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Extended Backus–Naur form

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In computer science, Extended Backus–Naur Form (EBNF) is a family of metasyntax notations used for expressing context-free grammars: that is, a formal way to describe computer programming languages and other formal languages. They are extensions of the basic Backus–Naur Form (BNF) metasyntax notation.

The earliest EBNF was originally developed by Niklaus Wirth. However, many variants of EBNF are in use. The International Organization for Standardization has adopted an EBNF standard (ISO/IEC 14977). This article uses EBNF as specified by the ISO for examples applying to all EBNF:s. Other EBNF variants use somewhat different syntactic conventions.

Basics

A code, e.g. source code of a computer program consisting of terminal symbols—that is, visible characters, digits, punctuation marks, white space characters, etc.

The EBNF defines production rules where sequences of symbols are respectively assigned to a nonterminal: 

digit excluding zero = "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ;
digit                = "0" | digit excluding zero ;

This production rule defines the nonterminal digit which is on the left side of the assignment. The vertical bar represents an alternative and the terminal symbols are enclosed with quotation marks followed by a semicolon as terminating character. Hence a Digit is a 0 or a digit excluding zero that can be 1 or 2 or 3 and so forth until 9.

A production rule can also include a sequence of terminals or nonterminals, each separated by a comma: 

twelve                          = "1" , "2" ;
two hundred one                 = "2" , "0" , "1" ;
three hundred twelve            = "3" , twelve ;
twelve thousand two hundred one = twelve , two hundred one ;

Expressions that may be omitted or repeated can be represented through curly braces { ... }: 

natural number = digit excluding zero , { digit } ;

In this case, the strings 1, 2, ...,10,...,12345,... are correct expressions. To represent this, everything that is set within the curly braces may be repeated arbitrarily often, including not at all.

An option can be represented through squared brackets [ ... ]. That is, everything that is set within the square brackets may be present just once, or not at all: 

integer = "0" | [ "-" ] , natural number ;

Therefore an integer is a zero (0) or a natural number that may be preceded by an optional minus sign.

EBNF also provides, among other things, the syntax to describe repetitions of a specified number of times, to exclude some part of a production, or to insert comments in an EBNF grammar.

Motivation to extend the BNF

The BNF had the problem that options and repetitions could not be directly expressed. Instead, they needed the use of an intermediate rule or alternative production defined to be either nothing or the optional production for option, or either the repeated production or itself, recursively, for repetition. The same constructs can still be used in EBNF.

Option: 

signed number = [ sign ] , number ;

can be defined in BNF style as: 

<signed number> ::= <sign> <number> | <number> ;

or 

<signed number> ::= <optional sign> <number> ;
<optional sign> ::= ε | <sign> ; (* epsilon ("ε") is used to denote more clearly an empty production *)

Repetition: 

number = { digit } ;

can be defined in BNF style as: 

<number> ::= ε | <number> <digit> ;

Note: while EBNF (ISO) uses comma ',' for concatenation, BNF uses no operator at all to express concatenation.

Other additions and modifications

The EBNF eliminates some of the BNF's flaws:

  • The BNF uses the symbols (<, >, |, ::=) for itself. When these appear in the language that is to be defined, the BNF cannot be used without modifications and explanation.
  • A BNF syntax can only represent a rule in one line.

The EBNF solves these problems:

  • Terminals are strictly enclosed within quotation marks ("..." or '...'). The angle brackets ("<...>") for nonterminals can be omitted.
  • A terminating character, the semicolon, marks the end of a rule.

Furthermore there are mechanisms for enhancements, defining the number of repetitions, excluding alternatives (e.g. all characters excluding quotation marks), comments, etc.

Despite all enhancements, the EBNF is not "more powerful" than the BNF in the sense of the language it can define. As a matter of principle any grammar defined in EBNF can also be represented in BNF. However, this often leads to a considerably larger representation.

The EBNF has been standardized by the ISO under the code ISO/IEC 14977:1996(E).

Under some circumstances, any extended BNF is referred to as EBNF. For example, the W3C uses one EBNF to specify XML.

