Lec 8 : Computation Theory
Chomsky Normal Form
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Chomsky Normal Form
Definition Chomsky Normal Form (CNF)
A grammar is in Chomsky Normal Form (CNF), if it has only
productions of the form:
A → B C or
A→a
with A, B, C V and aT.
This means all rules are consisting of either a single terminal on the
production, or they are binary, consisting of two variables.
Conversion of a CFG into Chomsky Normal Form
Here is the algorithm to convert a given CFG into CNF.
1. Make the initial grammar G free of λ-productions, unit-productions,
and useless variables and productions.You have rules now, which
contain one terminal on the productions or at least two symbols from
V  T on the productions.
2. We substitute terminals ai, which appear in complex PRODUCTIONS
(with more than one symbol), with new variables C1, C2,...
Add these C1, C2,... to V to form V'.
Lec 8 : Computation Theory
Chomsky Normal Form
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We have to add matching new productions: Ci → ai to P to form P'.
Productions in P' have either a single terminal on the productions, or at
least 2 variables:
A → a or A → V1 ... Vn with Vi  V' for i=1,...,n and n≥2
3. We split up the non-terminal productions in P' and make them "binary".
We introduce new variables D1, D2, ... which we add to V'.
Every A → V1 ... Vn
in P' is now substituted with a set of new
productions:
A → V1 D1
D1 → V2 D2
D2 → V3 D3
...
Dn-1 → Vn-1 Vn
The grammar is now in Chomsky NF:
Example 1: Transform a CFG into Chomsky Normal Form the grammar G with
V={S, A, B}, T={a, b} and productions P
S→A
Lec 8 : Computation Theory
Chomsky Normal Form
A→aBa|a
B → b A b | b |D
S→aBa|a
B→bAb|b
S → C1 B C1 | a
B → C2 A C2| b
C1→a
C2→b
S → C1 D1 | a
B → C2 D2 | b
D1→ B C1
D2 → A C2
C1→ a
C2→ b
H.W : Transform a CFG into Chomsky normal form (CNF)
S →Y X Z \ Y
Y →0Y1\01
X →aXb\
Z →bZ
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