The following code

let x = 42

has an expression in it (42) but is not itself an expression. Rather, it is a definition. Definitions bind values to names, in this case the value 42 being bound to the name x. The OCaml manual describes definitions (see the third major grouping titled "definition" on that page), but that manual page is again primarily for reference not for study. Definitions are not expressions, nor are expressions definitions—they are distinct syntactic classes. But definitions can have expressions nested inside them, and vice-versa.

For now, let's focus on one particular kind of definition, a function definition. Non-recursive functions are defined like this:

let f x = ...

Recursive functions are defined like this:

let rec f x = ...

The difference is just the rec keyword. It's probably a bit surprising that you explicitly have to add a keyword to make a function recursive, because most languages assume by default that they are. OCaml doesn't make that assumption, though.

One of the best known recursive functions is the factorial function. In OCaml, it can be written as follows:

(* requires: n >= 0 *)
(* returns: n! *)
let rec fact n = 
  if n=0 then 1 else n * fact (n-1)

We provided a specification comment above the function to document the precondition (requires) and postcondition (returns) of the function.

Note that, as in many languages, OCaml integers are not the "mathematical" integers but are limited to a fixed number of bits. The manual specifies that (signed) integers are at least 31 bits, but they could be wider. As architectures have grown, so has that size. In current implementations, OCaml integers are 63 bits. So if you test on large enough inputs, you might begin to see strange results. The problem is machine arithmetic, not OCaml. (For interested readers: why 31 or 63 instead of 32 or 64? The OCaml garbage collector needs to distinguish between integers and pointers. The runtime representation of these therefore steals one bit to flag whether a word is an integer or a pointer.)

Here's another recursive function:

(* requires: y>=0 *)
(* returns: x to the power of y *)
let rec pow x y = 
  if y=0 then 1 
  else x * pow x (y-1)

Note how we didn't have to write any types in either of our functions: the OCaml compiler infers them for us automatically. The compiler solves this type inference problem algorithmically, but we could do it ourselves, too. It's like a mystery that can be solved by our mental power of deduction:

  • Since the if expression can return 1 in the then branch, we know by the typing rule for if that the entire if expression has type int.

  • Since the if expression has type int, the function's return type must be int.

  • Since y is compared to 0 with the equality operator, y must be an int.

  • Since x is multiplied with another expression using the * operator, x must be an int.

If we wanted to write down the types for some reason, we could do that:

(* requires: y>=0 *)
(* returns: x to the power of y *)
let rec pow (x:int) (y:int) : int = 
  if y=0 then 1 
  else x * pow x (y-1)

The parentheses are mandatory when we write the type annotations for x and y. We will generally leave out these annotations, because it's simpler to let the compiler infer them. There are other times when you'll want to explicitly write down types. One particularly useful time is when you get a type error from the compiler that you don't understand. Explicitly annotating the types can help with debugging such an error message.

Syntax. The syntax for function definitions:

let rec f x1 x2 ... xn = e

The f is a metavariable indicating an identifier being used as a function name. These identifiers must begin with a lowercase letter. The remaining rules for lowercase identifiers can be found in the manual. The names x1 through xn are metavariables indicating argument identifiers. These follow the same rules as function identifiers. The keyword rec is required if f is to be a recursive function; otherwise it may be omitted.

Note that syntax for function definitions is actually simplified compared to what OCaml really allows. We will learn more about some augmented syntax for function definition in the next couple weeks. But for now, this simplified version will help us focus.

Mutually recursive functions can be defined with the and keyword:

let rec f x1 ... xn = e1
and g y1 ... yn = e2

For example:

(* [even n] is whether [n] is even.
 * requires: [n >= 0] *)
let rec even n = 
  n=0 || odd (n-1) 

(* [odd n] is whether [n] is odd.
 * requires: [n >= 0] *)
and odd n = 
  n<>0 && even (n-1);;

The syntax for function types:

t -> u
t1 -> t2 -> u
t1 -> ... -> tn -> u

The t and u are metavariables indicating types. Type t -> u is the type of a function that takes an input of type t and returns an output of type u. We can think of t1 -> t2 -> u as the type of a function that takes two inputs, the first of type t1 and the second of type t2, and returns an output of type u. Likewise for a function that takes n arguments.

Dynamic semantics. There is no dynamic semantics of function definitions. There is nothing to be evaluated. OCaml just records that the name f is bound to a function with the given arguments x1..xn and the given body e. Only later, when the function is applied, will there be some evaluation to do.

Static semantics. The static semantics of function definitions:

  • For non-recursive functions: if by assuming that x1:t1 and x2:t2 and ... and xn:tn, we can conclude that e:u, then f : t1 -> t2 -> ... -> tn -> u.
  • For recursive functions: if by assuming that x1:t1 and x2:t2 and ... and xn:tn and f : t1 -> t2 -> ... -> tn -> u, we can conclude that e:u, then f : t1 -> t2 -> ... -> tn -> u.

Note how the type checking rule for recursive functions assumes that the function identifier f has a particular type, then checks to see whether the body of the function is well-typed under that assumption. This is because f is in scope inside the function body itself (just like the arguments are in scope).

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