LL1 grammar for arithmetic expressions and assignments

I would like to design an LL1 grammar for arithmetic equations and variable assignments. I began with this grammar:

I have a nonambiguous grammar for arithmetic expressions:

E → T E’
E’ →  | + E
T → id T’
T’ →  | * T

However, I'm not sure how to incorporate variable assignments into the E productions.

How I tried previously was:

    stmt -> assignment SEMI | RETURN stmt_prime 
   | LBRACE stmt_list RBRACE 
   | IF LPAREN assignment RPAREN stmt stmt_prime_prime 
   | FOR LPAREN assignment SEMI assignment SEMI assignment RPAREN stmt | 
    stmt_prime -> SEMI | -> assignment SEMI
    stmt_prime_prime -> NOELSE 
    | ELSE stmt
    assignment -> ID typ ASSIGN expr | expr  
    expr -> TE* 
    E* -> + TE* | -TE* | epsilon
    T -> FT*
    T* -> * FT* | / FT* | epsilon
    F -> (E) | int_literal | TRUE | FALSE

Solution

assignment -> ID ASSIGN expr | expr

(I'm ignoring the typ part because I assume it got there by accident)

The problem here is that both ID ASSIGN expr and expr could start with an ID (or at least they could if T contained ID as an option, which I assume was the intention), so this isn't LL(1). It is LL(2) though, so if you're fine with that, you can just add an andalso next_token = ASSIGN to the if condition in your parser and be done with it.

If you do need it to be LL(1), you'll have to adjust the language allowed by your parser, I'm afraid (that is, there is no LL(1) grammar that matches exactly the same language as your current grammar). One easy way out would be to simply add a keyword like SET before assignment, though admittedly that's ugly.

The other alternative would be to allow arbitrary expressions as the left operand of =, making your assignment rule:

assignment -> exp (ASSIGN exp)?

Which is LL(1). The downside of this is that it allows a lot of non-sensical code such as 1+2 := 42, but you can fix that outside of the grammar. That is, your code for parsing assignments can simply call parse_exp and then, if the next token is an ASSIGN and the expression returned by parse_exp is not just an identifier, raise an error that the left-side of an assignment must be an identifier.