Chomsky normal form

Introduce a new start symbol S₀, and a new rule

S₀ → S,

where S is the previous start symbol. This does not change the grammar's produced language, and S₀ will not occur on any rule's right-hand side.

To eliminate each rule

A → X₁ ... a ... X_n

with a terminal symbol a being not the only symbol on the right-hand side, introduce, for every such terminal, a new nonterminal symbol N_a, and a new rule

N_a → a.

Change every rule

A → X₁ ... a ... X_n

to

A → X₁ ... N_a ... X_n.

If several terminal symbols occur on the right-hand side, simultaneously replace each of them by its associated nonterminal symbol. This does not change the grammar's produced language.[4]^: 92

Replace each rule

A → X₁ X₂ ... X_n

with more than 2 nonterminals X₁,...,X_n by rules

A → X₁ A₁,

A₁ → X₂ A₂,

... ,

A_n-2 → X_n-1 X_n,

where A_i are new nonterminal symbols. Again, this does not change the grammar's produced language.[4]^: 93

An ε-rule is a rule of the form

A → ε,

where A is not S₀, the grammar's start symbol.

To eliminate all rules of this form, first determine the set of all nonterminals that derive ε. Hopcroft and Ullman (1979) call such nonterminals nullable, and compute them as follows:

• If a rule A → ε exists, then A is nullable.
• If a rule A → X₁ ... X_n exists, and every single X_i is nullable, then A is nullable, too.

Obtain an intermediate grammar by replacing each rule

A → X₁ ... X_n

by all versions with some nullable X_i omitted. By deleting in this grammar each ε-rule, unless its left-hand side is the start symbol, the transformed grammar is obtained.[4]^: 90

For example, in the following grammar, with start symbol S₀,

S₀ → AbB | C

B → AA | AC

C → b | c

A → a | ε

the nonterminal A, and hence also B, is nullable, while neither C nor S₀ is. Hence the following intermediate grammar is obtained:[note 3]

S₀ → AbB | AbB | AbB | AbB   |   C

B → AA | AA | AA | AεA   |   AC | AC

C → b | c

A → a | ε

In this grammar, all ε-rules have been "inlined at the call site".[note 4] In the next step, they can hence be deleted, yielding the grammar:

S₀ → AbB | Ab | bB | b   |   C

B → AA | A   |   AC | C

C → b | c

A → a

This grammar produces the same language as the original example grammar, viz. {ab,aba,abaa,abab,abac,abb,abc,b,bab,bac,bb,bc,c}, but has no ε-rules.

A unit rule is a rule of the form

A → B,

where A, B are nonterminal symbols. To remove it, for each rule

B → X₁ ... X_n,

where X₁ ... X_n is a string of nonterminals and terminals, add rule

A → X₁ ... X_n

unless this is a unit rule which has already been (or is being) removed. The skipping of nonterminal symbol B in the resulting grammar is possible due to B being a member of the unit closure of nonterminal symbol A.[9]

Mutual preservation
of transformation results

Transformation X always preserves (Y) resp. may destroy (N) the result of Y:
Y X	START	TERM	BIN	DEL	UNIT
START
TERM
BIN
DEL
UNIT				(Y)*
*UNIT preserves the result of DEL   if START had been called before.

When choosing the order in which the above transformations are to be applied, it has to be considered that some transformations may destroy the result achieved by other ones. For example, START will re-introduce a unit rule if it is applied after UNIT. The table shows which orderings are admitted.

Moreover, the worst-case bloat in grammar size[note 5] depends on the transformation order. Using |G| to denote the size of the original grammar G, the size blow-up in the worst case may range from |G|² to 2^{2 |G|}, depending on the transformation algorithm used.[8]^: 7 The blow-up in grammar size depends on the order between DEL and BIN. It may be exponential when DEL is done first, but is linear otherwise. UNIT can incur a quadratic blow-up in the size of the grammar.[8]^: 5 The orderings START,TERM,BIN,DEL,UNIT and START,BIN,DEL,UNIT,TERM lead to the least (i.e. quadratic) blow-up.

Another way[4]^: 92[10] to define the Chomsky normal form is:

A formal grammar is in Chomsky reduced form if all of its production rules are of the form:

${\displaystyle A\rightarrow \,BC}$ or

${\displaystyle A\rightarrow \,a}$,

where ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ are nonterminal symbols, and ${\displaystyle a}$ is a terminal symbol. When using this definition, ${\displaystyle B}$ or ${\displaystyle C}$ may be the start symbol. Only those context-free grammars which do not generate the empty string can be transformed into Chomsky reduced form.

In a letter where he proposed a term Backus–Naur form (BNF), Donald E. Knuth implied a BNF "syntax in which all definitions have such a form may be said to be in 'Floyd Normal Form'",

${\displaystyle \langle A\rangle$ ::=\,\langle B\rangle \mid \langle C\rangle } or

${\displaystyle \langle A\rangle$ ::=\,\langle B\rangle \langle C\rangle } or

${\displaystyle \langle A\rangle$ ::=\,a} ,

where ${\displaystyle \langle A\rangle }$, ${\displaystyle \langle B\rangle }$ and ${\displaystyle \langle C\rangle }$ are nonterminal symbols, and ${\displaystyle a}$ is a terminal symbol, because Robert W. Floyd found any BNF syntax can be converted to the above one in 1961.[11] But he withdrew this term, "since doubtless many people have independently used this simple fact in their own work, and the point is only incidental to the main considerations of Floyd's note."[12] While Floyd's note cites Chomsky's original 1959 article, Knuth's letter does not.