CSI 3104 Winter 2006: Introduction to Formal Languages Chapter 17: ContextFree Languages

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CSI 3104 /Winter 2006 Introduction to Formal
Languages Chapter 17 Context-Free Languages

• Chapter 17 Context-Free Languages
• I. Theory of Automata
• ? II. Theory of Formal Languages
• III. Theory of Turing Machines

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Chapter 17 Context-Free Languages

• Theorem. The set of context-free languages is
  closed under union, concatenation, and Kleene
  closure.
• Union. L1L2
• L1 and L2 are generated by two context-free
  grammars G1 and G2. We replace each nonterminal X
  in G1 by X1, and each nonterminal X in G2 by X2.
  We add the productions
• S ? S1 S ? S2
• L1L2 is the language generated by this new CFG.

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Chapter 17 Context-Free Languages

• Example L1 PALINDROME S ? aSa bSb a
  b ?
• L2 anbn S ? aSb ?
• L1L2 S ? S1 S2
• S1 ? aS1a bS1b a b
  ?
• S2 ? aS2b ?
• Example L1 aa,bb S ? aA bB A ? a
  B ? b
• L2 ? S ? ?
• L1L2 S ? S1 S2
• S1 ? aA1 bB1 A1 ?
  a B1 ? b
• S2 ? ?

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Chapter 17 Context-Free Languages
Proof by machines
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Chapter 17 Context-Free Languages
(begin with a)
(contain aa)
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Chapter 17 Context-Free Languages
L1 L2
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Chapter 17 Context-Free Languages

• Concatenation. L1L2
• Similar to union except we add
• S ? S1S2
• Example
• L1 palindrome S ? aSa bSb a b ?
• L2 anbn S ? aSb ?
• L1L2 S ? S1S2
• S1 ? aS1a bS1b a b ?
• S2 ? aS2b ? proof by PDAs?

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L

• Kleene star. L
• We replace S by S1 and add
• S ? S1S ?
• S ? S1S ? S1S1S ? S1S1S1S ?
• Example L S ? aSa bSb a b ?
• L S ? S1S ?
• S1 ? aS1a bS1b a b ?

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Chapter 17 Context-Free Languages

• Theorem. The intersection of two context-free
  languages may or may not be context-free.
• L1 anbnam
• S ? XA
• X ? aXb / ab
• A ? aA / a
• L2 anbmam
• S ? AX
• X ? bXa / ba
• A ? aA / a
• L1 n L2 anbnan (is not a context-free
  language )

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Chapter 17 Context-Free Languages

• Theorem. The intersection of a context-free
  language and a regular language is always
  context-free.
• Proof By constructive algorithm using pushdown
  automata, similar to the intersection algorithm
  for two finite automata.
• (details not covered)

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Example use the theorem

• L doubleword ww abbabb,
• Assume L is CFL
• L n abab anbmanbm
• is then context free (n regular lang.)
• But we have proven that anbmanbm is not CFL
• ? contradiction.

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Chapter 17 Context-Free Languages

• Theorem. The complement of a context-free
  language may or may not be context-free.
• When the PDA is deterministic with other
  properties, we could use for the complement a
  similar technique as for FA

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Chapter 17 Context-Free Languages
PALINDROME-X
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Chapter 17 Context-Free Languages
For the complement We interchange the states
ACCEPT and REJECT. It does not work all the
time.
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Complement of a CFL may not be a CFL

• Proof by indirect arguments
• Suppose complement of every CFL is a CFL.
• L1 and L2 CFLs ? L1 and L2 CFLs ? L1L2 CFLs
  ? (L1L2) L1 ? L2 is CFL
• Contradicts intersection theorem.

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Nondeterminism

• L accepts w if some paths lead to accept (some
  paths may lead to reject)
• L rejects w if all paths lead to reject
• Exchange accept and reject states
• L rejects w if some paths lead to reject ??
• L accepts w if all paths lead to accept ??
• Nondeterminism does not support complement

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Example

• Lanbnan is not CFL but L is CFL
• LMpqMqpMprMrpMqrMrqM
• Mpqapbqar pgtq aanbna CFL
• Mqpapbqar qgtp anbnba CFL
• Mprapbqar pgtr aanban CFL
• Mqr, Mrq, Mrp similarly CFL
• Mabc complement of a regular language

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Deterministic PDA lt Nondeterministic PDA

• Proof Complement of anbncn is CFL and cannot
  have deterministic PDA since otherwise its
  complement has deterministic PDA and would be CFL
  (accept ?reject )