Chomsky normal form
where , and are nonterminal symbols, α is a terminal symbol (a symbol that represents a constant value), is the start symbol, and ε is the empty string. Also, neither nor 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. Several algorithms for performing such a transformation are known. Transformations are described in most textbooks on automata theory, such as Hopcroft and Ullman, 1979. As pointed out by Lange and Leiß, the drawback of these transformations is that they can lead to an undesirable bloat in grammar size. The size of a grammar is the sum of the sizes of its production rules, where the size of a rule is one plus the length of its right-hand side. Using to denote the size of the original grammar , the size blow-up in the worst case may range from to , depending on the transformation algorithm used.
Another way to define Chomsky normal form (e.g., Hopcroft and Ullman 1978,and Hopcroft et al. 2006) is:
A formal grammar is in Chomsky reduced form if all of its production rules are of the form:
where , and are nonterminal symbols, and α is a terminal symbol. When using this definition, or may be the start symbol. Only those context-free grammars which do not generate the empty string can be transformed into Chomsky reduced form.
Converting a grammar to Chomsky Normal Form
- Introduce a new start variable, and a new rule where is the previous start variable.
- Eliminate all rules
- rules are rules of the form where and where is the CFG's variable alphabet.
- Remove every rule with on its right hand side (RHS). For each rule with in its RHS, add a set of new rules consisting of the different possible combinations of replaced or not replaced with . If a rule has as a singleton on its RHS, add a new rule unless has already been removed through this process. For example, examine the following grammar :
- has one rule. When the is removed, we get the following:
- Notice that we have to account for all possibilities of and so we actually end up adding 3 rules.
- Eliminate all unit rules
- After all the rules have been removed, you can begin removing unit rules, or rules whose RHS contains one variable and no terminals (which is inconsistent with CNF).
- To remove
- , where is a string of variables and terminals, add rule unless this is a unit rule which has already been removed.
- Clean up remaining rules that are not in Chomsky normal form.
- Replace with where are new variables.
- If , replace in above rules with some new variable and add rule .
- * John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation, 3rd Edition, Addison-Wesley, 2006. ISBN 0-321-45536-3 (see subsection 7.1.5, page 272.)
- Lange, Martin and Leiß, Hans. To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm. Informatica Didactica 8, 2009.
- John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X. (See chapter 4.)
- 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.)
- 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 A. Harrison (1978). Introduction to Formal Language Theory. Addison-Wesley. ISBN 978-0-201-02955-0. (Pages 103–106.)
- Cole, Richard. Converting CFGs to CNF (Chomsky Normal Form), October 17, 2007. (pdf)
- Sipser, Michael. Introduction to the Theory of Computation, 2nd edition.