15-150: Principles of Functional Programming

Lecture 2: Functions

Today, we mainly discussed functions.

Recall also from last time:

Extensional equivalence.
Extensional equivalence is an equivalence relation on well-typed SML expressions, relating well-typed expressions of the same type.

Two expressions e and e' of the same base type (such as int, bool, char, etc.) are extensionally equivalent, written e ≅ e', whenever one of the following is true: (i) evaluation of e produces the same value as does evaluation of e', or (ii) evaluation of e raises the same exception as does evaluation of e', or (iii) evaluation of e and evaluation of e' both loop forever. Observe that two values of a given base type are equivalent if and only if they are identical. This definition generalizes naturally to more complicated types (such as products and lists) constructed from base types but not involving function types.

We further say that two function values  f : t -> t'  and  g : t -> t' of the same type are extensionally equivalent whenever  f(v) and g(w) are extensionally equivalent for all extensionally equivalent argument values v and w of type t.   Formally,  f ≅ g  if and only if  f(v) ≅ g(w)  for all values  v : t  and  w : t  for which  v ≅ w. Observe that function values can be equivalent without being identical (see also below).

We can then generalize our definition of equivalence to say, recursively, that two expressions e and e' of the same type are extensionally equivalent whenever evaluation of e and e': (i) produces extensionally equivalent values, or (ii) raises extensionally equivalent exceptions, or (iii) loops forever for both expressions.

Totality.
We say that a function  f : t -> t'  is  total  if and only if:   (i)  f  reduces to a value   and   (ii)  f(v)  reduces to a value for every possible argument value  v  of type  t.
(When reading condition (i) bear in mind that  f  could be a general expression of type  t -> t' .)

Frequently, a proof of correctness may require establishing that some expression reduces to a value. In order to accomplish that, it may be useful to know that some function is total.
For instance, if one wants to establish that the expression  x + f(y)  reduces to a value, one approach is to show that  x  and  y  reduce to values and that  f  is a total function (possibly also that  +  is total, though we usually take that for granted in this course).
Totality is an important concept, at the core of what it means to actually perform computations. Many mathematical functions are not computable by total functions.

Functions as values.
A function value consists of an anonymous lambda expression along with a (possibly empty) environment of bindings for any nonlocal variables that appear in the body of the function. The combination of a lambda expression and an environment is called a closure.

Example 1: The anonymous lambda expression (fn (x:int) => x*x) is a function that squares its argument. (The only nonlocal variable here is the symbol *. Technically, the environment includes a binding of symbol * to an internal multiplication function. For simplicity, we generally do not specify bindings of such built-in functions in this course, so we did not write an environment.)
Example 2: [3.14159265358979/pi](fn (r:real) => pi*r*r) is an environment and an anonymous function that together provide a function for computing the area of a disk of radius r. (Again, for simplicity, we omit writing a binding for the symbol *, but it too appears in the environment.)

Comment: We generally think of a function closure as consisting of a lambda expression and the environment present at the time of definition of the function. That is slightly different from the definition just given, since the environment at the time of definition may contain more bindings than are necessary to resolve the values of any nonlocal variables appearing in the body of the function. However, the two definitions amount to the same thing operationally, since those extra bindings never matter.

NOTE: A lambda expression (plus any bindings of nonlocal variables) is a value. A value is a value; one cannot reduce a value further.
So one would not say that  [2/s, 3/r](fn (x:int) => (s+r)*x)  reduces to  (fn (x:int) => 5*x).
However, it is true that these two function values are extensionally equivalent, i.e., [2/s, 3/r](fn (x:int) => (s+r)*x) ≅ (fn (x:int) => 5*x).
Why? Because one may make the extensionally equivalent substitutions of 2 for s and 3 for r in the lambda expression (fn (x:int) => (s+r)*x) to obtain the extensionally equivalent lambda expression (fn (x:int) => 5*x) and then (by referential transparency) one may use this simpler lambda expression in place of the original.
One important point here is that SML does not perform the mathematical operations (s+r)*x when the function is written, only later when the function is called on (applied to) an argument (see below).

Be aware: When SML prints a function value in the REPL, it will merely print   fn   (plus the type of the function), not all the internal details of the closure. Only in proofs or similar reasoning do we use the written representation shown above. Similarly, in code one simply writes function declarations or lambda expressions directly. One does not and cannot write out the environment as we have here, which we did purely for reasoning purposes. Instead, the environment arises from the declarations and function calls appearing in the code. It is implicit in the textual arrangement of the code via SML's static lexical scoping rules. If you want bindings in the environment, you have to create those using declarations. However, you can write anonymous lambda expressions inline in your code, as for instance in this expression (fn (x:int) => x*x)(7), which applies the function (fn (x:int) => x*x) to argument 7.

We discussed function application. In order to evaluate   e2 e1:

1. Reduce e2 to a function value f of the form fn(x:t)=>e (along with the environment env that existed at the time of definition of the function).
2. Reduce e1 to a value v.
3. Locally extend environment env with a binding of variable x to value v.
4. Evaluate the body e of f in the resulting environment.
   If that evaluation produces a value, return the value in the calling environment.

          fun square (x:int) : int = x * x

          square(3+4)
      ==> env(fn(x:int) => x*x)(3+4)
      ==> env(fn(x:int) => x*x)(7)
      ==> env[7/x]x*x
      ==> env[7/x]7*x
      ==> env[7/x]7*7
      ==> 49

See also again the evaluation notes from the previous lecture.

Five-Step Methodology.
As a reminder, here again is the 5-step methodology, useful for writing functions:

1. In the first line of comments, write the name and type of the function.
2. In the second line of comments, specify via a REQUIRES clause any assumptions about the arguments passed to the function.
3. In the third line of comments, specify via an ENSURES clause what the function computes (what it returns).
4. Implement the function.
5. Test your code.