Normal Forms

1
Normal Forms

• Chomsky Normal Form
• Griebach Normal Form

2

• Language preserving transformations
• Improve parsing efficiency
• Prove properties about languages and derivations

Shorter derivations
3
Elimination of l-rules
Reduces the length of the derivation
4

• Aim Restrict the grammar such that
• Approach
• Introduce S

5

• Add rules to capture the effect of l-rules to be
  deleted.
• (Ensures non-contracting rules)

6
Example
7
Determination of nullable non-terminals
Bottom-up flow of information
8
Algorithm Nullable Nonterminals

• NULL A A-gtl e P
• repeat
• PREV NULL
• foreach A e V do
• if there is an A-rule A-gtw
• and w e PREV
• then NULL NULL U A
• until NULL PREV

9
Proof of correctness

• Soundness
• If A e NULL(final) then Agt l.
• Induction on the number of iterations of the
  loop.
• Completeness
• If Agt l then A e NULL(final).
• Induction on the minimal derivation of the null
  string from a non-terminal.
• Termination
• Bounded by the number of non-terminals.

10
Elimination of Chain rules
Removing renaming rules redundant procedure
calls.
Top-down flow of information
11
Construction of Chain(A)

• Chain(A) A PREV f
• repeat
• NEW Chain(A) - PREV
• PREV Chain(A)
• foreach B e NEW do
• if there is a rule B-gtC
  
• then Chain(A) Chain(A) U C
• until Chain(A) PREV

12
Examples
13
Elimination of useless symbols

• A variable is useful if it occurs in a derivation
  that begins with the start symbol and generates a
  terminal string.
• Reachable from S
• Derives terminal string

14

• Construction of the set of variables that derive
  terminal string.
• Bottom-up flow of information
• Similar to the computation of nullable variables.
• Construction of the set of variables that are
  reachable
• Top-down flow of information
• Similar to the computation of chained variables.

15
Examples
B does not derive terminal string C unreachable.
A unreachable.
Empty set of productions
Non-termination
16
Chomsky Normal Form

• A CFG is in Chomsky Normal Form if each rule is
  of the form
• Theorem There is an algorithm to construct a
  grammar G in CNF that is equivalent to a CFG G.

17
Construction

• Obtain an equivalent grammar that does not
  contain l-rules, chain rules, and useless
  variables.
• Apply following conversion on rules of the form
  

18
Significance of CNF

• Length of derivation of a string of length n in
  CNF (2n-1)
• (Cf. Number of nodes of a strictly binary tree
  with n-leaves)
• Maximum depth of a parse tree n
• Minimum depth of a parse tree

19
Removal of direct left recursion

• Causes infinite loop in top-down (depth-first)
  parsers.
• Approach Generate string from left to right.

20
Note that absence of direct left recursion does
not imply absence of left recursion.
21
(Cf. Gaussian Elimination)
22
Griebach Normal Form

• ( Constructs terminal prefixes that
  facilitates
• discovery of dead-ends )
• A CFG is in Griebach Normal Form if each rule is
  of the form
• Theorem There is an algorithm to construct a
  grammar G in GNF that is equivalent to a CFG G.

23
Analogy solving linear simultaneous equations
What are the values of x,y, and z?
(Solving for z and then back substituiting.)
24
Example conversion to GNF
Introducing terminals as first element on RHS
Eliminating left recursion
25

• The size of the equivalent GNF can be large
  compared to the original grammar.
• Example CFG has 5 rules, but the corresponding
  GNF has 24 rules!!
• Length of the derivation in GNF
• Length of the string.
• GNF is useful in relating CFGs (generators) to
  pushdown automata (recognizers/acceptors).