Functions & Function Values

PLAI §4

Now that we have a form for local bindings, which forced us to deal with proper substitutions and everything that is related, we can get to functions. The concept of a function is itself very close to substitution, and to our with form. For example, when we write:

then the {* x x} body is itself parametrized over some value for x. If we take this expression and take out the 5, we’re left with something that has all of the necessary ingredients of a function — a bunch of code that is parameterized over some input identifier:

We only need to replace with and use a proper name that indicates that it’s a function:

Now we have a new form in our language, one that should have a function as its meaning. As we have seen in the case of with expressions, we also need a new form to use these functions. We will use call for this, so that

will be the same as the original with expression that we started with — the fun expression is like the with expression with no value, and applying it on 5 is providing that value back:

Of course, this does not help much — all we get is a way to use local bindings that is more verbose from what we started with. What we’re really missing is a way to name these functions. If we get the right evaluation rules, we can evaluate a fun expression to some value — which will allow us to bind it to a variable using with. Something like this:

In this expression, we say that x is the formal parameter (or argument), and the 5 and 6 are actual parameters (sometimes abbreviated as formals and actuals). Note that naming functions often helps, but many times there are small functions that are fine to specify without a name — for example, consider a two-stage addition function, where there is no apparent good name for the returned function:

Implementing First Class Functions

PLAI §6 (uses some stuff from PLAI §5, which we do later)

This is a simple plan, but it is directly related to how functions are going to be used in our language. We know that {call {fun {x} E1} E2} is equivalent to a with expression, but the new thing here is that we do allow writing just the {fun ...} expression by itself, and therefore we need to have some meaning for it. The meaning, or the value of this expression, should roughly be “an expression that needs a value to be plugged in for x”. In other words, our language will have these new kinds of values that contain an expression to be evaluated later on.

There are three basic approaches that classify programming languages in relation to how the deal with functions:

1. First order: functions are not real values. They cannot be used or returned as values by other functions. This means that they cannot be stored in data structures. This is what most “conventional” languages used to have in the past. (You will be implementing such a language in homework 4.)

   An example of such a language is the Beginner Student language that is used in HtDP, where the language is intentionally first-order to help students write correct code (at the early stages where using a function as a value is usually an error). It’s hard to find practical modern languages that fall in this category.

2. Higher order: functions can receive and return other functions as values. This is what you get with C and modern Fortran.

3. First class: functions are values with all the rights of other values. In particular, they can be supplied to other functions, returned from functions, stored in data structures, and new functions can be created at run-time. (And most modern languages have first class functions.)

The last category is the most interesting one. Back in the old days, complex expressions were not first-class in that they could not be freely composed. This is still the case in machine-code: as we’ve seen earlier, to compute an expression such as

you have to do something like this:

In other words, every intermediate value needs to have its own name. But with proper (“high-level”) programming languages (at least most of them…) you can just write the original expression, with no names for these values.

With first-class functions something similar happens — it is possible to have complex expressions that consume and return functions, and they do not need to be named.

What we get with our fun expression (if we can make it work) is exactly this: it generates a function, and you can choose to either bind it to a name, or not. The important thing is that the value exists independently of a name.

This has a major effect on the “personality” of a programming language as we will see. In fact, just adding this feature will make our language much more advanced than languages with just higher-order or first-order functions.

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Quick Example: the following is working JavaScript code, that uses first class functions.

Note that the above definition of foo does not use an anonymous “lambda expression” — in Racket terms, it’s translated to

The returned function is not anonymous, but it’s not really named either: the bar name is bound only inside the body of foo, and outside of it that name no longer exists since it’s not its scope. It gets used in the printed form if the function value is displayed, but this is merely a debugging aid. The anonymous lambda version that is common in Racket can be used in JavaScript too:

Side-note: GCC includes extensions that allow internal function definitions, but it still does not have first class functions — trying to do the above is broken:

#include <stdio.h>
typedef int(*int2int)(int);
int2int foo(int x) {
  int bar(int y) { return x + y; }
  return bar;
}
int main() {
  int2int f = foo(1);
  int2int g = foo(10);
  printf(">> %d, %d\n", f(2), g(2));
}

Side-note: how important is it to have anonymous functions?

You’ll see many places where people refer to the feature of first-class functions as the ability to create anonymous functions, but this is a confusion and it’s not accurate. Whether a function has a name or not is not the important question — instead, the important question is whether functions can exist with no bindings that refers to them.

As a quick example in Racket:

in Javascript:

and in Python:

In all three of these, we have a foo function that returns a function named bar — but the bar name, is only available in the scope of foo. The fact that the name is displayed as part of the textual rendering of the function value is merely a debugging feature.

The FLANG Language

Now for the implementation — we call this new language FLANG.

First, the BNF:

And the matching type definition:

The parser for this grammar is, as usual, straightforward:

We also need to patch up the substitution function to deal with these things. The scoping rule for the new function form is, unsurprisingly, similar to the rule of with, except that there is no extra expression now, and the scoping rule for call is the same as for the arithmetic operators:

And the matching code:

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Now, before we start working on an evaluator, we need to decide on what exactly do we use to represent values of this language. Before we had functions, we had only number values and we used Racket numbers to represent them. Now we have two kinds of values — numbers and functions. It seems easy enough to continue using Racket numbers to represent numbers, but what about functions? What should be the result of evaluating

? Well, this is the new toy we have: it should be a function value, which is something that can be used just like numbers, but instead of arithmetic operations, we can call these things. What we need is a way to avoid evaluating the body expression of the function — delay it — and instead use some value that will contain this delayed expression in a way that can be used later.

To accommodate this, we will change our implementation strategy a little: we will use our syntax objects for numbers ((Num n) instead of just n), which will be a little inconvenient when we do the arithmetic operations, but it will simplify life by making it possible to evaluate functions in a similar way: simply return their own syntax object as their values. The syntax object has what we need: the body expression that needs to be evaluated later when the function is called, and it also has the identifier name that should be replaced with the actual input to the function call. This means that evaluating:

now yields

and a number (Num 5) evaluates to (Num 5).

In a similar way, (Fun 'x (Num 2)) evaluates to (Fun 'x (Num 2)).

Why would this work? Well, because call will be very similar to with — the only difference is that its arguments are ordered a little differently, being retrieved from the function that is applied and the argument.

The formal evaluation rules are therefore treating functions like numbers, and use the syntax object to represent both values:

Note that the last rule could be written using a translation to a with expression:

And alternatively, we could specify with using call and fun:

There is a small problem in these rules which is intuitively seen by the fact that the evaluation rule for a call is expected to be very similar to the one for arithmetic operations. We now have two kinds of values, so we need to check the arithmetic operation’s arguments too:

The corresponding code is:

Note that the Call case is doing the same thing we do in the With case. In fact, we could have just generated a With expression and evaluate that instead:

The arith-op function is in charge of checking that the input values are numbers (represented as FLANG numbers), translating them to plain numbers, performing the Racket operation, then re-wrapping the result in a Num. Note how its type indicates that it is a higher-order function.

It uses the following function to convert FLANG numbers to Racket numbers. (Note that else is almost always a bad idea since it can prevent the compiler from showing you places to edit code — but this case is an exception since we never want to deal with anything other than Nums.) The reason that this function is relatively trivial is that we chose the easy way and represented numbers using Racket numbers, but we could have used strings or anything else.

We can also make things a little easier to use if we make run convert the result to a number:

Adding few simple tests we get:

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