Recitation 2: Tuples, records and datatypes


Every function in SML takes exactly one value and returns exactly one result.  For instance, our square_root function takes one real value and returns one real value.  The advantage of always taking one argument and returning one result is that the language is extremely uniform.   Later, we'll see that this buys us a lot when it comes to composing new functions out of old ones.

But it looks like we can write functions that take more than one argument!  For instance, we may write:

fun max(r1:real, r2:real):real =
  if r1 < r2 then r2
  else r1
max(3.1415, 2.718)

and it appears as if max takes two arguments.  In truth max takes one argument that is a 2-tuple (also known as an ordered pair.) 

In general, an n-tuple is an ordered sequence of n values written in parenthesis and separated by commas as (expr, expr, ..., expr).  For instance, (42, "hello", true) is a 3-tuple that contains the integer 42 as its first component, the string "hello" as its second component, and the boolean value true as its third component.  As another example, () is the empty tuple.  This is called "unit" in SML.

When you call a function in SML, if it takes more than one argument, then you have to pass it a tuple of the arguments.  For instance, when we write:

max(3.1415, 2.718)

we're passing the 2-tuple (3.1415, 2.718) to the function max.   We could just as well write:

val args = (3.1415, 2.178);
max args  (* evaluates to 3.1415 *)

The type of an n-tuple is written ( t1*...*tn ).  For instance, the type of args above is (real*real).  This notation is based on the Cartesian product in mathematics (i.e., the plane is R^2 = R * R).

Similarly, the 3-tuple (42, "hello", true) has type (int * string * bool).   Notice that max has type (real*real)->real, indicating that it takes one argument (a 2-tuple of reals) and returns one result (a real).

The above grammar for types is a simplification, in fact. A more complete grammar is as follows:

t ::= int  |  real  |  bool  |  string  |  char  |   t1*...*tn  |  t1->t2  |  ( t )

So the types (real*real)->real and real*real->real are exactly the same type.

You can extract the components of a tuple by using the form "#n e" where n is a number between 1 and the size of the tuple.  For instance, #2 (1, "hello", true) evaluates to "hello", whereas #1 (3.1415, 2.178) evaluates to 3.1415

So, for instance, we can rewrite the max function as follows:

fun max(pair: real*real):real =
    if (#1 pair) < (#2 pair) then
  (#2 pair) else (#1 pair);

and this is completely equivalent to the first definition.    This emphasizes that max really does take just one argument -- a pair of real numbers.  But of course, it's a lot less readable than the first definition.  We can get closer to the first definition by declaring local values r1 and r2 and bind them to the appropriate components of the pair:

fun max(pair: real*real):real =
  let val r1 = #1 pair
      val r2 = #2 pair
    if r1 < r2 then r2 else r1

Pattern-matching tuples

This is a little better because we avoid re-computing the same expressions over and over again.  However, it's still not as succinct as our first definition of max.   This is because the first definition uses pattern matching to implicitly de-construct the 2-tuple and bind the components to variables r1 and r2.  You can use pattern matching in a val declaration or in a function definition to deconstruct a tuple.  A tuple pattern is always of the form (x1:t1x2:t2,..., xn:tn).  For instance, here is yet another version of max that uses a pattern in a val declaration to deconstruct the pair:

fun max(pair: real*real):real =
  let val (r1:real, r2:real) = pair
    if r1 < r2 then r2 else r1

In the example above, the val declaration matches the pair against the tuple-pattern (r1:real, r2:real).   This binds r1 to the first component of the pair (#1 pair) and r2 to the second component (#2 pair).  A similar thing happens when you write a function using a tuple-pattern as in the original definition of max:

fun max(r1:real, r2:real):real = if r1 < r2 then r2 else r1

Here, when we call max with the pair (3.1415, 2.718), the tuple is matched against the pattern (r1:real, r2:real) and r1 is bound to the 3.1415 and r2 to 2.718.  As we'll see later on, SML uses pattern matching in a number of places to simplify expressions.

