How can I correctly express in BNF this condition? - language-design

Question

I am looking for a way to express the following types of conditions in BNF:
if(carFixed) { }
if(carFixed = true) {}
if(cars >= 4) { }
if(cars != 15) { }
if(cars < 3 && cars > 1) {}
Note:
* denotes 0 or more instances of something.
I have replaced normal BNF ::= with :.
I presently am using the following code, and am not sure if it's correct:
conditionOperator: "=" | "!=" | "<=" | ">=" | "<" | ">" | "is";
logicalAndOperator: "&&";
condition: (booleanIdentifier ((conditionOperator booleanIdentifier)* (logicalAndOperator | logicalOrOperator) booleanIdentifer (conditionOperator booleanIdentifier)*)*);

Answer 1

There are several approaches and they usually rely on the capabilities of the parser to indicate precedence and associativty. One that is typically used with recursive-descent parsers is to recreate the precedence of the operators by using the hierarchy provided by the bnf (or, in this case, pseudo-bnf) structure.
(In the examples bellow, CONDITIONAL_OP are the likes of <, != etc and LOGICAL_OP are &&, || etc)
Something in the lines of:
condition: logicalExpr
logicalExpr: conditionalExpr (LOGICAL_OP conditionalExpr)*
conditionalExpr: primary (CONDITIONAL_OP primary)*
primary: NUMBER | IDENTIFIER | BOOLEAN_LITERAL | '(' condition ')'
The problem with the above solution is that the left-associativity of the operators is lost and requires special measures to restore it while parsing.
For parsers able to deal with left recursion, a more 'correct' notation could be:
condition: logicalExpr
logicalExpr: logicalExpr LOGICAL_OP conditionalExpr
| conditionalExpr
conditionalExpr: conditionalExpr CONDITIONAL_OP primary
| primary
primary: NUMBER | IDENTIFIER | BOOLEAN_LITERAL | '(' condition ')'
Finally, some parsers allow a special notation to indicate precedence and associativity. Something like (note that this is a completely invented syntax):
%LEFT LOGICAL_OP
%LEFT CONDITIONAL_OP
condition: condition CONDITIONAL_OP condition
| condition LOGICAL_OP condition
| '(' condition ')'
| NUMBER
| IDENTIFIER
| BOOLEAN_LITERAL
Hope this points you the right direction.