I'm trying to learn about shift-reduce parsing. Suppose we have the following grammar, using recursive rules that enforce order of operations, inspired by the ANSI C Yacc grammar:

S: A;

P
    : NUMBER
    | '(' S ')'
    ;

M
    : P
    | M '*' P
    | M '/' P
    ;

A
    : M
    | A '+' M
    | A '-' M
    ;

And we want to parse 1+2 using shift-reduce parsing. First, the 1 is shifted as a NUMBER. My question is, is it then reduced to P, then M, then A, then finally S? How does it know where to stop?

Suppose it does reduce all the way to S, then shifts '+'. We'd now have a stack containing:

S '+'

If we shift '2', the reductions might be:

S '+' NUMBER
S '+' P
S '+' M
S '+' A
S '+' S

Now, on either side of the last line, S could be P, M, A, or NUMBER, and it would still be valid in the sense that any combination would be a correct representation of the text. How does the parser "know" to make it

A '+' M

So that it can reduce the whole expression to A, then S? In other words, how does it know to stop reducing before shifting the next token? Is this a key difficulty in LR parser generation?