Every function in OCaml takes exactly one value and returns exactly one result.
For instance, our squareRoot function takes one float value and returns
one float 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:
let max1(r1, r2):float = if r1 < r2 then r2 else r1max1(3.1415, 2.718)
and it appears as if max1 takes two arguments. In truth max1 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 OCaml.
When you call a function in OCaml, if it takes more than one argument, then you have to pass it a tuple of the arguments. For instance, when we write:
max1(3.1415, 2.718)
we're passing the 2-tuple (3.1415, 2.718) to the function max1.
We could just as well write:
val args = (3.1415, 2.178);max1 args (* evaluates to 3.1415 *)
The type of an n-tuple is written ( t1*...*tn
). For instance, the type of args above is (float*float).
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 max1 has type (float*float)->float, indicating that it
takes one argument (a 2-tuple of floats) and returns one result (a float).
Combining what we've seen so far, we can write a grammar for the basic types as follows:
t ::=
int|float|bool|string|char| t1*...*tn | t1->t2 |(t)
There are a few tricky parts to this. The two most important are that -> has lower precedence than *,
so the types (float*float)->float and float*float->float
are exactly the same type. The second is that -> is right-associative, which will
come up when higher-order functions are discussed.
You can extract the first two components of a tuple by using the fst and snd
operators, which retreive the first and second elements of a tuple, respectively. However,
there are no analogous functions for the other elements of a tuple.
So, for instance, we can rewrite the max function as follows:
let max1(pair:float*float):float = if (fst pair) < (snd pair) then (snd pair) else (fst pair)
and this is completely equivalent to the first definition. This emphasizes that max really does take just one argument -- a pair
of floating point 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:
let max1(pair:float*float):float = let r1 = fst pair in let r2 = snd pair in if r1 < r2 then r2 else r1
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:t1, x2:t2,..., xn:tn).
Pattern matching (either in a let or in the function declaration) is the encouraged method of extracting elements from tuples. It is often more efficient, and is almost always more concise and easier to read.
For instance, here is yet another version of max that uses a pattern in a val
declaration to deconstruct the pair:
let max1(pair:float*float):float = let (r1, r2) = pair in if r1 < r2 then r2 else r1
In the example above, the val declaration
matches the pair against the tuple-pattern (r1, r2).
This binds r1 to the first component of the pair (fst pair) and
r2 to the second component (snd pair). A similar thing
happens when you write a function using a tuple-pattern as in the original
definition of max:
let max1(r1, r2):float = 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, r2) and r1 is bound to the 3.1415
and r2 to 2.718. As we'll see later on, OCaml 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:
let minmax(a, b):float*float = if a < b then (a, b) else (b, a)
let (mn, mx) = minmax(2.0, 1.0)
This binds mn to 1.0 and mx to 2.0. The
type of minmax is (float*float)->(float*float), which we can
write without the parentheses because * has a higher precedence than ->
when writing type expressions.
In summary:
(e1, ..., en) creates an
n-tuple.*...*tnfst e and snd e extract the first and second elements of a tuple
val(x1:t1, x2:t2,..., xn:tn)=
e matches the tuple expression e
against the tuple-pattern (x1:t1, x2:t2,..., xn:tn)
and binds the identifiers
in the pattern to the appropriate components of the tuple.fun
y(x1:t1, x2:t2,..., xn:tn)= e
is a function declaration that takes an n-tuple as an argument and
matches the tuple against the tuple-pattern (x1:t1, x2:t2,..., xn:tn).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.
Before making a record, however, you must give it's type a name, using the type keyword. type allows
the construction of custom types and gives OCaml much of flexibility, and will be covered in full detail later. To declare a record
type with it, the type must be enclosed in braces and each field must be given a name and a type of it's own. For example, the following
declares an account type:
type account = {first:string; last:string; age:int; balance:float}
Once you have declared the type, you may create records of that type. For example, given the declaration of the account type, 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 exp.id 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 OCaml top-level, it sorts the fields into a canonical order, based on the order in the declaration of the type:
# let pers = { last = "Doe"; age = 150; balance = 0.12; first = "John" };; val pers : account = {first = "John"; last = "Doe"; age = 150; balance = 0.12}
The type of a record is written as {x1:t1, x2: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:
let {first=first;last=last;age=age;balance=balance} = pers
and OCaml responds with:
val first : string = "John" val last : string = "Doe" val age : int = 150 val balance : float = 0.12
thereby binding the identifiers first, last, age, and balance
to the respective components of the record. You can also
write functions where the argument is a record using a record pattern. For
example:
let full_name{first=first;last=last;age=age;balance=balance}:string = first ^ " " ^ last (* ^ is the string concatenation operator *)
Calling full_name and passing it the record pers 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:
type before making them{x1
= e1, x2
= e2, ...,
xn = en}.{x1:t1, x2:t2,..., xn:tn}..x, where x is the name of
the field.{x1=id1, x2=id2,..., xn=idn}.We'll cover more kinds of types (datatypes) and more pattern matching constructs next week.