Suppose we wanted to extract both the minimum and the maximum of two numbers in a single function. With tuples, this is easy: we just return a tuple containing both results. Using a let, we can extract both results into separate variables conveniently, too:

fun minmax(a: real, b: real): real*real =
  if a < b then (a, b) else (b, a)
val (mn: real, mx: real) = minmax(2.0,1.0)

This binds mn to 1.0 and mx to 2.0. The type of minmax is (real*real)->(real*real), which we can write without the parentheses because * has a higher precedence than -> when writing type expressions.

In summary:


Records are similar to tuples in that they are data structures for holding multiple values.  However, they are different from tuples in that they carry an unordered collection of labeled values.  In general, record expressions are of the form {x1=e1,...,xn=en} where the identifiers x are labels.  For example, the expression 

{first = "John", last = "Doe", age = 150, balance = 0.12

is a record with four fields named first, last, age, and balance.  You can extract a field from a record by using #id expr where exp is the record and id is the field that you want to extract.  For instance, applying #age to the record above yields 150, whereas applying #balance yields 0.12. 

When creating a record, it does not matter in what order you give the fields.  So the record

{balance = 0.12, age = 150, first = "John", last = "Doe"}

is equivalent to the example above.  Note that when you type in one of these records to the SML top-level, it sorts the fields into a canonical order:

- val pers = { first = "John", last = "Doe",
              age = 150, balance = 0.12 };
val pers = {age=150,balance=0.12,first="John",last="Doe"}
: {age:int,balance:real,first:string,last:string}

The type of a record is written as {x1:t1x2:t2,..., xn:tn}

Just as you can use pattern-matching to extract the components of a tuple, you can use pattern matching to extract the fields of a record.  For instance, you can write:

val {first:string,last:string,age:int,balance:real} = jgm

and SML responds with:

val age = 150 : int
val balance = 0.12 : real
val first = "Greg" : string
val last = "Morrisett" : string

thereby binding the identifiers age, balance, first, and last to the respective components of the record.  You can also write functions where the argument is a record using a record pattern. For example:

fun full_name{first:string,last:string,age:int,balance:real}:string =
  first ^ " " ^ last (* ^ is the string concatenation operator *)

Calling full_name and passing it the record jgm yields "John Doe" as an answer.

It turns out that we can think of tuples as short-hand for records.  In particular, the tuple expression (3.14, "Greg", true) is like the record expression {1=3.14, 2="Greg", 3=true}.  So in some sense, tuples are just syntactic sugar for records.

In summary:

We'll cover more kinds of types (datatypes) and more pattern matching constructs next week.

Simple Datatypes and Case Expressions

Datatypes are used for two basic purposes which we'll describe by example.  The first example of a datatype declaration is as follows:

datatype mybool = Mytrue | Myfalse

This definition declares a new type (mybool) and two constructors (Mytrue and Myfalse) for creating values of type mybool.  In otherwords, after entering this definition into SML, we can use Mytrue or Myfalse as values of type mybool and indeed, these are the only values of type mybool.  So one purpose of datatypes is to introduce new types into the language and to introduce ways of creating values of this new type.  In fact, the builtin bool type is simply defined as:

datatype bool = true | false

Notice that a datatype definition is a lot like a BNF grammar.  For instance, we can think of bool as consisting of true or false.   We'll use this built-in grammar fracility in SML to good effect when we start building implementations of languages.

Side note: the logical operators for conjunction and disjunction are as follows:

exp ::= ... | e1 andalso e2 | e1 orelse e2

Note that and is not for logical conjunction, although it is a keyword.  These appear to be like binary operators; however, they are different from infix functions as all the other binary operators evaluate both expressions.  These two logical constructs have a special capability called short-circuiting.  If the result of the logical formula can be determined by evaluating the left-hand expression, the right-hand expression will remain unevaluated.

