Can't get my head around creating unambiguous grammar for if else

Question

This is from a tutorial our professor wrote and I can't get my head around it. I can come up with derivation but I can't come up with grammar by just analyzing the derivation.

What does "matched" refer to in this context?

Can you explain how matched_if, matched_stmt, unmatched_if works in simple words :) ?

The following is an unambiguous grammar for the problem:

stmt → if_stmt | nonif_stmt
if_stmt → matched_if | unmatched_if
matched_if → 'if' logical_expr 'then' matched_stmt 'else' matched_stmt
matched_stmt → mathced_if | nonif_stmt
unmatched_if → 'if' logical_expr 'then' stmt
 | 'if' logical_expr 'then' matched_stmt 'else' unmatched_if
logical_expr → id '==' lit
nonif_stmt → assgn_stmt
assgn_stmt → id '=' expr
expr → expr '+' term | term
term → '(' expr ')' | id
id → 'A' | 'B' | 'C'
lit → '0' | '1' | '2'


Consider the following input:
if A == 0 then
  if B == 1 then
    C = A + B
  else
    B = C

Let us do a leftmost derivation for the input:

stmt
=> if_stmt
=> unmatched_if
=> 'if' logical_expr 'then' stmt
=> 'if' id '==' lit 'then' stmt
=> 'if' 'A' '==' lit 'then' stmt
=> 'if' 'A' '==' '0' 'then' stmt
=> 'if' 'A' '==' '0' 'then' if_stmt
=> 'if' 'A' '==' '0' 'then' matched_if
=> 'if' 'A' '==' '0' 'then' 'if' logical_expr 'then' matched_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' id '==' lit 'then' matched_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' lit 'then' matched_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' matched_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' nonif_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' assgn_stmt 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' id '=' expr 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' expr '+' term 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' term '+' term 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' id '+' term 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + term 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' matched_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' nonif_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' assgn_stmt
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' id '=' expr
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' 'B' '=' expr
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' 'B' '=' term
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' 'B' '=' id
=> 'if' 'A' '==' '0' 'then' 'if' 'B' '==' '1' 'then' 'C' '=' 'A' + 'B' 'else' 'B' '=' 'C'

Answer 1

"Matched" means that every then is matched with an else.

It's not necessary for an if statement to have a matching else, but if it doesn't, it cannot be inside an matched if statement, because that would imply that some else matches an outer then instead of an inner then).

All the grammar does is formalize the above.

A similar problem is writing a grammar for ordinary arithmetic expressions with the additional rule that you can leave out trailing close parentheses. (So you could write (1+2*(1+2*(1+2, for example.) That language is clearly unambiguous, but when you are writing the grammar for it, you need to deal with unmatched parentheses and thus expressions containing unmatched parentheses. That's the same use of the word "matched"· (and the solution would be somewhat similar).