How does one make an unambiguous context-free grammar for arithmetic expressions? Say I have a context-free grammar defined by the following rule.

$$
\langle EXPR\rangle \rightarrow \langle EXPR\rangle + \langle EXPR\rangle~|~\langle EXPR\rangle \times \langle EXPR\rangle~|~(\langle EXPR \rangle)~|~x
$$

This grammar is ambiguous since, for instance, I can generate the string $x + x \times x$ via more than 1 leftmost derivation.

How could I make this grammar unambiguous? Should I make sure that no $\langle EXPR\rangle + \langle EXPR\rangle$ is evaluated after a $\langle EXPR\rangle \times \langle EXPR\rangle$ as such:

$$
\langle EXPR\rangle \rightarrow \langle EXPR\rangle + \langle EXPR\rangle~|~\langle MUL\_EXPR\rangle \times \langle MUL\_EXPR\rangle~|~(\langle EXPR \rangle)~|~x \\
\langle MUL\_EXPR \rangle \rightarrow \langle EXPR\rangle \times \langle EXPR\rangle~|~(\langle EXPR \rangle)~|~x \\
$$
Indeed you made a step to resolve ambiguity, but your solution does not give a fully unmabiguous grammar yet. The string $x+x+x$ can be parsed in two different ways, like $x + [x+x]$ or like $[x+x]+x$, where the brackets indicate parses. As far as I see your solution resolves semantical ambiguities (it fixes the relative order of + and x) but not the syntactical ambiguity. So obtaining an umambiguous grammar is not the only goal for an example like this: we want to respect meaning (here operator precedence). (Perhaps there is more official terminology for that)

Your example is a familiar one, and is used in wikipedia/Syntax diagram:

&lt;expression&gt; ::= &lt;term&gt; | &lt;expression&gt; "+" &lt;term&gt;
&lt;term&gt;       ::= &lt;factor&gt; | &lt;term&gt; "*" &lt;factor&gt;
&lt;factor&gt;     ::= &lt;constant&gt; | &lt;variable&gt; | "(" &lt;expression&gt; ")"


Probably you will know that not every grammar has an unambiguous equivalent, so no general approach is possible.