Another example of a datatype declaration is as follows:

datatype day = Sun | Mon | Tue | Wed | Thu | Fri | Sat

This declaration defines a new type (day) and 7 new constructors for that type (Sun-Sat).  So, for example, we can write a function which maps a number to a day of the week:

fun int_to_day(i: int):day =
  if i mod 7 = 0 then Sun else
  if i mod 7 = 1 then Mon else
  if i mod 7 = 2 then Tue else
  if i mod 7 = 3 then Wed else
  if i mod 7 = 4 then Thu else
  if i mod 7 = 5 then Fri else Sat

This sequence of if expressions where we test the value i is rather tedious.  A more concise way to write this is to use a case expression:

fun int_to_day(i: int):day =
  (case i mod 7 of
     0 => Sun
   | 1 => Mon
   | 2 => Tue
   | 3 => Wed
   | 4 => Thu
   | 5 => Fri
   | _ => Sat)

The case expression is similar to the switch statement in languages such as Java or C.  In the example above, we perform a case on the value of (i mod 7) and match it against a set of number patterns (i.e., 0, 1, 2, etc.)   The last pattern is a wildcard and matches any value.  In Java, we would write the above as something like:

switch (i % 7) {
  case 0: return Sun;
  case 1: return Mon;
  case 2: return Tue;
  case 3: return Wed;
  case 4: return Thu;
  case 5: return Fri;
  default: return Sat;

So much for mapping integers to days.  How about mapping days to integer? 

fun day_to_int(d: day):int =
  (case d of
     Sun => 0
   | Mon => 1
   | Tue => 2
   | Wed => 3
   | Thu => 4
   | Fri => 5
   | Sat => 6)

With case expressions lying around, we technically don't need an if expression form.  In particular, an expression of the form if exp1 then exp2 else exp3 is equivalent to:

case exp1 of
  true => exp2
| false => exp3

In fact it turns out that with the general form of datatypes and case expressions, we can encode a lot of things that appear to be built in to the language.  This is a good thing because it simplifies the number of special forms that we have to reason about. 

In summary:

Algebraic Datatypes and Even More Pattern Matching:

A record (or tuple) is logically like an "and".  For instance, a tuple of type (int,real,string) is an object that contains an int and a real and a string.   Datatypes, in the most general form, are used for defining "or" types -- when something needs to be one type or another.  In particular, suppose we want to define a new type "number" that includes both ints and reals.  This can be accomplished in SML by the following datatype definition:

datatype num = Int_num of int | Real_num of real

This declaration gives us a new type (num) and two constructors Int_num and Real_num.  The Int_num constructor takes an int as an argument and returns a num, while the Real_num constructor takes a real as an argument and returns a num.  In this fashion, we can create a type that is the (disjoint) union of two other types. 

But how do we use a value of type num?  We can't apply an operation such as + to it, because + is only defined for either ints or reals.  In order to use a value of type num, we have to use pattern matching to deconstruct it.   For example, the following function computes the maximum of two nums:

fun num_to_real(n:num):real =
  (case n of
     Int_num(i) => Real.fromInt(i)
   | Real_num(r) => r)
fun max(n1:num, n2:num):num =
  let val r1:real = num_to_real(n1)
      val r2:real = num_to_real(n2)
    if r1 >= r2 then Real_num(r1) else Real_num(r2)

The strategy is simple:  convert the numbers to reals and then compare the two real numbers, returning the larger of the two.  In order to make the return value a num (as opposed to a real), we have to put the result in a Real_num constructor.  We could just have well written this as:

fun max2(n1:num, n2:num):num =
  let val r1:real = num_to_real(n1)
      val r2:real = num_to_real(n2)
    Real_num(if r1 >= r2 then r1 else r2)

Here, we've wrapped the whole if-expression with the Real_num constructor.  This is one advantage of treating if as an expression as opposed to a statement.  

Notice that in the function num_to_real, we use a case-expression to determine whether the number n is an integer or real.  The pattern Int_num(i) matches n if and only if n was created using the Int_num data constructor, whereas Real_num(r) matches n if and only if it was created with the Real_num data constructor.  Also, notice that in the Int_num(i) case, we have bound the underlying integer carried by the data constructor to the variable i and that this is used in the expression Real.fromInt(i).  Similarly, the Real_num(r) pattern extracts the underlying real value carried by the data constructor and binds it to r.  

