Chomsky normal form

Not to be confused withconjunctive normal form.

Informal languagetheory, acontext-free grammar,G,is said to be inChomsky normal form(first described byNoam Chomsky)if all of itsproduction rulesare of the form:

A→BC,or

A→a,or

S→ ε,

whereA,B,andCarenonterminal symbols,the letterais aterminal symbol(a symbol that represents a constant value),Sis the start symbol, and ε denotes theempty string.Also, neitherBnorCmay be thestart symbol,and the third production rule can only appear if ε is inL(G), the language produced by the context-free grammarG.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into anequivalentonewhich 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 onautomata theory. The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).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 symbolS₀,and a new rule

S₀→S,

whereSis the previous start symbol. This does not change the grammar's produced language, andS₀will not 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 symbolabeing not the only symbol on the right-hand side, introduce, for every such terminal, a new nonterminal symbolN_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.

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

Replace each rule

A→X₁X₂...X_n

with more than 2 nonterminalsX₁,...,X_nby rules

A→X₁A₁,

A₁→X₂A₂,

...,

A_n-2→X_n-1X_n,

whereA_iare new nonterminal symbols. Again, this does not change the grammar's produced language.

DEL: Eliminate ε-rules

An ε-rule is a rule of the form

A→ ε,

whereAis notS₀,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 nonterminalsnullable,and compute them as follows:

• If a ruleA→ ε exists, thenAis nullable.
• If a ruleA→X₁...X_nexists, and every singleX_iis nullable, thenAis nullable, too.

Obtain an intermediate grammar by replacing each rule

A→X₁...X_n

by all versions with some nullableX_iomitted. By deleting in this grammar each ε-rule, unless its left-hand side is the start symbol, the transformed grammar is obtained.

For example, in the following grammar, with start symbolS₀,

S₀→AbB|C

B→AA|AC

C→b|c

A→a| ε

the nonterminalA,and hence alsoB,is nullable, while neitherCnorS₀is. Hence the following intermediate grammar is obtained:

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 "inlinedat the call site ". 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,ba,baa,bab,bac,bb,bc,c}, but has no ε-rules.

UNIT: Eliminate unit rules

A unit rule is a rule of the form

A→B,

whereA,Bare nonterminal symbols. To remove it, for each rule

B→X₁...X_n,

whereX₁...X_nis 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 symbolBin the resulting grammar is possible due toBbeing a member of the unit closure of nonterminal symbolA.

Order of transformations

Mutual preservation
of transformation results

TransformationXalways preserves(Y) resp.may destroy(N) the result ofY:
Y X	START	TERM	BIN	DEL	UNIT
START
TERM
BIN
DEL
UNIT				(Y)*
*UNITpreserves the result ofDEL ifSTARThad 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,STARTwill re-introduce a unit rule if it is applied afterUNIT.The table shows which orderings are admitted.

Moreover, the worst-case bloat in grammar sizedepends on the transformation order. Using |G| to denote the size of the original grammarG,the size blow-up in the worst case may range from |G|²to 2^{2 |G|},depending on the transformation algorithm used.The blow-up in grammar size depends on the order betweenDELandBIN.It may be exponential whenDELis done first, but is linear otherwise.UNITcan incur a quadratic blow-up in the size of the grammar.The orderingsSTART,TERM,BIN,DEL,UNITandSTART,BIN,DEL,UNIT,TERMlead to the least (i.e. quadratic) blow-up.

Example

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

The following grammar, with start symbolExpr,describes a simplified version of the set of all syntactical valid arithmetic expressions in programming languages likeCorAlgol60.Bothnumberandvariableare considered terminal symbols here for simplicity, since in acompiler front endtheir internal structure is usually not considered by theparser.The terminal symbol "^" denotedexponentiationin Algol60.

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

In step "START" of theaboveconversion algorithm, just a ruleS₀→Expris added to the grammar. After step "TERM", the grammar looks like this:

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

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

S0	→Expr
Expr	→Term	|ExprAddOp_Term	|AddOpTerm
Term	→Factor	|TermMulOp_Factor
Factor	→Primary	|FactorPowOp_Primary
Primary	→number	|variable	|OpenExpr_Close
AddOp	→ +	| −
MulOp	→ *	| /
PowOp	→ ^
Open	→ (
Close	→ )
AddOp_Term	→AddOpTerm
MulOp_Factor	→MulOpFactor
PowOp_Primary	→PowOpPrimary
Expr_Close	→ExprClose

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

S0	→number	|variable	|OpenExpr_Close	|FactorPowOp_Primary	|TermMulOp_Factor	|ExprAddOp_Term	|AddOpTerm
Expr	→number	|variable	|OpenExpr_Close	|FactorPowOp_Primary	|TermMulOp_Factor	|ExprAddOp_Term	|AddOpTerm
Term	→number	|variable	|OpenExpr_Close	|FactorPowOp_Primary	|TermMulOp_Factor
Factor	→number	|variable	|OpenExpr_Close	|FactorPowOp_Primary
Primary	→number	|variable	|OpenExpr_Close
AddOp	→ +	| −
MulOp	→ *	| /
PowOp	→ ^
Open	→ (
Close	→ )
AddOp_Term	→AddOpTerm
MulOp_Factor	→MulOpFactor
PowOp_Primary	→PowOpPrimary
Expr_Close	→ExprClose

TheN_aintroduced in step "TERM" arePowOp,Open,andClose. TheA_iintroduced in step "BIN" areAddOp_Term,MulOp_Factor,PowOp_Primary,andExpr_Close.

Alternative definition

Chomsky reduced form

Another wayto define the Chomsky normal form is:

Aformal grammaris inChomsky reduced formif 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 aterminal symbol.When using this definition,$B$or$C$may be the start symbol. Only those context-free grammars which do not generate theempty stringcan be transformed into Chomsky reduced form.

Floyd normal form

In a letter where he proposed a termBackus–Naur form(BNF),Donald E. Knuthimplied 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, becauseRobert W. Floydfound any BNF syntax can be converted to the above one in 1961.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."While Floyd's note cites Chomsky's original 1959 article, Knuth's letter does not.

Application

Besides its theoretical significance, CNF conversion is used in some algorithms as a preprocessing step, e.g., theCYK algorithm,abottom-up parsingfor context-free grammars, and its variant probabilistic CKY.

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

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.ISBN978-0-07-232200-2.(Pages 237–240 of section 6.6: simplified forms and normal forms.)
• Michael Sipser(1997).Introduction to the Theory of Computation.PWS Publishing.ISBN978-0-534-94728-6.(Pages 98–101 of section 2.1: context-free grammars. Page 156.)
• Charles D. Allison (2021) (20 August 2021).Foundations of Computing: An Accessible Introduction to Formal Language.Fresh Sources, Inc.ISBN9780578944173.{{cite book}}:CS1 maint: numeric names: authors list (link)(pages 171-183 of section 7.1: Chomsky Normal Form)
• Sipser, Michael.Introduction to the Theory of Computation,2nd edition.
• Alexander Meduna (6 December 2012).Automata and Languages: Theory and Applications.Springer Science & Business Media.ISBN978-1-4471-0501-5.