Datatypes are used for two basic purposes which we'll describe by example. The first example of a datatype declaration is as follows:
type 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 OCaml, 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:
type 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 OCaml to good effect when we start building implementations of languages.
exp ::= ... | e1 && e2 | e1 || 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:
type 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:
let 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 match expression:
let int_to_day(i: int):day = (match i mod 7 with 0 -> Sun | 1 -> Mon | 2 -> Tue | 3 -> Wed | 4 -> Thu | 5 -> Fri | _ -> Sat)
The match 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?
let day_to_int(d: day):int = (match d with Sun -> 0 | Mon -> 1 | Tue -> 2 | Wed -> 3 | Thu -> 4 | Fri -> 5 | Sat -> 6)
With match 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:
match exp1 with 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:
A record (or tuple) is logically like an "and". For instance, a tuple of type (int*float*string) is an object that contains an int and a float 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 floats. This can be accomplished in OCaml by the following datatype definition:
type num = Int_num of int | Real_num of float
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 float 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 floats. 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:
let num_to_float(n:num):float = (match n with Int_num(i) -> float_of_int(i) | Real_num(r) -> r)let max1(n1, n2):num = let r1:float = num_to_float(n1) in let r2:float = num_to_float(n2) in if r1 >= r2 then Real_num(r1) else Real_num(r2)
The strategy is simple: convert the numbers to floats and then compare the two floating point numbers, returning the larger of the two. In order to make the return value a num (as opposed to a float), we have to put the result in a Real_num constructor. We could just have well written this as:
let max2(n1, n2):num = let r1:float = num_to_float(n1) in let r2:float = num_to_float(n2) in 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_float, we use a case-expression to determine whether the number n is an integer or float. The pattern Int_num(i) matches n if and only if n was created using the Int_num data constructor, and 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 float_of_int(i). Similarly, the Real_num(r) pattern extracts the underlying float value carried by the data constructor and binds it to r.
So, for instance, calling num_to_float(Int_num(3)) matches the first pattern, binds i to 3, and then returns float_of_int(i) = float_of_int(3) = 3.0. Calling num_to_float(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.
let rec max2(n1, n2):num = match (n1, n2) with (Real_num(r1), Real_num(r2)) -> Real_num(max r1 r2) | (Int_num(i1), Int_num(i2)) -> Int_num(max i1 i2) | (_, Int_num(i2)) -> max2(n1, Real_num(float_of_int(i2))) | (Int_num(i1), _) -> max2(n2, Real_num(float_of_int(i1)))
Notice that that match 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 floats.
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 float 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 float 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:
let rec max3(n1, n2):num = match (n1, n2) with (Int_num(i1), Int_num(i2)) -> Int_num(max i1 i2) | (_, Int_num(i2)) -> max3(n1, Real_num(float_of_int(i2))) | (Int_num(i1), _) -> max3(n2, Real_num(float_of_int(i1)))
Now there is no case for when n1 and n2 are both floats. If you type this in to OCaml, then it will complain that the pattern match is not exhaustive. 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 not exhaustive 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:
let rec max2(n1, n2):num = match(n1, n2) with (Real_num(r1), Real_num(r2)) -> Real_num(max r1 r2) | (Int_num(i1), Int_num(i2)) -> Int_num(max i1 i2) | (_, Int_num(i2)) -> max2(n1, Real_num(float_of_int(i2))) | (Int_num(i1), _) -> max2(n2, Real_num(float_of_int(i1))) | (_, _) -> n1
Then OCaml complains again that the last match case is unsued (i.e., it 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 OCaml 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 OCaml 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.