#### 8.1` `Functions as Data

It’s interesting to consider how expressive the little programming we’ve learned so far can be. To illustrate this, we’ll work through a few exercises of interesting concepts we can express using just functions as values. We’ll write two quite different things, then show how they converge nicely.

##### 8.1.1` `A Little Calculus

\begin{equation*}{\frac{d}{dx}} x^2 = 2x\end{equation*}

First, let’s take on the two expressions; we’ll discuss one, and the discussion will cover the other as well. The correct response to “what does \(x^2\) mean?” is, of course, an error: it doesn’t mean anything, because \(x\) is an unbound identifier.

```
fun sq(x :: Number) -> Number: x * x end
fun dbl(x :: Number) -> Number: 2 * x end
```

`sq`

is `dbl`

.We’re
assuming functions of arity one in the variable that is changing.`d-dx :: ((Number -> Number) -> (Number -> Number))`

Let us now implement `d-dx`

. We’ll implement numerical
differentiation, though in principle we could also implement
symbolic differentiation—

\begin{equation*}\frac{f(x + \epsilon) - f(x)}{\epsilon}\end{equation*}

`epsilon = 0.00001`

```
d-dx-at :: (Number -> Number), Number -> Number
fun d-dx-at(f, x):
(f(x + epsilon) - f(x)) / epsilon
end
```

```
check:
d-dx-at(sq, 10) is-roughly dbl(10)
end
```

`epsilon`

so that the
default tolerance `is-roughly`

works for this example.```
fun d-dx(f):
(f(x + epsilon) - f(x)) / epsilon
end
```

Do Now!

What’s the problem with the above definition?

`x`

isn’t
bound. Indeed, what is `x`

? It’s the point at which we’re trying
to compute the numeric derivative. That is, `d-dx`

needs to
return not a number but a function (as the type indicates) that
will consume this `x`

:“Lambdas are relegated to
relative obscurity until Java makes them popular by not having
them.”—```
fun d-dx(f):
lam(x):
(f(x + epsilon) - f(x)) / epsilon
end
end
```

```
fun d-dx(f):
lam(x :: Number) -> Number:
(f(x + epsilon) - f(x)) / epsilon
end
end
```

`num-floor`

to avoid numeric precision
issues from making our tests appear to fail):
```
d-dx-sq = d-dx(sq)
check:
ins = [list: 0, 1, 10, 100]
for map(n from ins):
num-floor(d-dx-sq(n))
end
is
for map(n from ins):
num-floor(dbl(n))
end
end
```

`d-dx(lam(x): x * x end) = lam(x): 2 * x end`

\begin{equation*}{\frac{d}{dx}} [x \rightarrow x^2] = [x \rightarrow 2x]\end{equation*}

##### 8.1.2` `A Helpful Shorthand for Anonymous Functions

`{(a): b}`

`a`

is zero or more arguments and `b`

is the body. For
instance, we can write `lam(x): x * x end`

as
`{(x): x * x}`

`end`

, because the braces take the place of
showing where the expression begins and ends. Similarly, we could have
written `d-dx`

as
```
fun d-dx-short(f):
{(x): (f(x + epsilon) - f(x)) / epsilon}
end
```

`lam`

makes clear that `d-dx`

returns
an (anonymous) function, whereas this syntax obscures it. Therefore,
we will usually only use this shorthand syntax for “one-liners”.##### 8.1.3` `Streams From Functions

People typically think of a function as serving one purpose: to parameterize an expression. While that is both true and the most common use of a function, it does not justify having a function of no arguments, because that clearly parameterizes over nothing at all. Yet functions of no argument also have a use, because functions actually serve two purposes: to parameterize, and to suspend evaluation of the body until the function is applied. In fact, these two uses are orthogonal, in that one can employ one feature without the other. Below, we will focus on delay without abstraction (the other shows up in other computer science settings).

Let’s consider the humble list. A list can be only finitely long. However, there are many lists (or sequences) in nature that have no natural upper bound: from mathematical objects (the sequence of natural numbers) to natural ones (the sequence of hits to a Web site). Rather than try to squeeze these unbounded lists into bounded ones, let’s look at how we might represent and program over these unbounded lists.

```
fun nats-from(n):
link(n, nats-from(n + 1))
end
```

Do Now!

Does this program have a problem?

While this represents our intent, it doesn’t work: running it—`nats-from(0)`

—`nats-from`

for every subsequent natural number. In other words,
we want to write something very like the above, but that doesn’t recur
until we want it to, i.e., on demand. In other words, we want
the rest of the list to be lazy.

This is where our insight into functions comes in. A function, as we
have just noted, delays evaluation of its body until it is
applied. Therefore, a function would, in principle, defer the
invocation of `nats-from(n + 1)`

until it’s needed.

