BNF grammar for left-associative operators

Question

I have the following EBNF grammar for simple arithmetic expressions with left-associative operators:

expression:
    term {+ term}
    
term:
    factor {* factor}
    
factor:
    number
    ( expression )

How can I convert this into a BNF grammar without changing the operator associativity? The following BNF grammar does not work for me, because now the operators have become right-associative:

expression:
    term
    term + expression
    
term:
    factor
    factor * term
    
factor:
    number
    ( expression )

Wikipedia says:

Several solutions are:

rewrite the grammar to be left recursive, or

rewrite the grammar with more nonterminals to force the correct precedence/associativity, or

if using YACC or Bison, there are operator declarations, %left, %right and %nonassoc, which tell the parser generator which associativity to force.

But it does not say how to rewrite the grammar, and I don't use any parsing tools like YACC or Bison, just simple recursive descent. Is what I'm asking for even possible?

Puppy · Accepted Answer · 2012-07-11T14:22:36.523

10

expression
    : term 
    | expression + term;

Just that simple. You will, of course, need an LR parser of some description to recognize a left-recursive grammar. Or, if recursive descent, recognizing such grammars is possible, but not as simple as right-associative ones. You must roll a small recursive ascent parser to match such.

Expression ParseExpr() {
    Expression term = ParseTerm();
    while(next_token_is_plus()) {
        consume_token();
        Term next = ParseTerm();
        term = PlusExpression(term, next);
    }
    return term;
}

This pseudocode should recognize a left-recursive grammar in that style.

edited Jul 11 '12 at 14:22

answered Jul 11 '12 at 14:14

Puppy

144,682
38
256
465

The last line will send my recursive descent parser into an infinite recursion. – fredoverflow Jul 11 '12 at 14:17
1

As I said, you must use an LR parser (or recursive ascent) to recognize a left-recursive grammar. – Puppy Jul 11 '12 at 14:20

score 1 · Answer 2 · answered Dec 08 '16 at 16:12

1

What Puppy suggests can also be expressed by the following grammar:

expression: term opt_add
opt_add: '+' term opt_add
       | /* empty */

term:  factor opt_mul
opt_mul: '*' factor opt_mul
       | /* emtpty */

factor: number
      | '(' expression ')

answered Dec 08 '16 at 16:12

Branco Medeiros

729
4
10

1

Removing left recursion like this makes the operators right associative. – Indrajit Banerjee Feb 18 '20 at 11:42

BNF grammar for left-associative operators

2 Answers2