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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 16: Non-Context-Free Languages

CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 16: Non-Context-Free Languages. Chapter 16: Non-Context-Free Languages I. Theory of Automata  II. Theory of Formal Languages III. Theory of Turing Machines …. Chapter 16: Non-Context-Free Languages.

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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 16: Non-Context-Free Languages

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  1. CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 16: Non-Context-Free Languages Chapter 16: Non-Context-Free Languages I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines … Zaguia/Stojmenovic

  2. Chapter 16: Non-Context-Free Languages • Theorem. Let G be a context-free grammar in Chomsky normal form. Let L be the subset of words generated by G which have derivations such that each production of the form: Nonterminal  Nonterminal Nonterminal is used at most once. Then L is a finite set of words. Proof.At each step in a derivation, a nonterminal is replaced by either 2 nonterminals or one terminal. Thus, if there are p productions of the form: Nonterminal  Nonterminal Nonterminal a word in L contains at most p+1 letters. The set of words that contain at most p+1 letters is finite Zaguia/Stojmenovic

  3. Chapter 16: Non-Context-Free Languages Example: S  AB | a A  XY | a Y  SXX  a | b B b • S  AB  aB  ab • S  AB  XYB  bYB  bSXB  baXB  baaB  baab (1) (2) (3) • S  AB  XYB  XSXB  bSXB  baXB  baaB  baab (1) (2) (3) Zaguia/Stojmenovic

  4. S Z A a B B Z A … b S A A C B B B … b Chapter 16: Non-Context-Free Languages a branch: a path between the root and a leaf of a derivation tree S  AZ Z  BB B  ZA A  a B  b S  A Z  aZ  aBB abB  ab Z A… the same nonterminal twice on the same branch (The 2nd is a tree descendant of the 1st. The nonterminal is self-embedded.) S  AA A  BC C  BB A  a B  b S  AA  B CA  bCA b B BA… two different branches Remark: derivation trees are binary Zaguia/Stojmenovic

  5. Chapter 16: Non-Context-Free Languages • Theorem. Let G be a context-free grammar in Chomsky normal form that has p productions of the form: Nonterminal  Nonterminal Nonterminal. Let w be a word such that length(w)2p. Then in every derivation tree for w there exists some nonterminal Z that appears twice on the same branch. Zaguia/Stojmenovic

  6. S X Y A C B D X B C C A Y D A Chapter 16: Non-Context-Free Languages Example 1st raw 20 nodes 2nd raw 21 nodes 3rd raw 22 nodes 4th raw 23 nodes p=3 p+1 rows 2p leaves, maximum Zaguia/Stojmenovic

  7. p+1 nonterminals p nonterminals 1 terminal 1 terminal Chapter 16: Non-Context-Free Languages Proof. A tree that has more than 2p leaves has more than p+1 rows. (A tree that has p+1 rows has at most 2p leaves.) Zaguia/Stojmenovic

  8. S X Y B Y a S X X Y X a Y a X Y B Y b B Y b b a a b Chapter 16: Non-Context-Free Languages S  XY B  b X  BY X  a Y  XY Y  a B  SX Y  b • p=4 • the branch that ends in a has 6 nonterminals (5 applications of productions containing only nonterminals) • at least one production is used more than once Zaguia/Stojmenovic

  9. ………… 1st row S ……… 2nd row X A …… 3rd row a A S …… 4th row a X A …… 5th row S A a …… 6th row b a Chapter 16: Non-Context-Free Languages 26 = 64 Example: PALINDROME – {Λ} S  AX|BY|a|b|AA|BB X  SA Y  SB A  a B  b Zaguia/Stojmenovic

  10. S S X A X A a S A a A S a X A a X A A S a S A a a X A b a S A a b a Chapter 16: Non-Context-Free Languages aabaa aaabaaa Zaguia/Stojmenovic

  11. Chapter 16: Non-Context-Free Languages The pumping lemma (for context-free languages) Let G be a context-free grammar in Chomsky normal form that has p productions of the form: Nonterminal  Nonterminal Nonterminal. Let w be a word such that length(w)2p. Then w can be decomposed into 5 factors: w = uvxyzsuch that • x is not  • at least one of v and y is not  • for all n  1, uvnxynz is in the language generated by G. Zaguia/Stojmenovic

  12. S P u z Q P R v y x Chapter 16: Non-Context-Free Languages • Proof. From the previous theorem, there exists a non terminal P that occurs at least twice on the same branch. • w = uvxyz • x  L • either v  L, or y  L Zaguia/Stojmenovic

  13. S S A a S A a b Chapter 16: Non-Context-Free Languages • Note: u, z, and at least one of v and y could be L. • u = L v = L x = ba y = a z = L Zaguia/Stojmenovic

  14. S P S u z Q P R P v y u z Q Q P R P R v y v y Q Q P R P R v y v y x x Chapter 16: Non-Context-Free Languages uvvxyyz In general: uvnxynz uvvvxyyyz Zaguia/Stojmenovic

  15. S P u z Q P R v y x Algebraic proof • Alternatively: • S  uPz P  vPy P  x • S  uPz  uvPyz uvxyz = w • S  uPz  uvPyz uvvPyyz  uvvvPyyyz …  uvnxynz • w = uvxyz x  L • either v  L, or y  L means generated after several steps Zaguia/Stojmenovic

  16. START PUSH a ACCEPT a PUSH a D READ READ READ POP POP POP b b a D a a a Chapter 16: Non-Context-Free Languages anbnan? Zaguia/Stojmenovic

  17. anbnan is not context free language • p= # live productions:Nonterm.  Nonterm. Nonterm. • Let 3n > 2p by theorem w= anbnan = uvxyz • w contains exactly one abv and y have no ab • Similarly v and y have no ba • v=at or bt, y=aq or bq • v=at y=aq ? v=at y=bq ? v=bt, y=aq ? v=bt, y=bq ? • Problem: two out of three parts an bn an are growing, while the third part does not grow • But it supposed to be always of equal size … • Is anbncn a context free language ? Zaguia/Stojmenovic

  18. Length(vxy)  2p • More precise theorem: live production repeats already within length(vxy) up to 2p • Example: L={anbmanbm} is not context free • Proof: follow similar discussion, four parts: anbmanbm one case is: • v=at grows part 1, y=aq grows part 3 • but for m>2p this case does not exists in the more precise theorem  part imbalance Zaguia/Stojmenovic

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