context-Free Grammars Total language tree is infinite since S can call itself recursively S  aSb | bS | a Zaguia/Stojmenovic

  26. S b X aX aaX aaaX . . Chapter 12: Context-Free Grammars S  X | b X  aX Total language tree is infinite but language is finite Zaguia/Stojmenovic

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