In computer science, BNF (Backus Normal Form or Backus–Naur Form) is one of the two [1] main notation techniques for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document formats, instruction sets and communication protocols. The other main technique for writing context-free grammars being the van Wijngaarden form. They are applied wherever exact descriptions of languages are needed, for instance, in official language specifications, in manuals, and in textbooks on programming language theory.
Many extensions and variants of the original Backus–Naur notation are used; some are exactly defined, including Extended Backus–Naur Form (EBNF) and Augmented Backus–Naur Form (ABNF).
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The idea of describing the structure of language with rewriting rules can be traced back to at least the work of Pāṇini (about the 4th century BC), who used it in his description of Sanskrit word structure. American linguists such as Leonard Bloomfield and Zellig Harris took this idea a step further by attempting to formalize language and its study in terms of formal definitions and procedures (around 1920–1960).
Meanwhile, string rewriting rules as formal, abstract systems were introduced and studied by mathematicians such as Axel Thue (in 1914), Emil Post (1920s–1940s) and Alan Turing (1936). Noam Chomsky, teaching linguistics to students of information theory at MIT, combined linguistics and mathematics, by taking what is essentially Thue's formalism as the basis for the description of the syntax of natural language; he also introduced a clear distinction between generative rules (those of context-free grammars) and transformation rules (1956).[2][3]
John Backus, a programming language designer at IBM, proposed "metalinguistic formulas"[4][5] to describe the syntax of the new programming language IAL, known today as ALGOL 58 (1959), using the BNF notation.
Further development of ALGOL led to ALGOL 60; in its report (1963), Peter Naur named Backus's notation Backus Normal Form, and simplified it to minimize the character set used. However, Donald Knuth argued that BNF should rather be read as Backus–Naur Form, as it is "not a normal form in any sense",[6] unlike, for instance, Chomsky Normal Form.
A BNF specification is a set of derivation rules, written as
<symbol> ::= __expression__
where <symbol> is a nonterminal, and the __expression__ consists of one or more sequences of symbols; more sequences are 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. On the other hand, symbols that appear on a left side are non-terminals and are always enclosed between the pair <>.
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-suffix-part> <EOL> | <personal-part> <name-part> <personal-part> ::= <first-name> | <initial> "." <street-address> ::= <house-num> <street-name> <opt-apt-num> <EOL> <zip-part> ::= <town-name> "," <state-code> <ZIP-code> <EOL> <opt-suffix-part> ::= "Sr." | "Jr." | <roman-numeral> | ""
This translates into English as:
Note that many things (such as the format of a first-name, apartment specifier, ZIP-code, and Roman numeral) are left unspecified here. If necessary, they may be described using additional BNF rules.
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> | "" <!-- "" is empty string, i.e. no 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> "'" <!-- actually, the original BNF did not use quotes -->
This assumes that no whitespace is necessary for proper interpretation of the rule. <EOL> represents 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.
In the U.S. postal address example above, the entire block-quote is a syntax. Each line or unbroken grouping of lines is a rule; for example one rule begins with "<name-part> ::=". The other part of that rule (aside from a line-end) is an expression, which consists of two lists separated by a pipe "|". These two lists consists of some terms (three terms and two terms, respectively). Each term in this particular rule is a rule-name.
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 regular expression 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 and RBNF are other extensions commonly used to describe Internet Engineering Task Force (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:
This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.