Operands vs Operators Precedence in C/C++

Описание: We say that an operator op1 has greater precedence than another operator op2 if op1 must be evaluated before op2 whenever both operators are in the same expression. For instance, it is a usual convention that we evaluate divisions before subtractions in arithmetic expressions that contain both operators. Thus, we generally consider that 4 - 4 / 2 = 2. However, if we were to use a different convention, then we could also consider that 4 - 4 / 2 = (4 - 4) / 2 = 0.

As a guideline, the farther from the starting symbol is the production rule, the deeper its nodes will be nested in the derivation tree. Consequently, operators that are generated by production rules that are more distant from the starting symbol of the grammar tend to have higher precedence. This, of course, only applies if our evaluation algorithm starts by computing values from the leaves of the derivation tree towards its root. Going back to the example above, we can only build the derivation tree of the expression 4 - 4 / 2 in one unique way.


In arithmetics, mathematicians have adopted the convention that the leftmost subtraction must be solved first. Again, this is just a convention: mathematics would still work, albeit in a slightly different way, had we decided, a couple hundred years ago, that sequences of subtractions should be solved right-to-left. In terms of syntax, we can modify our grammar to always nest more deeply the leftmost subtractions, as well as the leftmost divisions. The grammar below behaves in this fashion. This grammar is no longer ambiguous. Any string that it can generate has only one derivation tree.

Because subtractions are more deeply nested towards the left side of the derivation tree, we say that this operator is left-associative. In typical programming languages, most of the operators are left-associative. However, programming languages also have binary operators that are right-associative. A well-known example is the assignment in C. An assignment command, such as int a = 2 modifies the state of variable a. However, this command is also an expression: it return the last value assigned. In this case, the assignment expression returns the value 2. This semantics allows programmers to chain together sequences of assignments, such as int a = b = 2;. Another example of right-associative operator is the list constructor in ML. This operator, which we denote by :: receives an element plus a list, and inserts the element at the beginning of the list. An expression such as 1::2::3::nil is equivalent to 1::(2::(3::nil)). It could not be different: the type of the operand requires the first operator to be an element, and the second to be a list. Had we evaluated it in a different way, e.g., 1::2::3::nil = ((1::2)::3)::nil, then we would have two elements paired together, which would not pass through the ML type system.