2.8 normal forms gnf & problems

Greibach
Normal Form
(GNF)
-Sampath Kumar S,
AP/CSE, SECE

Greibach Normal Form (GNF):
 A CFG is in Greibach Normal Form (GNF), if the
right-hand side of each rule has one terminal
followed by zero or more non-terminals:
A → a B, where a ∈ T and B ∈ V*.
 For converting the given grammar to GNF, we use
2 lemmas.
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Sampath Kumar S, AP/CSE, SECE
2
Zero or more non-terminal symbols
One terminal symbol

Example of a grammar not in GNF
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S → aBc
B → b
Not in Greibach Normal Form
Terminal at end is not allowed

Lemma 1: Substitution Rule
Let G = {V, T, P, S} be a CFG. Let A → αBγ be a
production in P and B is
B → β1|β2|β3|β4
The equivalent grammar can be obtained by
substituting B in A. Then the resulting grammar
is
A → αβ1γ|αβ2γ|αβ3γ|αβ4γ
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Lemma 2: Elimination of Left recursion
Grammar of the form A → Aα|β is called left
recursive grammar. To eliminate left recursion, rewrite
the grammar as
A → βZ
Z → αZ|ε
If we eliminate ε production, then we get
A → βZ|β
Z → αZ|α
We can generalize this grammar. If there is a CFG as
A → Aα1|Aα2|Aα3|….Aαn|β1|β2|β3……βn
The equivalent grammar without left recursion is
A → β1Z|β2Z|β3Z|…|β1|β2|β3 ……
Z → α1Z|α2Z|α3Z|……|α1|α2|α3
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Procedure for converting to GNF:
1. Simplify the CFG (i.e., eliminate null production, unit
production and useless symbols) and convert to Chomsky
Normal Form (CNF).
2. Convert the rules to ascending order. Rename the Non
Terminals as A1, A2 …. with S = A1.
3. For each production of the form Ai → Ajα, apply the
following:
(a) if j>i – Leave the production as it is.
(b) if j=i – Apply elimination of left recursion rule.
(c) if j<i – Apply substitution rule.
4. For each production of the form Ai → Ajα, where j>i,
apply the substitution lemma if Aj is in GNF, to bring Ai to
GNF.
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Problems to discuss:
104. Convert the following CFG to GNF
S → AA|a
A→ SS|b
Solution:
Step 1: Simplify the CFG and convert to CNF
– Given grammar is in CNF form.
Step 2: Rename the variables as S=A1, A=A2
A1 → A2A2|a ….(1)
A2 → A1A1|b ….(2)
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Step 3.1: In (1) j>i so leave the production as it is.
Step 3.2: In (2) j<i so apply substitution rule.
A2 → A2A2A1|aA1|b ….(new 2)
Step 3.3: In (new 2) j=i so apply elimination of left
recursion rule.
A2 → aA1A3|bA3|aA1|b ….(new 2)
A3 → A2A1A3|A2A1 ….(3)
Step 3.4: In (3) j<i so apply substitution rule.
A3 → aA1A3A1A3|bA3A1A3|aA1A1A3|bA1A3|
aA1A3A1|bA3A1|aA1A1|bA1 ….(new 3)
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Step 4: Now (1) is in Ai → Ajα where j<i so we
apply substitution rule to convert it to GNF.
A1 → aA1A2|bA2|aA1A3A2|bA3A2|a …(new 1)
Final production rule:
A1 → aA1A2|bA2|aA1A3A2|bA3A2|a
A2 → aA1A3|bA3|aA1|b
A3 → aA1A3A1A3|bA3A1A3|aA1A1A3|bA1A3|
aA1A3A1|bA3A1|aA1A1|bA1
Now the given CFG is converted to GNF.
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நன்றி
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2.8 normal forms gnf & problems

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