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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 13: Grammatical Format

CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 13: Grammatical Format. Chapter 13: Grammatical Format I. Theory of Automata  II. Theory of Formal Languages III. Theory of Turing Machines …. Chapter 13: Grammatical Format.

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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 13: Grammatical Format

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  1. CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 13: Grammatical Format Chapter 13: Grammatical Format I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines … Dr. Nejib Zaguia CSI3104-W06

  2. Chapter 13: Grammatical Format Theorem. All regular languages are context-free languages. Proof. We show that for any FA, there is a CFG such that the language generated by the grammar is the same as the language accepted by the FA. By constructive algorithm. Input: a finite automaton. Output: a CF grammar. Dr. Nejib Zaguia CSI3104-W06

  3. a X X Y a Chapter 13: Grammatical Format • The alphabet  of terminals is the alphabet of the FA. • Nonterminals are the state names. (The start state is renamed S.) • For every edge, create a production: X  aY X  aX • For every final state, create a production: • X Λ Dr. Nejib Zaguia CSI3104-W06

  4. b a,b b a b b a a b a a S → S → M → S → S → M → F → F → F S– F+ a M b Chapter 13: Grammatical Format Example: S  aM | bS M  aF | bS F  aF | bF | Λ babbaaba S bS  baM  babS babbSbabbaMbabbaaF  babbaabF babbaabaF babbaaba Dr. Nejib Zaguia CSI3104-W06

  5. Chapter 13: Grammatical Format Definition: A semiword is a sequence of terminals (possibly none) followed by exactly one nonterminal. (terminal)(terminal)…(terminal)(Nonterminal) Definition: A CFG is a regular grammar if all of its productions have the form: Nonterminal  semiword or Nonterminal  word (a sequence of terminals or Λ) Dr. Nejib Zaguia CSI3104-W06

  6. wy wq Nx Np Nz + Chapter 13: Grammatical Format Theorem. All languages generated by regular grammars are regular. Proof. By constructive algorithm. We build a transition graph. • The alphabet  of the transition graph is the set of terminals. • One state for each nonterminal. The state named S is the start state. We add one final state +. • Transitions: Nx wyNz Np wq Dr. Nejib Zaguia CSI3104-W06

  7. aa L (aa+bb)* – + bb wq Np + Chapter 13: Grammatical Format Example: S  aaSS  bbSS L Alternative Algorithm: Np wq whenever wq is not Λ For each transition of the form NΛ, we mark the state for N with +.. Dr. Nejib Zaguia CSI3104-W06

  8. aa,bb aa,bb ab ba X – ab L ba + Chapter 13: Grammatical Format Example S  aaS | bbS|abX | baX | Λ X  aaX | bbX | abS | baS EVEN-EVEN Dr. Nejib Zaguia CSI3104-W06

  9. Chapter 13: Grammatical Format • If the empty word is in the language, a production of the form NΛ(called a Λ-production) is necessary. • The existence of a production of the form N Λ does not necessarily mean that Λ is a part of the language. S  aXS  aX Λ Theorem. Let L be a language generated by a CFG. There exists a CFG without productions of the form X Λ such that: • If L, L is generated by the new grammar. • If L, all words of L except for  are generated by the new grammar. Dr. Nejib Zaguia CSI3104-W06

  10. Proof: For each N Λ and all X  smt1 N smt2 include production X  smt1 smt2 and eliminate N Λ. • Example: S  aSa | bSb | Λ • L-{Λ} generated with • S  aSa | bSb | aa | bb Dr. Nejib Zaguia CSI3104-W06

  11. Problem: Creating new Λ productions • Example: S  a | Xb | aYa X  Y | Λ Y  b | X • S  a | Xb | b | aYa X  Y Y  b | X | Λ • Definition: N nullable if N … Λ • In example, X and Y are nullable Y XΛ • Modified replacement rule • Include X  smt1 smt2 for each • X  smt1 N smt2 and N nullable, • Delete all (old and new) Λ-productions • Ex: S  a | Xb | aYa | b | aa X  Y Y  b | X • How to find all nullable nonterminals in a CFG? Dr. Nejib Zaguia CSI3104-W06