So, for instance, calling num_to_real(Int_num(3)) matches the first pattern, binds i to 3, and then returns Real.fromInt(i) = Real.fromInt(3) = 3.0.  Calling num_to_real(Real_num(4.5)) fails to match the first pattern, succeeds in matching the second pattern, binds the r to 4.5, and then returns r = 4.5.  

Here is an alternative definition of max on numbers.

fun max2(n1:num, n2:num):num =
  (case (n1, n2) of
     (Real_num(r1), Real_num(r2)) =>
   | (Int_num(i1), Int_num(i2)) =>
   | (_, Int_num(i2)) => 
     max2(n1, Real_num(num_to_real(i2))
   | (Int_num(i1), _) => 
     max2(n2, Real_num(num_to_real(i1)))

Notice that that case expression in max2 matches a tuple of the numbers n1 and n2.  Thus, all of the patterns in the case expressions are of the tuple form.  For example, the pattern (Real_num(r1), Real_num(r2)) matches if and only if both the numbers are reals.  

In the third and fourth patterns, we've used a "wildcard" (or default) pattern.  For instance, the third pattern (_, Int_num(i2)) matches iff the first number is anything, but the second is an integer.  In this case, we simply convert the integer to a real and then call ourselves recursively.  Similarly, the fourth pattern (Int_num(i1), _) the first number is an integer and the second number is anything.  In this case, we convert the first number to a real and call ourselves recursively.

Now suppose we call max2 with two integers max2(Int_num(3), Int_num(4)).  It appears as if this matches any of the last three cases, so which one do we select? The answer is that we try the matches in order.  So the second pattern will succeed and the other patterns won't even be tried.

Another question is, how do we know if there is a case for every situation?  For instance, suppose we accidentally wrote:

fun max3(n1:num, n2:num):num =
  (case (n1, n2) of
     (Int_num(i1), Int_num(i2)) =>
   | (_, Int_num(i2)) => 
     max3(n1, Real_num(num_to_real(i2))
   | (Int_num(i1), _) => 
     max3(n2, Real_num(num_to_real(i1)))

Now there is no case for when n1 and n2 are both reals.  If you type this in to SML, then it will complain that the pattern match is inexhaustive.  This is wonderful because it tells you your code might fail since you forgot a case!  In general, we will not accept code that has a match inexhaustive warning.  That is, you must make sure you never turn in code that doesn't cover all of the cases.

What happens if we put in too many cases?  For instance, suppose we wrote:

fun max2(n1:num, n2:num):num =
  (case(n1, n2) of
     (Real_num(r1), Real_num(r2)) =>
   | (Int_num(i1), Int_num(i2)) =>
   | (_, Int_num(i2)) => 
     max2(n1, Real_num(num_to_real(i2))
   | (Int_num(i1), _) => 
     max2(n2, Real_num(num_to_real(i1)))
   | (_, _) => n1

Then SML complains again that the last case will never be reached.  Again, this is wonderful because it tells us there's some useless code that we should either trim away, or reexamine (in its context) to see why it will never be executed.  Again, we will not accept code that has redundant patterns.  

So how can the SML type-checker determine that a pattern match is exhaustive and that there are no dead cases?  The reason is that patterns can only test a finite number of things (there are no loops in patterns), the tests are fairly simple (e.g., is this a Real_num or an Int_num?) and the set of datatype constructors for a given type is closed.  That is, after defining a datatype, we can't simply add new data constructors to it.  Note that if we could, then every pattern match would be potentially inexhaustive.  

At first, this seems to be a shortcoming of the language.  Adding new constructors is something that happens all the time, just as adding new subclasses happens all the time in Java programs.  The difference in SML is that, if you add a new data constructor to a datatype declaration, then the compiler will tell you where you need to examine or change your code through "match inexhaustive" errors.  This makes pattern matching an invaluable tool for maintaining and evolving programs. 

So sometimes, by limiting a programming language we gain some power.  In the case of the pattern-matching sub-language of ML, the designers have restricted the set of tests that can be performed so that the compiler can automatically tell you where you need to look at your code to get it in synch with your definitions.