Handle; reduction of id + id * id with grammar $E\to E+E \mid E*E \mid \text{id}$

What is a handle?

A handle of a (right) sentential form is a substring that matches the right-hand side of some production, and whose reduction to the left-hand-side nonterminal is a step in a rightmost (reverse) derivation — that is, the rightmost handle that, when reduced, produces a valid step toward the start symbol.

In shift–reduce parsing we repeatedly prune handles: shift symbols onto the stack until the top stack symbols match a production RHS, then reduce (replace the handle by its nonterminal).

Find the handles for id + id * id

Grammar: $E\to E+E \mid E*E \mid id$. The (unique, rightmost) reduction sequence:

Stack (top shown last)	Input	Action	Handle reduced
id	+ id * id	shift id	—
E	+ id * id	reduce id → E	handle id
E +	id * id	shift +	—
E + id	* id	shift id	—
E + E	* id	reduce	handle id (2nd)
E + E *	id	shift *	—
E + E * id	(empty)	shift	—
E + E * E		reduce	handle id (3rd)
E + E		reduce E*E → E	handle E*E
E		reduce E+E → E	handle E+E

Handles (in reduction order), with the actual substrings:

• id (first id on the left)
• id (the middle id)
• id (the id after *)
• E * E (the redex E*E → E)
• E + E (the redex E+E → E)

Written against the original string, the reduction is a reverse of the rightmost derivation:
$$id + id * id \;\xrightarrow{\text{reduce id}} E + id*id \xrightarrow{\text{reduce id}} E+E*id \xrightarrow{} E+E*E \xrightarrow{E+E} \xrightarrow{E}$$
The seeds/handles reduced, going left-to-right on the string: the *ids**, then EE, then E + E`.