  12. Chapter 13: Grammatical Format Definition. A production of the form: Nonterminal  Nonterminalis called a unit production. Theorem. Let L be a language generated by a CFG that has no Λ-productions. Then there is another CFG without Λ -productions and without unit productions that generates L. Proof: A  B B  S1 | S2 |.. Replace with A  S1 | S2 | … Does it always work ? Example: S  A | bb A  B | b Dr. Nejib Zaguia CSI3104-W06

  13. Modified elimination rule • Whenever A  X1 X2 … B introduce • A  S1|S2|S3 | …for all B  S1|S2|S3 |.. • And eliminate all unit productions • Example: S  A | bb A  B | b B  S | a • S  bb | b | a A  b | a | bb B  a | bb | b Dr. Nejib Zaguia CSI3104-W06

  14. Theorem. Let L be a language generated by a CFG. Then there exists another CFG that generates all words of L (except Λ) such that all the productions are of the form: Nonterminal  sequence of Nonterminals Nonterminal  one terminal • Proof: Introduce Aa Bb … and replace a, b,… with A, B,.. in all productions • Example: SX|XaY|aSb|b XYY|b YaY|aaX • becomes? Dr. Nejib Zaguia CSI3104-W06

  15. Chapter 13: Grammatical Format Definition. A CFG in is said to be in Chomsky Normal Form (CNF) if all the productions have the form: Nonterminal  (Nonterminal)(Nonterminal) Nonterminal  terminal Theorem. Let L be a language generated by a CFG. There there is another grammar which is in CNF that generates all the words of L (except Λ). Dr. Nejib Zaguia CSI3104-W06

  16. CNF construction • Proof: following previous theorem, one can: • Eliminate all Λ-productions • Eliminate all unit productions • Reduce to X  X1X2…Xn and Xterminal • Replace first type, if n>2, with • XX1R1 R1X2R2 R2X3R3 … Rn-3Xn-2Rn-2 Rn-2 Xn-1Xn Dr. Nejib Zaguia CSI3104-W06

  17. Chapter 13: Grammatical Format Definition. In a sequence of terminals and nonterminals in a derivation (called a working string), if there is at least one nonterminal, then the first one is called the leftmost nonterminal. Definition.A leftmost derivation is a derivation where at each step, a production is applied to the leftmost nonterminal in the working string. Example.S  aSX | b X  Xb | a S aSX  aaSXX  aabXX aabXbX aababX aababa Dr. Nejib Zaguia CSI3104-W06

  18. S S X Y X Y X X Y Y X X Y Y X X X X a a a b b a a a b b Chapter 13: Grammatical Format S  XY X  XX | a Y  YY | b Example aaabb Derivation I Derivation II Dr. Nejib Zaguia CSI3104-W06

  19. S X Y X X Y Y X X a a a b b Chapter 13: Grammatical Format S XY  XXY  aXY  aXXY  aaXY  aaaY  aaaYY  aaabY  aaabb Dr. Nejib Zaguia CSI3104-W06

  20. S X Y X X Y Y X X a a a b b Chapter 13: Grammatical Format S XY  XXY  XXXY  aXXY  aaXY  aaaY  aaaYY  aaabY  aaabb Dr. Nejib Zaguia CSI3104-W06

  21. S ) ( S  S S ) ( p S  S S S ~ q p Chapter 13: Grammatical Format Theorem. Any word that is in the language generated by a CFG has a leftmost derivation. Proof: do a preorder traversal in the derivation tree Example.S  (S) | S  S | ~S | p | q S (S)  (S  S)  (p  S)  (p  (S)) (p  (S  S))  (p  (~S  S))  (p  (~p  S))  (p  (~p  q)) Dr. Nejib Zaguia CSI3104-W06

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