How to tell the precedence of operators in a context free grammar

Question

How can we know which of the following logical operations ( or, and, not ) in the bellow context free grammar have higher precedence? Is there a general approach to this kind of problems?

X → X or Y | Y

Y → Y and Z | Z

Z → not Z | (X) | true | false

Eudemonia

Answer 1

Here's an example approach:

expr -> addExpr;
addExpr -> multExpr (('+'|'-') multExpr)*;
multExpr -> terminalExpr (('*'|'/') terminalExpr)*;
terminalExpr -> integer | variable | '(' expr ')';

But the associativity is ambiguous. Here's a more explicit way in BNF:

expr -> addExpr;
addExpr -> addExpr '+' multExpr | addExpr '-' multExpr | multExpr;
multExpr -> multExpr '*' terminalExpr | multExpr '/' terminalExpr | terminalExpr;
terminalExpr -> integer | variable | '(' expr ')';

These grammars define the operators * and / as having more precedence as + and -. You declare the operation with higher precedence deeper in the parse tree, so they will be tried first by the parser.

Benzol

good example. thanks. so what it means is that the operation with higher precedence should be lower in the parsing tree. maybe u can add that to your answer and id plus one it! – Eudemonia

@rici huh I don't know what I was thinking :-\ Thanks for noticing this, I've fixed the answer. – Benzol

Thanks. So if I want to add an exponential(**) operator to the above grammar(which is having the highest precedence) then I need to add this operation in the second last level of your grammar right? – Autumnautumnal

@Autumnautumnal Since exponentiation has a higher precedence than multiplication, you need to add a separate level to the grammar (between multExpr and terminalExpr) in order to handle it correctly. – Benzol