CNF (Chomsky Normal Form) restricts the shapes of a context-free grammar’s production rules. BNF (Backus–Naur Form) is a notation for writing grammar rules. They serve different roles: CNF is a constrained grammar form, while BNF is a way to present a grammar.
CNF and BNF compared
| Question | CNF | BNF |
|---|---|---|
| Full name | Chomsky Normal Form | Backus–Naur Form |
| What it is | A restricted form of a context-free grammar | A notation for expressing grammar productions |
| Typical form | A → BC or A → a, where uppercase symbols represent variables and a is a terminal |
Named nonterminals, alternatives, and terminals; often written with ::= and | |
| Why use it | To work with a uniform grammar structure in formal procedures and proofs | To present language syntax in a readable form |
The University of Maryland, Baltimore County describes CNF’s rule patterns and connects them to the CYK membership algorithm; Virginia Tech’s OpenDSA describes BNF as a popular notation for context-free grammars. UMBC: Formal Language Definitions; Virginia Tech OpenDSA: BNF.
What BNF looks like
A BNF-style rule for a sequence of digits could be:
<digits> ::= <digit> | <digits> <digit>It says that a digit sequence can consist of one digit, or of an existing digit sequence followed by another digit. The notation makes the alternatives readable; it does not make the rule CNF.
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BNF is commonly used to describe programming-language syntax. GNU Bison calls it a common human-readable way to present language rules and notes its historical association with specifying ALGOL 60. BNF is named for John Backus and Peter Naur. GNU Bison: Language and Grammar; University of Geneva: About BNF notation.
What makes a grammar CNF
CNF concerns the structure of the productions, not the punctuation used to write them. In the usual rule patterns, a variable produces either two variables, A → BC, or one terminal, A → a. A grammar written with BNF-style ::= and | could still describe productions arranged in CNF; changing the notation alone does not convert a grammar to that form.
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Converting a context-free grammar to CNF generally means rewriting its productions and, where needed, introducing helper variables so the rules meet the required shapes. Definitions vary in how they handle the empty string and the start symbol, so conversion claims need those qualifications. The University of Manchester’s overview distinguishes BNF notation from a normal form such as CNF. University of Manchester: Notations for context-free grammars.
When each is useful
Use BNF to communicate syntax
BNF is useful when people need to read or write a language specification. Its production notation shows the available grammatical structures and alternatives in a compact, familiar format.
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Use CNF for formal methods
CNF is useful when a formal-language procedure or proof benefits from a restricted, uniform set of production shapes. For example, the CYK algorithm tests whether a string belongs to a context-free language represented by a grammar in CNF; UMBC describes its running time as cubic in the input string’s length. That complexity statement is about the algorithm’s input-string length, not a general performance claim about all grammar processing. UMBC: Formal Language Definitions.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What neither one tells you
A grammar describes syntax: the strings or structures a language permits. It does not by itself explain what a program means or what it will do when executed. Syntax and semantics are separate concerns. Virginia Tech OpenDSA: BNF.
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The short version
- BNF is a notation for writing grammar productions.
- CNF is a restricted arrangement of productions in a context-free grammar.
- They are not competing grammar classes: BNF concerns presentation, while CNF concerns rule structure.
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