# Introduction to Programming Languages/Precedence and Associativity

### Precedence and Associativity

The semantics of a programming language is not defined by its syntax. There are, however, some aspects of a program's semantics that are completely determined by how the grammar of the programming language is organized. One of these aspects is the order in which operators are applied to their operands. This order is usually defined by the precedence and the associativity between the operators. Most of the algorithms that interpreters or compilers use to evaluate expressions tend to analyze first the operators that are deeper in the derivation tree of that expression. For instance, lets consider the following C snippet, taken from Modern Programming Languages: `a = b < c ? * p + b * c : 1 << d ()`. The side figure shows the derivation tree of this assignment.

Given this tree, we see that the first star, e.g, `*p` is a unary operator, whereas the second, e.g, `*c` is binary. We also see that we will first multiply variables b and c, instead of summing up the contents of variable p with b. This evaluation order is important not only to interpret the expression, but also to type check it or even to produce native code for it.

Precedence: 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.

An ambiguous grammar might compromise the exact meaning of the precedence rules in a programming language. To illustrate this point, we will use the grammar below, that recognizes expressions containing subtractions and divisions of numbers:

```<exp> ::= <exp> - <exp>
| <exp> / <exp>
| (<exp>)
| <number>
```

According to this grammar, the expression 4 - 4 / 2 has two different derivation trees. In the one where the expression 4 - 4 is more deeply nested, we have that 4 - 4 / 2 = 0. On the other hand, in the tree where we have 4/2 more deeply nested, we have that 4 - 4 / 2 = 2. This figure illustrates the ambiguity that occurs because the underlying grammar does not provide any type of associativity to the minus operator

It is possible to re-write the grammar to remove this ambiguity. Below we have a slightly different grammar, in which division has higher precedent than subtraction, as it is usual in mathematics:

```<exp> ::= <exp> - <exp>
| <mulexp>
<mulexp> ::= <mulexp> / <mulexp>
| (<exp>)
| <number>
```

As a guideline, the farther the production rule is from the starting symbol, 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. This figure shows the derivation tree for the expression 4 - 4 / 2. There is only one way to derive a tree for this expression, given the underlying grammar.

By adding the `mulexp` node into our grammar, we have given division higher precedence over subtraction. However, we might still have problems to evaluate parsing trees unambiguously. These problems are related to the associativity of the operators. As an example, the expression 4 - 3 - 2 can be interpreted in two different ways. We might consider 4 - 3 - 2 = (4 - 3) - 2 = -1, or we might consider 4 - 3 - 2 = 4 - (3 - 2) = 3. The two possible derivation trees that we can build for this expression are shown below:

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. Thus, there is only one way to build a parsing tree for our example 4 - 3 - 2. This figure shows the derivation tree for the expression 4 - 3 - 2. In this grammar, subtraction has been made left-associative; thus, this expression is equivalent to (4-3)-2, given an interpreter that starts evaluating these trees from the leaves up.
```<exp> ::= <exp> - <mulexp>
| <mulexp>
<mulexp> ::= <mulexp> / <rootexp>
| <rootexp>
<rootexp> ::= ( <exp> )
| <number>
```

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.