Editing Ambiguous grammar (section)

=== Addition and subtraction ===
The [[context free grammar]]
: A → A + A | A − A | a
is ambiguous since there are two leftmost derivations for the string a + a + a:

{| border="0"
|-----
| &nbsp;&nbsp;&nbsp;&nbsp; || A || → A + A
| &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;&nbsp;
| A || → A + A
|-----
| &nbsp;&nbsp;&nbsp;&nbsp; ||  || → a + A
| &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;&nbsp;
| || → A + A + A (First A is replaced by A+A. Replacement of the second A would yield a similar derivation.)
|-----
| &nbsp;&nbsp;&nbsp;&nbsp; ||  || → a + A + A
| &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;&nbsp;
| || → a + A + A
|-----
| &nbsp;&nbsp;&nbsp;&nbsp; ||  || → a + a + A
| &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;&nbsp;
| || → a + a + A
|-----
| &nbsp;&nbsp;&nbsp;&nbsp; ||  || → a + a + a
| &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;&nbsp;&nbsp;&nbsp;
| || → a + a + a
|}

As another example, the grammar is ambiguous since there are two [[parse tree]]s for the string a + a &minus; a:
: [[Image:Leftmostderivations jaredwf.svg|Leftmostderivations jaredwf.svg|400px|class=skin-invert]]

The language that it generates, however, is not inherently ambiguous; the following is a non-ambiguous grammar generating the same language:
: A → A + a | A − a | a