Module 32

1
Module 32

• Chomsky Normal Form (CNF)
• 4 step process

2
Chomsky Normal Form

• A CFG is in Chomsky normal form (CNF) if every
  production is one of these two types
• A ? BC
• A ? a
• Key ideas
• Eliminating ?-productions (e.g. S ? ?)
• Eliminating unit productions (e.g. A ? B)

3
Nullable Variables

• A variable A in a CFG G (V, S, S, P) is defined
  as nullable if
• Base case P contains the production A ? ?
• Recursive case P contains the production
• A ? B1B2 Bn and B1 through Bn are nullable
• No other variables are nullable

4
Finding Nullable Variables

• Initialize N0 to be the set of nullable variables
  by the base case definition
• i 0
• do
• i i1
• Ni Ni-1 union A P contains A ? a where a is
  a string in Ni-1
• while Ni ? Ni-1
• The final Ni is the set of nullable variables

5
Eliminating ?-productions

• Given CFG G (V, S, S, P), construct a CFG G1
  (V, S, S, P1) as follows.
• Initialize P1 P
• Find set of all nullable variables N in G
• For every production A ? a in P, add to P1 every
  production that can be obtained from this one by
  deleting from a one or more of the occurrences of
  nullable variables in a
• Example A ? BBCD where B and C are nullable
  leads to
• A ? BCD BCD BBD BD BD CD D
• Clean up
• Delete all ?-productions from P1
• Delete any duplicate productions
• Delete any productions of the form A ? A
• Thm L(G1) L(G) ?

6
A-derivable Variables

• B is A-derivable in a CFG G if and only if
• A gtG B
• Recursive definition A variable B in a CFG G
  (V, S, S, P) is defined as A-derivable if
• Base case P contains the production A ? B or B
  A
• Recursive case Variable C is A-derivable and P
  contains the production C ? B
• No other variables are A-derivable
• Easy to make into algorithm

7
Eliminating unit productions

• Given CFG G (V, S, S, P), construct a CFG G1
  (V, S, S, P1) as follows.
• Initialize P1 P
• Find each A in V, find set of A-derivable
  variables in V
• For each pair (A,B) such that B is A-derivable
  and every non-unit production B ? a in P, add
  production A ? a in P1
• Clean up
• Delete all unit productions from P1
• Delete any duplicate productions
• Thm L(G1) L(G) if G did not have any
  ?-productions

8
Making CNF grammar

• First eliminate ?-productions
• Then eliminate unit productions
• For each terminal a in S, introduce a variable Xa
  with production rule Xa ? a
• For each production of the form A ? a where
  terminal a appears and a gt 1, replace a with Xa
• Finally, replace productions of the form A ? B1B2
  Bn with a series of productions
• A ?B1Y1
• Y1 ?B21Y2
• ....
• Yn-2 ?Bn-1Bn
• Other methods can be used for this last step

9
Example Eliminate ?-productions

• S ? AACD
• A ? aAb ?
• C ? aC a
• D ? aDa bDb ?
• Nullable variables A D
• New grammar
• S ? AACD ACD AAC CD AC C
• A ? aAb ab
• C ? aC a
• D ? aDa bDb aa bb

10
Example Eliminate unit productions

• S ? AACD ACD AAC CD AC C
• A ? aAb ab
• C ? aC a
• D ? aDa bDb aa bb
• New grammar
• S ? AACD ACD AAC CD AC aC a
• A ? aAb ab
• C ? aC a
• D ? aDa bDb aa bb

11
Example Add Xa and Xb

• S ? AACD ACD AAC CD AC aC a
• A ? aAb ab
• C ? aC a
• D ? aDa bDb aa bb
• New grammar
• S ? AACD ACD AAC CD AC XaC a
• A ? XaAXb XaXb
• C ? XaC a
• D ? XaDXa XbDXb XaXa XbXb
• Xa? a
• Xb? b

12
Example Shorten long productions

• S ? AACD ACD AAC CD AC XaC a
• A ? XaAXb XaXb
• C ? XaC a
• D ? XaDXa XbDXb XaXa XbXb
• Xa? a
• Xb? b
• Example replacement
• S ? AACD becomes S ? AT1, T1 ? AT2, T2 ? CD

13
Observations

• Consider a derivation from a CNF grammar G that
  begins
• S gtG ABCD
• How short can the final derived terminal string
  be?
• Why?

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Observation 2

• A path in a parse tree has length x if it
  contains x variables
• Consider a parse tree T for a string x and a CNF
  grammar G with m variables.
• Suppose the longest path in T has length k. How
  long can this string x be?
• Suppose string x has length 2m. How short can the
  longest path in T be?