Backus–Naur form

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The Backus–Naur form (BNF) is a metasyntax used to express context-free grammars: that is, a formal way to describe formal languages. John Backus and Peter Naur developed a context free grammar to define the syntax of a programming language by using two sets of rules: i.e., Lexical rules and Syntactic rules. BNF is widely used as a notation for the grammars of computer programming languages, instruction sets and communication protocols, as well as a notation for representing parts of natural language grammars. Many textbooks for programming language theory and/or semantics document the programming language in BNF. There are many extensions of and variants on BNF.

1 History
2 Introduction
3 Example
4 Further examples
5 Variants
6 See also
7 References
8 External links

History

John Backus created the notation in order to express the grammar of ALGOL. At the first World Computer Congress, which took place in Paris in 1959, Backus presented "The syntax and semantics of the proposed international algebraic language of the Zurich ACM-GAMM Conference", a formal description of the IAL which was later called ALGOL 58. The formal language he presented was based on Emil Post's production system. Generative grammars were an active subject of mathematical study, e.g. by Noam Chomsky, who was applying them to the grammar of natural language.^[1]^[2] Peter Naur (ALGOL 60, 1963) identified Backus's notation as Backus Normal Form, and simplified it to minimize the character set used, and, at the suggestion of Donald Knuth, his name was added in recognition of his contribution, his initial replacing the N for "Normal" since, Knuth argued, the BNF is "not a normal form in any sense".^[3] The Backus–Naur Form or BNF grammars have significant similarities to Pāṇini's grammar rules, and the notation is sometimes also referred to as Panini–Backus Form.^[4]

Introduction

A BNF specification is a set of derivation rules, written as

<symbol> ::= <expression with symbols>

where <symbol> is a nonterminal, and the expression consists of sequences of symbols and/or sequences separated by the vertical bar, '|', indicating a choice, the whole being a possible substitution for the symbol on the left. Symbols that never appear on a left side are terminals.

Example

As an example, consider this possible BNF for a U.S. postal address:

<postal-address> ::= <name-part> <street-address> <zip-part>

     <name-part> ::= <personal-part> <last-name> <opt-jr-part> <EOL> 
                   | <personal-part> <name-part> <EOL>
 
 <personal-part> ::= <first-name> | <initial> "." 

<street-address> ::= <opt-apt-num> <house-num> <street-name> <EOL>

      <zip-part> ::= <town-name> "," <state-code> <ZIP-code> <EOL>

<opt-jr-part> ::= "Sr." | "Jr." | <roman numeral>

<roman numeral> := "III" |  <five> | <tens>
:<ones>  :: = "I"{1,3}
:<five>   = "V"<ones>? | "IV"
:<tens>   = X{1,3}<five>? | X{0,3}IX<five>

This translates into English as:

A postal address consists of a name-part, followed by a street-address part, followed by a zip-code part.
A name-part consists of either: a personal-part followed by a last name followed by an optional "jr-part" (Jr., Sr., or dynastic number) and end-of-line, or a personal part followed by a name part (this rule illustrates the use of recursion in BNFs, covering the case of people who use multiple first and middle names and/or initials).
A personal-part consists of either a first name or an initial followed by a dot.
A street address consists of an optional apartment specifier, followed by a house number, followed by a street name, followed by an end-of-line.
A zip-part consists of a town-name, followed by a comma, followed by a state code, followed by a ZIP-code followed by an end-of-line.

Note that many things (such as the format of a first-name, apartment specifier, or ZIP-code) are left unspecified here. If necessary, they may be described using additional BNF rules.

Further examples

BNF's syntax itself may be represented with a BNF like the following:

<syntax> ::= <rule> | <rule> <syntax>
<rule> ::= <opt-whitespace> "<" <rule-name> ">" <opt-whitespace> "::=" 
           <opt-whitespace> <expression> <line-end>
<opt-whitespace> ::= " " <opt-whitespace> | ""  
<expression> ::= <list> | <list> "|" <expression>
<line-end> ::= <opt-whitespace> <EOL> | <line-end> <line-end>
<list> ::= <term> | <term> <opt-whitespace> <list>
<term> ::= <literal> | "<" <rule-name> ">"
<literal> ::= '"' <text> '"' | "'" <text> "'"

This assumes that no whitespace is necessary for proper interpretation of the rule. <EOL> to be the appropriate line-end specifier (in ASCII, carriage-return and/or line-feed, depending on the operating system). <rule-name> and <text> are to be substituted with a declared rule's name/label or literal text, respectively.

Variants

There are many variants and extensions of BNF, generally either for the sake of simplicity and succinctness, or to adapt it to a specific application. One common feature of many variants is the use of regexp repetition operators such as * and +. The Extended Backus-Naur form (EBNF) is a common one. In fact the example above is not the pure form invented for the ALGOL 60 report. The bracket notation "[]" was introduced a few years later in IBM's PL/I definition but is now universally recognised. ABNF is another extension commonly used to describe IETF protocols. Parsing expression grammars build on the BNF and regular expression notations to form an alternative class of formal grammar, which is essentially analytic rather than generative in character. Many BNF specifications found online today are intended to be human readable and are non-formal. These often include many of the following syntax rules and extensions:

Optional items enclosed in square brackets. E.g. [<item-x>]
Items repeating 0 or more times are enclosed in curly brackets or suffixed with an asterisk. E.g. <word> ::= <letter> { <letter> }
Items repeating 1 or more times are followed by a '+'
Terminals may appear in bold and NonTerminals in plain text rather than using italics and angle brackets
Alternative choices in a production are separated by the ‘|’ symbol. E.g., <alternative-A> | <alternative-B>
Where items need to be grouped they are enclosed in simple parenthesis

References

This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.

^ Chomsky, Noam (1956). "Three Models for the Description of Language". IRE Transactions on Information Theory Vol. 2 (No. 2): pp. 113-123.
^ Chomsky, Noam (1957). Syntactic Structures. The Hague: Mouton.
^ Knuth, Donald E. (1964). "Backus Normal Form vs. Backus Naur Form". Communications of the ACM 7 (12): pp. 735-736.
^ P.Z. Ingerman (1967)