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.{{rp|87–94}}{{cite book |last1=Hopcroft |first1=John E. |last2=Motwani |first2=Rajeev |last3=Ullman |first3=Jeffrey D. |date=2006 |title=Introduction to Automata Theory, Languages, and Computation |edition=3rd |publisher=Addison-Wesley |isbn=978-0-321-45536-9 |url-access=registration |url=https://archive.org/details/introductiontoau0000hopc }} Section 7.1.5, p.272{{cite book |last=Rich |first=Elaine |author-link=Elaine Rich|date=2007 |title=Automata, Computability, and Complexity: Theory and Applications |publisher=Prentice-Hall |edition=1st |page=169 |section=11.8 Normal Forms |url=https://www.cs.utexas.edu/~ear/cs341/automatabook/AutomataTheoryBook.pdf |archive-url=https://archive.today/20230117061906/https://www.cs.utexas.edu/~ear/cs341/automatabook/AutomataTheoryBook.pdf |archive-date=2023-01-17 |isbn=978-0132288064}}{{cite book |last=Wegener |first=Ingo |date=1993 |title=Theoretische Informatik - Eine algorithmenorientierte Einführung |language=de |series=Leitfäden und Mongraphien der Informatik |publisher=B. G. Teubner |location=Stuttgart |isbn=978-3-519-02123-0}} Section 6.2 "Die Chomsky-Normalform für kontextfreie Grammatiken", p. 149–152

The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).{{cite journal |last1=Lange |first1=Martin |last2=Leiß |first2=Hans |date=2009 |title=To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm |journal=Informatica Didactica |volume=8 |url=http://ddi.cs.uni-potsdam.de/InformaticaDidactica/LangeLeiss2009.pdf |archive-url=https://web.archive.org/web/20110719111029/http://ddi.cs.uni-potsdam.de/InformaticaDidactica/LangeLeiss2009.pdf |archive-date=2011-07-19 |url-status=live }}For example, Hopcroft, Ullman (1979) merged TERM and BIN into a single transformation. Each of the following transformations establishes one of the properties required for 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.{{rp|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.{{rp|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.{{rp|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:indicating a kept and omitted nonterminal N by {{color|#006000|N}} and {{color|#ffc0c0|N}}, respectively

: S₀ → {{color|#006000|A}}b{{color|#006000|B}} | {{color|#006000|A}}b{{color|#ffc0c0|B}} | {{color|#ffc0c0|A}}b{{color|#006000|B}} | {{color|#ffc0c0|A}}b{{color|#ffc0c0|B}}   |   C

: B → {{color|#006000|AA}} | {{color|#ffc0c0|A}}{{color|#006000|A}} | {{color|#006000|A}}{{color|#ffc0c0|A}} | {{color|#ffc0c0|A}}ε{{color|#ffc0c0|A}}   |   {{color|#006000|A}}C | {{color|#ffc0c0|A}}C

: C → b | c

: A → a | ε

In this grammar, all ε-rules have been "inlined at the call site".If the grammar had a rule S₀ → ε, it could not be "inlined", since it had no "call sites". Therefore it could not be deleted in the next step.

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.

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.{{Cite book |last=Allison |first=Charles D. |title=Foundations of Computing: An Accessible Introduction to Automata and Formal Languages |publisher=Fresh Sources, Inc. |year=2022 |isbn=9780578944173 |pages=176 |language=en}}

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 sizei.e. written length, measured in symbols 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.{{rp|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.{{rp|5}} The orderings START,TERM,BIN,DEL,UNIT and START,BIN,DEL,UNIT,TERM lead to the least (i.e. quadratic) blow-up.

File:Syntax tree of arithmetic expression wrt Chomsky normal form grammar.gif 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.

:

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:

:

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

:

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:

:

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.

Another way{{rp|92}}Hopcroft et al. (2006){{page needed|date=November 2014}} 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:

: $A\; \backslash rightarrow\backslash ,\; BC$ or

: $A\; \backslash rightarrow\backslash ,\; a$,

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.

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'",

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

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

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

where $\backslash langle\; A\; \backslash rangle$, $\backslash langle\; B\; \backslash rangle$ and $\backslash langle\; C\; \backslash rangle$ 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.{{cite journal | url=https://core.ac.uk/download/pdf/82003923.pdf |archive-url=https://web.archive.org/web/20210305050258/https://core.ac.uk/download/pdf/82003923.pdf |archive-date=2021-03-05 |url-status=live | author=Floyd, Robert W. | title=Note on mathematical induction in phrase structure grammars. | journal=Information and Control | volume=4 | pages=353–358 | year=1961 | issue=4 | doi=10.1016/S0019-9958(61)80052-1 | doi-access=free }} Here: p.354 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."{{cite journal |last=Knuth |first=Donald E. |date=December 1964 |title=Backus Normal Form vs. Backus Naur Form |journal=Communications of the ACM |doi=10.1145/355588.365140 |volume=7 |issue=12 |pages=735–736|s2cid=47537431 |doi-access=free }} While Floyd's note cites Chomsky's original 1959 article, Knuth's letter does not.

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.{{cite book |last1=Jurafsky |first1=Daniel |last2=Martin |first2=James H. |date=2008 |title=Speech and Language Processing |edition=2nd |publisher=Pearson Prentice Hall |isbn=978-0-13-187321-6 |page=465}}

• 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

{{reflist|group=note}}

• Cole, Richard. Converting CFGs to CNF (Chomsky Normal Form), October 17, 2007. [http://cs.nyu.edu/courses/fall07/V22.0453-001/cnf.pdf (pdf)] — uses the order TERM, BIN, START, DEL, UNIT.
• {{cite book |author=John Martin |year=2003 |url=https://archive.org/details/introductiontola0000mart |title=Introduction to Languages and the Theory of Computation |publisher=McGraw Hill |isbn=978-0-07-232200-2 |url-access=registration}} (Pages 237–240 of section 6.6: simplified forms and normal forms.)
• {{cite book

| author-link = Michael Sipser

| author = Michael Sipser

| year = 1997

| title = Introduction to the Theory of Computation

| publisher = PWS Publishing

| isbn = 978-0-534-94728-6

| url = https://archive.org/details/introductiontoth00sips

}} (Pages 98–101 of section 2.1: context-free grammars. Page 156.)

• {{cite book|author=Charles D. Allison (2021)|title = Foundations of Computing: An Accessible Introduction to Formal Language| date=20 August 2021 |publisher=Fresh Sources, Inc.|isbn = 9780578944173}} (pages 171-183 of section 7.1: Chomsky Normal Form)
• Sipser, Michael. Introduction to the Theory of Computation, 2nd edition.
• {{cite book|author=Alexander Meduna|title=Automata and Languages: Theory and Applications|url=https://books.google.com/books?id=a-rjBwAAQBAJ&q=%22Chomsky+normal+form%22|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4471-0501-5}}

{{Noam Chomsky}}

Category:Formal languages

Category:Noam Chomsky