## CSE 428: Lecture 20

#### Parentheses

Due to the higher-order features of ML, the rules about parentheses in expressions might be, in certain cases, a bit different than in other languages. Consider for instance the following definition of the identity function:
```   - fun f(x) = x;
val f = fn : 'a -> 'a
```
Now, suppose that we want to write the expression
```   not(f(true));
```
in a language like C or Pascal you could eliminate the external parentheses, and write
```   not f(true);
```
In ML, however, if we write such a thing, we get
```   - not f(true);
stdIn:117.1-117.12 Error: operator and operand don't agree [tycon mismatch]
operator domain: bool
operand:         'Z -> 'Z
in expression:
not f
```
The reason is that ML tries to interpret the above expression as
```   (not f)(true)
```
This is because in an expression of the form
```   op1 op2 const
```
it might be the case that op1 is an higher order function, which takes as an argument a function op2, and gives as result something that can be applied to const. In a language without higher order this interpretation would not make sense, and therefore the rules of priority are usually defined so that op1 op2 const is parsed as op1 (op2 const)

In general, in ML, an expression of the form

```   e1 e2 e3 ... en
```
is parsed as
```   (...((e1 e2) e3) ... en)
```
Of course this is just the default. In ML, like in other languages, there are some priority rules (like the precedence of "*" over "+") that may alter this order. In general, if you are unsure of how an expression is parsed, just put explicit parentheses to make sure that it is parsed the way you want.

One simplification that is always allowed in ML, however, is the elimination of the innermost parentheses around a "token". In other words, we can always write f x instead of f(x). Hence, for instance, we can write the identity function as

```   fun f x = x;
```
and the expression not(f(true)) as
```   not (f true);
```

#### Match nonexhaustive

When you define a function by pattern matching, you might get a warning of the form "match nonexhaustive". This means that, in the patterns, you have not covered all possible cases, i.e. all possible patterns that the input data may present.

For instance, consider the following function which gives the maximum of a list of integers:

```   fun maxlist [x] = x
| maxlist (x::l) = let val y = maxlist l
in  if x < y then y else x
end;
```
if you compile this definition in SML, you'll get a non-exhaustive matching warning. In fact, the case of emptylist is missing: If you write the expression
```   maxlist []
```
you will get a run-time error.

When the missing cases corresponds to arguments for which the value of the function is undefined, the correct way to eliminate the warnings would be by introducing exceptions (ML allows to handle exceptions in a way similar to C++ and Java).

However, if you are sure that the missing cases do not correspond to interesting cases (i.e. to cases that may present in input) then you can just ignore the warnings of non-ehaustive matching.

For instance, suppose that you want to write a function which construct a balanced tree from a list, and that to this purpose you need to define an internal auxiliary function

```   half: int * 'a list -> 'a list * 'a list
```
such that half(n,l) gives as result the two lists obtained by dividing l in two lists of equal length (plus or minus 1), and suppose that, in your intention, n represents the length of the original list l.

Then you probably will give a definition of the following form:

```   fun half(0, nil) = ...
| half(n,(x::l)) = ... ;
```
this will give you a non-exhaustive matching warning. (since the case (0,non-empty) is missing.) However, if you make sure that in your main program you always call your auxiliary function with an expression of the form
```   half(n,l);
```
where n is defined as the length of l, then your program will never give a runtime error and you don't need to worry about the warning.

#### Equality types

Consider the following two declarations in ML, the first defining a function which checks whether two lists have the same number of elements, and the second checking that they are also the same elements.
```   - fun same_length (nil,nil) = true
| same_length (x::l,y::k) = same_length(l,k)
| same_length(l,k) = false;
val same_length = fn : 'a list * 'b list -> bool
-
- fun same_list(nil,nil) = true
| same_list(x::l,y::k) = x=y andalso same_list(l,k)
| same_list(l,k) = false;
val same_list = fn : ''a list * ''a list -> bool
```
As we can see, the ML type inference mechanism gives for same_list a type different from the one given for same_length, i.e. we get a parameter ''a instead of 'a. This is due to the fact that, in the definition of same_list, we make an equality test between two elements of the lists (x and y). The double quote notation, in ML, is used to represent a type where equality is defined (equality type).

Examples of equality types are integers, booleans, characters, reals, and any other structure (predefined or used-defined) made by equality types. For instance, pairs of equality types, lists of equality types, trees of equality types etc.

Examples of types on which equality is not defined are functional types and everything constructed with functional types. Thus, if we call the two functions above with lists of functions as arguments, same_length will give an answer true or false while same_list will give a type error.

Examples:

```   - same_length([1,2],[2,3]);
val it = true : bool
- same_list([1,2],[2,3]);
val it = false : bool
- fun f x = x;
val f = fn : 'a -> 'a
- fun g x = x;
val g = fn : 'a -> 'a
- same_length([f],[g]);
val it = true : bool
- same_list([f],[g]);
stdIn:24.1-24.19 Error: operator and operand don't agree [equality type required]
operator domain: ''Z list * ''Z list
operand:         ('Y -> 'Y) list * ('X -> 'X) list
in expression:
same_list (f :: nil,g :: nil)
```

### Some exercises with binary trees

Consider the following definition of binary trees, which has the empty tree as base case:
```   datatype 'a btree = emptybt | consbt of 'a * 'a btree * 'a btree;
```

#### Visits

Below we define the three typical visits of a binary tree. Each function puts the result of the visit in a list.
• In_visit (first the left subtree, then the root, then the right subtree).
```   fun in_visit emptybt = []
| in_visit (consbt(x,t1,t2)) = (in_visit t1) @ [x] @ (in_visit t2);
```
• Pre_visit (first the root, then the left subtree, then the right subtree).
```   fun pre_visit emptybt = []
| pre_visit (consbt(x,t1,t2)) = [x] @ (pre_visit t1) @ (pre_visit t2);
```
• Post_visit (first the left subtree, then the right subtree, then the root).
```   fun post_visit emptybt = []
| post_visit (consbt(x,t1,t2)) = (post_visit t1) @ (post_visit t2) @ [x];
```
Examples of evaluations
```- in_visit (consbt(1,consbt(2,emptybt,emptybt),consbt(3,emptybt,emptybt)));
val it = [2,1,3] : int list
- pre_visit(consbt(1,consbt(2,emptybt,emptybt),consbt(3,emptybt,emptybt)));
val it = [1,2,3] : int list
- post_visit(consbt(1,consbt(2,emptybt,emptybt),consbt(3,emptybt,emptybt)));
val it = [2,3,1] : int list
```

#### Changing the value of the elements

Suppose we want to define a function which takes a binary tree of integers, and returns a binary tree with the same structure and the same positive nodes, while the negative ones are converted to 0's. The following is a possible definition:
```   fun convert_to_0 emptybt = emptybt
| convert_to_0 (consbt(x,t1,t2)) = let val u1 = convert_to_0(t1)
val u2 = convert_to_0(t2)
in if x < 0
then consbt(0,u1,u2)
else consbt(x,u1,u2)
end;
```