Chomsky normal form

In formal language theory, a context-free grammar G is said to be in Chomsky normal form (first described by Noam Chomsky)^[1] if all of its production rules are of the form:^{[2]:92–93,106}

A → BC,   or

A → a,   or

S → ε,

where A, B, and C are nonterminal symbols, a is a terminal symbol (a symbol that represents a constant value), S is the start symbol, and ε denotes the empty string. Also, neither B nor C may be the start symbol, and the third production rule can only appear if ε is in L(G), namely, the language produced by the context-free grammar G.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form and has a size no larger than the square of the original grammar's size.

Converting a grammar to Chomsky normal form

To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory.^{[2]:87–94[3][4][5]} The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).^{[6][note 1]} Each of the following transformations establishes one of the properties required for Chomsky normal form.

START: Eliminate the start symbol from right-hand sides

Introduce a new start symbol S₀, and a new rule

S₀ → S,

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

TERM: Eliminate rules with nonsolitary terminals

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 doesn't change the grammar's produced language.^[2]:92

BIN: Eliminate right-hand sides with more than 2 nonterminals

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 doesn't change the grammar's produced language.^[2]:93

DEL: Eliminate ε-rules

An ε-rule is a rule of the form

A → ε,

where A is not 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 each 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.^[2]: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 2]}

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 3]} 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 apparently has no ε-rules.

UNIT: Eliminate unit 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 removed.

Order of transformations

Mutual preservation of transformation results
Transformation X always perserves (✓) resp. may destroy (✗) the result of Y:
X\^Y	START	TERM	BIN	DEL	UNIT
START		✓	✓	✓	✗
TERM	✓		✗	✓	✓
BIN	✓	✓		✓	✓
DEL	✓	✓	✓		✗
UNIT	✓	✓	✓	(✓)^*
^*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 4]} 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.^[6]: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.^[6]:5 The orderings START,TERM,BIN,DEL,UNIT and START,BIN,DEL,UNIT,TERM lead to the least (i.e. quadratic) blow-up.

Example

Abstract syntax tree of the arithmetic expression "a^2+4*b" wrt. the example grammar (top) and its Chomsky normal form (bottom)

The following grammar, with start symbol Expr, describes a simplified version of the set of all syntactical valid arithmetic expressions in programming languages like C or Algol60. Both number and variable are considered terminal symbols here for simplicity, since in a compiler front-end their internal structure is usually not considered by the parser. The terminal symbol "^" denoted exponentiation in Algol60.

Expr	→ Term	| Expr AddOp Term	| AddOp Term
Term	→ Factor	| Term MulOp Factor
Factor	→ Primary	| Factor ^ Primary
Primary	→ number	| variable	| ( Expr )
AddOp	→ +	| −
MulOp	→ *	| /

In step "START" of the above conversion algorithm, just a rule S₀→Expr is added to the grammar. After step "TERM", the grammar looks like this:

S0	→ Expr
Expr	→ Term	| Expr AddOp Term	| AddOp Term
Term	→ Factor	| Term MulOp Factor
Factor	→ Primary	| Factor PowOp Primary
Primary	→ number	| variable	| Open Expr Close
AddOp	→ +	| −
MulOp	→ *	| /
PowOp	→ ^
Open	→ (
Close	→ )

After step "BIN", the following grammar is obtained:

S0	→ Expr
Expr	→ Term	| Expr AddOp_Term	| AddOp Term
Term	→ Factor	| Term MulOp_Factor
Factor	→ Primary	| Factor PowOp_Primary
Primary	→ number	| variable	| Open Expr_Close
AddOp	→ +	| −
MulOp	→ *	| /
PowOp	→ ^
Open	→ (
Close	→ )
AddOp_Term	→ AddOp Term
MulOp_Factor	→ MulOp Factor
PowOp_Primary	→ PowOp Primary
Expr_Close	→ Expr Close

Since there are no ε-rules, step "DEL" doesn't change the grammar. After step "UNIT", the following grammar is obtained, which is in Chomsky normal form:

