Examples of BNF Grammars

Here are some examples from a site on the internet.

A grammar for simple sentences. Things in this language are:


THE MAN BITES A DOG
A DOG PETS A DOG

Things not in this language are:


MAN BITES DOG

Here is the grammar:


<sentence> ::= <subject> <predicate>
<subject> ::= <article> <noun>
<predicate> ::= <verb> <direct-object>
<direct-object> ::= <article> <noun>
<article> ::= THE | A
<noun> ::= MAN | DOG
<verb> ::= BITES | PETS

An ambiguous grammar for binary numbers:


<binary-string> ::= 0 | 1 | <binary-string> <binary-string>

An unambiguous grammar for binary numbers:


<binary-string> ::= <binary-string> 0 | <binary-string> 1 | 0 | 1

An ambiguous grammar for some arithmatic expressions using + and - and variables X, Y and Z. This grammar also does not assert any precedence between + and - or associativity.
An unambiguous grammar for some arithmatic expressions using + and - and variables X, Y and Z. THis asserts * is performed before + and both * and + are done left to right.

The following shows the ^ operator performed right to left


<sentence> ::= <expression>
<expression> ::= <term> | <term> ^ <expression>
<term> ::= <identifier> | <term> * <identifier>
<identifier> ::= X | Y | Z

This is ambiguous. Why?


<sentence> ::= <expression>
<expression> ::= <term> | <expression> + <term>
<term> ::= <identifier> | <term> * <identifier> | <expression>
<identifier> ::= X | Y | Z

This solves the ambiguity but doesn't handle all cases that you might want.


<sentence> ::= <expression>
<expression> ::= <term> | <expression> + <term>
<term> ::= <identifier> | <term> * <identifier> | ( <expression> )
<identifier> ::= X | Y | Z

How is this language defined by this grammar different than the language above?


<sentence> ::= <expression>
<expression> ::= <term> | <expression> + <term>
<term> ::= <identifier> | <term> * <identifier>
<identifier> ::= X | Y | Z | ( <expression> )

This is a grammar for a language that has signed digits.


<signed> ::= + <nosign> | - <nosign>
<nosign> ::= <digit> <nosign> | <digit>
<digit> ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

This is a grammar for an even number of X's

This is a grammar for numbers with no leading zeros and signs allowed on nonzero numbers.


<number> ::= + <nosign> | - <nosign> | <nosign> | 0
<nosign> ::= <nonzero> <anydig> | <nonzero>
<anydig> ::= 0 | <nozero>
<nozero> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Some more grammars with errors:

This is ambiguous

This is ambiguous


<zot> ::= <zing> | a
<zing> ::= <zot> | b

This has a different sort of problem


<zot> ::= <zing> | a
<zing> ::= <zing> b