Another example

A pascal-like programming language that allows only assignments can be defined in EBNF as follows: 

(* a simple program in EBNF − Wikipedia *)
program = 'PROGRAM' , white space , identifier , white space ,
           'BEGIN' , white space ,
           { assignment , ";" , white space } ,
           'END.' ;
identifier = alphabetic character , { alphabetic character | digit } ;
number = [ "-" ] , digit , { digit } ;
string = '"' , { all characters − '"' } , '"' ;
assignment = identifier , ":=" , ( number | identifier | string ) ;
alphabetic character = "A" | "B" | "C" | "D" | "E" | "F" | "G"
                     | "H" | "I" | "J" | "K" | "L" | "M" | "N"
                     | "O" | "P" | "Q" | "R" | "S" | "T" | "U"
                     | "V" | "W" | "X" | "Y" | "Z" ;
digit = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ;
white space = ? white space characters ? ;
all characters = ? all visible characters ? ;

A syntactically correct program then would be: 

PROGRAM DEMO1
BEGIN
  A0:=3;
  B:=45;
  H:=-100023;
  C:=A;
  D123:=B34A;
  BABOON:=GIRAFFE;
  TEXT:="Hello world!";
END.

The language can easily be extended with control flows, arithmetical expressions, and Input/Output instructions. Then a small, usable programming language would be developed.

The following characters that are proposed in the standard as normal representation have been used:

Usage Notation
definition =
concatenation ,
termination ;
separation |
option [ ... ]
repetition { ... }
grouping ( ... )
double quotation marks " ... "
single quotation marks ' ... '
comment (* ... *)
special sequence ? ... ?
exception -

Conventions

1. The following conventions are used:

  • Each meta-identifier of Extended BNF is written as one or more words joined together by hyphens;
  • A meta-identifier ending with “-symbol” is the name of a terminal symbol of Extended BNF.

2. The normal character representing each operator of Extended BNF and its implied precedence is (highest precedence at the top): 

* repetition-symbol
- except-symbol
, concatenate-symbol
| definition-separator-symbol
= defining-symbol
; terminator-symbol

3. The normal precedence is overridden by the following bracket pairs: 

'  first-quote-symbol            first-quote-symbol  '
"  second-quote-symbol          second-quote-symbol  "
(* start-comment-symbol          end-comment-symbol *)
(  start-group-symbol              end-group-symbol  )
[  start-option-symbol            end-option-symbol  ]
{  start-repeat-symbol            end-repeat-symbol  }
?  special-sequence-symbol   special-sequence-symbol ?

The first-quote-symbol is the apostrophe as defined by ISO/IEC 646:1991, that is to say Unicode 0x0027 ('); however, the font used in ISO/IEC 14977:1996(E) renders it very much like the acute, Unicode 0x00B4 (´), so confusion sometimes arises. However, the ISO Extended BNF standard invokes ISO/IEC 646:1991, "ISO 7-bit coded character set for information interchange", as a normative reference and makes no mention of any other character sets, so formally, there is no confusion with Unicode characters outside the 7-bit ASCII range.

As examples, the following syntax rules illustrate the facilities for expressing repetition: 

aa = "A";
bb = 3 * aa, "B";
cc = 3 * [aa], "C";
dd = {aa}, "D";
ee = aa, {aa}, "E";
ff = 3 * aa, 3 * [aa], "F";
gg = {3 * aa}, "G";

Terminal strings defined by these rules are as follows: 

aa: A
bb: AAAB
cc: C AC AAC AAAC
dd: D AD AAD AAAD AAAAD etc.
ee: AE AAE AAAE AAAAE AAAAAE etc.
ff: AAAF AAAAF AAAAAF AAAAAAF
gg: G AAAG AAAAAAG etc.

EBNF Extensibility

According to the ISO 14977 standard EBNF is meant to be extensible, and two facilities are mentioned. The first is part of EBNF grammar, the special sequence, which is arbitrary text enclosed with question marks. The interpretation of the text inside a special sequence is beyond the scope of the EBNF standard. For example, the space character could be defined by the following rule: 

space = ? US-ASCII character 32 ?;

The second facility for extension is using the fact that parentheses cannot in EBNF be placed next to identifiers (they must be concatenated with them). The following is valid EBNF: 

something = foo, ( bar );

The following is not valid EBNF: 

something = foo ( bar );

Therefore, an extension of EBNF could use that notation. For example, in a Lisp grammar, function application could be defined by the following rule: 

function application = list( symbol , { expression } );

See also

References

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.

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