## 10.10.An Unambiguous Grammar for Expressions

It is possible to write a grammar for arithmetic expressions that

```1.   is unambiguous.
2.   enforces the precedence of * and / over + and -.
3.   enforces left associativity.
```

Here is one such grammar:

```1.   <Sum>:       <Sum> + <Product> | <Sum> - <Product> | <Product>
2.   <Product>:   <Product> * <N> | <Product> / <N> | <N>
3.   <N>:         ( <Sum> ) | number
```

If we attempt to build a parse tree for number + number * number, we see there is only one such tree:

The only parse tree is: This parse tree correctly represents left associativity by using recursion on the left. If we rewrote the grammar to use recursion on the right, we would represent right associativity.

Our grammar also correctly represents precedence levels by introducing a new non-terminal symbol for each precedence level. According to our grammar, expressions consist of the sum or difference of terms (or a single term), where a term consists of the product or division of factors (or a single factor), and a factor is a nested expression or a number. Add the power functionality to the grammar.

Here is the grammar:

```1.   <Sum>:           <Sum> + <Product>   (1)
|   <Sum> - <Product>   (2)
|   <Product>           (3)
2.   <Product>:       <Product> * <Pow>   (4)
|   <Product> / <Pow>   (5)
|   <Pow>               (6)
3.   <Pow>:           <N> ^ <Pow>         (7)
|   <N>                 (8)
4.   <N>:             ( <Sum> )           (9)
|   number              (10)
```
What is the problem with the following grammar?

```1.   <Sum>:           <Sum> + <Product>   (1)
|   <Sum> - <Product>   (2)
|   <Product>           (3)
2.   <Product>:       <Product> * <Pow>   (4)
|   <Product> / <Pow>   (5)
|   <Pow>               (6)
3.   <Pow>:           <Pow> ^ <N>         (7)
|   <N>                 (8)
4.   <N>:             ( <Sum> )           (9)
|   number              (10)
```