Lecture 22 Pumping Lemma for Context Free Languages

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Lecture 22Pumping Lemma forContext Free
Languages
CSCE 355 Foundations of Computation

• Topics
• Normal forms
• Pumping Lemma for CFLs
• Closure properties

November 19, 2008
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• Last Time
• Useless symbols
• generating symbols,
• useful symbols
• Algorithm for generating and reachable symbols
• Removal of useless symbols
• Removal of epsilon productions
• Removal of unit productions
• Chomsky normal form
• New
• Chomsky normal form
• Chomsky Hierarchy
• Pumping Lemma for Context Free Languages

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• Useless symbols
• generating symbols,
• useful symbols
• Algorithm for generating and reachable symbols
• Removal of useless symbols
• Removal of epsilon productions
• Removal of unit productions
• Chomsky normal form

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Chomsky Normal Form

• A CFG (Context Free Grammar) is in Chomsky Normal
  form if productions are one of the following two
  forms
• A ? BC
• A ? a
• References
• http//www.chomsky.info/

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Conversion to Chomsky Normal Form

• Remove e-productions, unit productions
• A ? BCDE
• A ? abc

• In general
• For each terminal a create a new non-terminal
  Na with Na ? a added as a production
• A ? B1B2Bk create a new non-terminals C1C2Ck
  and replace the production with
• A ? B1C1 and
• Ci ? Bi1Ci1 for i1,k-3
• Ck-2 ? Bk-1Bk

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Example
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Regular Grammars

• A CFG is regular if all productions are of the
  form
• A ? a or
• A ? aB
• Note sentential forms in a derivation based on a
  regular grammar have a unique form!
• What is it ?
• Grammar ? NFA construction
• Create a state for each nonterminal.
• A ? aB means d(A, a) B and
• A ? a means d(A, a) Qfinal and

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Example
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Chomsky Hierarchy

• http//en.wikipedia.org/wiki/Chomsky_hierarchy

Grammar Languages Automaton Production rules (constraints)
Type-0 Recursively enumerable Turing machine a ? ß no restrictions
Type-1 Context-sensitive Linear-bounded non-deterministic Turing machine aAß ? a?ß
Type-2 Context-free Non-deterministic pushdown automaton A ? a
Type-3 Regular DFA A ? a or A ? aB
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Chomsky Hierarchy Venn Diagram
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Backus Naur Form (BNF)

• Backus Naur Form
• N a ß (just a CFG)
• http//en.wikipedia.org/wiki/Backus-Naur_form
• John Backus
• Fortran compiler
• http//en.wikipedia.org/wiki/John_Backus
• Peter Naur
• http//en.wikipedia.org/wiki/Peter_Naur

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Greibach Normal Form

• Each production RHS starts with a terminal
• A ? aa or S? e
• http//en.wikipedia.org/wiki/Greibach_normal_form

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Showing Languages are not CFLs

• Recursive productions
• A ? a A b
• B ? B a b
• D ? aDb d
• A ? a A ß

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Pumping Lemma for CFLs

• Let L be a CFL. Then there exists a constant n
  such that if z is a string in L of length at
  least n, then we can write z uvwxy such that
• vwx lt n
• vx gt 0
• uviwxi y is in L for all i gt 0.

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Idea behind proof

• Assume CNF (or do for L(G)-e)
• Consider Parse Tree
• Sufficiently long string z, means the parse tree
  must be sufficiently big.

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Similarities to Pumping Lemma for Regular
Languages

• Given an arbitrary n.
• Carefully choose z in L (depending on n) with z
  gt n.
• Then for any partition z uvwxy that satisfies
• vx gt 0
• vwx lt n
• We must be able to pump, i.e.
• uviwxiy is in L for all i gt 0

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Example L anbncn n gt 0

• Given L as above, suppose we chose n for the
  Pumping Lemma (for CFLs).
• Choose z
• Consider arbitrary partition of z uvwxy
  satisfying
• vwx lt n
• vx gt 0
• Then show

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Example
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Homework

1. 7.1.4
2. 7.1.3
3. 7.1.6