Lecture 21 Properties of Context Free Languages

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Lecture 21Properties of Context Free Languages

CSCE 355 Foundations of Computation

• Topics
• Normal forms
• Closure properties

November 17, 2008
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• Last Time
• Test 2 not looked at
• Test 2 take-home
• New
• Equivalent Grammars, left factoring (review)
• Normal Forms for Grammars
• Useless symbols generating symbols, useful
  symbols
• Algorithm for generating and reachable symbols
• Epsilon productions unit productions
• Chomsky normal form
• Pumping Lemma for Context Free Languages

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Questions/Comments about Test 2?
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Homework

• 7.1.1
• 7.1.2
• 7.1.3
• 7.1.6

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Recall Equivalent Grammars?
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Left Factoring (recall)
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Useless Symbols

• A grammar symbol X is useful if
• S ? aXß ?w, where w is in T
• X is generating if
• X is reachable if

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Example 7.1

• S ? AB a
• A ? b
• Generating?
• Reachable?
• L(G)?

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Algorithm for Eliminating Useless Symbols

1. Find non-generating symbols
2. Eliminate all productions including
   non-generating symbols
3. Eliminate non-reachable symbols

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Example from 7.1.1

• S ? AB CA
• A ? a
• B ? BC AB
• C ? aB b
• Generating?
• A?
• B?
• C?
• S?
• Reachable?

• Grammar after eliminating non-generating symbols
• Equivalent grammar with no useless symbols

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Theorem 7.2
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Proof of Theorem 7.2 -

• First show G1 has no useless symbols
• Generating?
• Reachable?
• L(G1) is a subset of L(G)
• L(G) is a subset of L(G1)

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Recursive (inductive) algorithms for 1)
generating symbols 2) reachable symbols

• Based on inductive definitions.
• Basis step.
• Inductive step.

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algorithm for generating symbols
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Theorem 7.4 Alg. finds Generating Symbols( and no
more)
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algorithm for reachable symbols
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Theorem 7.6 Alg. finds Reachable Symbols (and no
more)
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Nullable Symbols
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Algorithm for Nullable
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Eliminating Epsilon Productions

• Modify the grammar to produce and equivalent one
  with no epsilon productions
• Note which non-terminals are nullable
• Eliminate epsilon productions
• Then suppose B ? CAD where A is/was nullable

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More Generally

• Consider A ? X1 X2 Xk-1 Xk
• Suppose than M are nullable
• Then we add the 2m productions with all possible
  subsets of the nullable non-terminals omitted
• Exception if m k we dont allow all to be
  eliminated

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Example
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Unit Productions

• E ? E T T
• Replace with
• E ? E T T F F
• But what if we have A ? B, B? C, C ? A
• Unit Pairs

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Unit Production removal

• Find unit pairs
• For each unit pair (A,B) add to P1 all reductions
  of the form A? a, where B? a is a non-unit
  production in P.
• Note AB is possible

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Example 7.12

• E ? ET T
• T ? FT F
• F ? (E) I

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Chomsky Normal Form
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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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Similarities to Pumping Lemma for Regular
Languages
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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