S0	→ number	| variable	| Open Expr_Close	| Factor PowOp_Primary	| Term MulOp_Factor	| Expr AddOp_Term	| AddOp Term
Expr	→ number	| variable	| Open Expr_Close	| Factor PowOp_Primary	| Term MulOp_Factor	| Expr AddOp_Term	| AddOp Term
Term	→ number	| variable	| Open Expr_Close	| Factor PowOp_Primary	| Term MulOp_Factor
Factor	→ number	| variable	| Open Expr_Close	| Factor PowOp_Primary
Primary	→ number	| variable	| Open Expr_Close
AddOp	→ +	| −
MulOp	→ *	| /
PowOp	→ ^
Open	→ (
Close	→ )
AddOp_Term	→ AddOp Term
MulOp_Factor	→ MulOp Factor
PowOp_Primary	→ PowOp Primary
Expr_Close	→ Expr Close

The N_a introduced in step "TERM" are PowOp, Open, and Close. The A_i introduced in step "BIN" are AddOp_Term, MulOp_Factor, PowOp_Primary, and Expr_Close.

Alternative definition

Chomsky reduced form

Another way^[2]:92[7] 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:

$A \rightarrow\, BC$ or

$A \rightarrow\, a$,

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

Floyd normal form

In a paper 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","

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

$\langle A \rangle ::= \, \langle B \rangle \langle C \rangle$ or

$\langle A \rangle ::=\, a$,

where $\langle A \rangle$, $\langle B \rangle$ and $\langle C \rangle$ are nonterminal symbols, and $a$ is a terminal symbol, because Robert W. Floyd found any BNF syntax can be converted to the above one in 1961.^[8] 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."^[8]

Application

Besides its theoretical significance, CNF conversion is used in some algorithms as a preprocessing step, e.g., the CYK algorithm, a bottom-up parsing for context-free grammars, and its variant probabilistic CKY.^[9]

See also

• Backus-Naur form
• CYK algorithm
• Greibach normal form
• Kuroda normal form
• Pumping lemma for context-free languages — its proof relies on the Chomsky normal form

Notes

1. For example, Hopcroft, Ullman (1979) merged TERM and BIN into a single transformation.
2. indicating a kept and omitted nonterminal N by N and N, respectively
3. If the grammar had a rule S₀ → ε, it could not be "inlined", since it had no "call sites". Therefore it couldn't be deleted in the next step.
4. i.e. written length, measured in symbols

References

1. Chomsky, Noam (1959). "On Certain Formal Properties of Grammars" (PDF). Information and Control 2: 137–167. doi:10.1016/S0019-9958(59)90362-6.
2. Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages and Computation. Reading, Massachusetts: Addison-Wesley Publishing. ISBN 0-201-02988-X.
3. Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2006). Introduction to Automata Theory, Languages, and Computation (3rd ed.). Addison-Wesley. ISBN 0-321-45536-3. Section 7.1.5, p.272
4. Rich, Elaine (2007). Automata, Computability, and Complexity: Theory and Applications (1st ed.). Prentice-Hall. ISBN 978-0132288064.
5. Wegener, Ingo (1993). Theoretische Informatik - Eine algorithmenorientierte Einführung. Leitfäden und Mongraphien der Informatik (in German). Stuttgart: B.G. Teubner. ISBN 978-3-519-02123-0. Section 6.2 "Die Chomsky-Normalform für kontextfreie Grammatiken", p.149-152
6. Lange, Martin; Leiß, Hans (2009). "To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm" (PDF). Informatica Didactica 8.
7. Hopcroft et al. (2006)
8. Knuth, Donald E. (December 1964). "Backus Normal Form vs. Backus Naur Form". Communications of the ACM 7 (12): 735–736. doi:10.1145/355588.365140.
9. Jurafsky, Daniel; Martin, James H. (2008). Speech and Language Processing (2nd ed.). Pearson Prentice Hall. p. 465. ISBN 978-0-13-187321-6.

Further reading

• Cole, Richard. Converting CFGs to CNF (Chomsky Normal Form), October 17, 2007. (pdf) — uses the order TERM, BIN, START, DEL, UNIT.
• John Martin (2003). Introduction to Languages and the Theory of Computation. McGraw Hill. ISBN 0-07-232200-4. (Pages 237–240 of section 6.6: simplified forms and normal forms.)
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. (Pages 98–101 of section 2.1: context-free grammars. Page 156.)
• Sipser, Michael. Introduction to the Theory of Computation, 2nd edition.