`link`

needs to be a list, and cannot be a function. Indeed,
because it must be a list, and every value that has been constructed
must be finite, every list is finite and eventually terminates in
`empty`

. Therefore, we need a new data structure to represent the
links in these lazy lists (also known as streams):
```
data Stream<T>:
| lz-link(h :: T, t :: ( -> Stream<T>))
end
```

`( -> Stream<T>)`

means a function from no
arguments (hence the lack of anything before `->`

),
also known as a thunk. Note that the way we have
defined streams they must be infinite, since we have provided
no way to terminate them.`ones = lz-link(1, lam(): ones end)`

`ones = link(1, ones)`

`ones`

is not defined at the point of
definition, so when Pyret evaluates `link(1, ones)`

, it complains
that `ones`

is not defined. However, it is being overly
conservative with our former definition: the use of `ones`

is
“under a `lam`

”, and hence won’t be needed until after the
definition of `ones`

is done, at which point `ones`

will be defined. We can indicate this to Pyret by using the
keyword `rec`

:
`rec ones = lz-link(1, lam(): ones end)`

`fun`

implicitly has a `rec`

beneath it, which is why we can
create recursive functions with aplomb.Exercise

Earlier we said that we can’t write`ones = link(1, ones)`

What if we tried to write`rec ones = link(1, ones)`

instead? Does this work and, if so, what value is`ones`

bound to? If it doesn’t work, does it fail to work for the same reason as the definition without the`rec`

?

`rec ones = lz-link(1, {(): ones})`

`{(): …}`

defines an anonymous function of no
arguments. You can’t leave out the `()`

! If you do, Pyret will
get confused about what your program means.`rec`

. Consider
this example:
```
fun nats-from(n :: Number):
lz-link(n, {(): nats-from(n + 1)})
end
```

`nats = nats-from(0)`

`nats`

is not recursive itself—`nats-from`

—`rec`

to define `nats`

.Do Now!

Earlier, we said that every list is finite and hence eventually terminates. How does this remark apply to streams, such as the definition of

`ones`

or`nats`

above?

`ones`

is still a finite one; it simply
represents the potential for an infinite number of values. Note
that:
A similar reasoning doesn’t apply to lists because the rest of the list has already been constructed; in contrast, placing a function there creates the potential for a potentially unbounded amount of computation to still be forthcoming.

That said, even with streams, in any given computation, we will create only a finite prefix of the stream. However, we don’t have to prematurely decide how many; each client and use is welcome to extract less or more, as needed.

`fun lz-first<T>(s :: Stream<T>) -> T: s.h end`

`fun lz-rest<T>(s :: Stream<T>) -> Stream<T>: s.t() end`

```
fun take<T>(n :: Number, s :: Stream<T>) -> List<T>:
if n == 0:
empty
else:
link(lz-first(s), take(n - 1, lz-rest(s)))
end
end
```

```
check:
take(10, ones) is map(lam(_): 1 end, range(0, 10))
take(10, nats) is range(0, 10)
take(10, nats-from(1)) is map((_ + 1), range(0, 10))
end
```

`(_ + 1)`

defines a Pyret function of
one argument that adds `1`

to the given argument.`map`

over
streams. For reasons that will soon become obvious, we’ll define a
version that takes two lists and applies the first argument to them
pointwise:
```
fun lz-map2<A, B, C>(
f :: (A, B -> C),
s1 :: Stream<A>,
s2 :: Stream<B>) -> Stream<C>:
lz-link(
f(lz-first(s1), lz-first(s2)),
{(): lz-map2(f, lz-rest(s1), lz-rest(s2))})
end
```

`map`

over
lists would have two cases, here we have only one case because the
data definition (<stream-type-def>) has only one case!
What is the consequence of this? In a traditional `map`

, one case
looks like the above, but the other case corresponds to the `empty`

input, for which it produces the same output. Here, because the stream
never terminates, mapping over it doesn’t either, and the structure of
the function reflects this.This raises a much subtler
problem: if the function’s body doesn’t have base- and
inductive-cases, how can we perform an inductive proof over it? The
short answer is we can’t: we must instead use
☛ coinduction.`lz-map2`

instead of `lz-map`

? Because it
enables us to write the following:
```
rec fibs =
lz-link(0,
{(): lz-link(1,
{(): lz-map2({(a :: Number, b :: Number): a + b},
fibs,
lz-rest(fibs))})})
```

```
check:
take(10, fibs) is [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
end
```

Exercise

Define the equivalent of

`map`

and`filter`

for streams.

Streams and, more generally, infinite data structures that unfold on demand are extremely valuable in programming. Consider, for instance, the possible moves in a game. In some games, this can be infinite; even if it is finite, for interesting games the combinatorics mean that the tree is too large to feasibly store in memory. Therefore, the programmer of the computer’s intelligence must unfold the game tree on demand. Programming it by using the encoding we have described above means the program describes the entire tree, lazily, and the tree unfolds automatically on demand, relieving the programmer of the burden of implementing such a strategy.

In some languages, such as Haskell, lazy evaluation is built in by default. In such a language, there is no need to use thunks. However, lazy evaluation places other burdens on the language, which you can learn about in a programming-languages class.

##### 8.1.4` `Combining Forces: Streams of Derivatives

When we defined `d-dx`

, we set `epsilon`

to an arbitrary, high
value. We could instead think of `epsilon`

as itself a stream that
produces successively finer values; then, for instance, when the
difference in the value of the derivative becomes small enough, we can
decide we have a sufficient approximation to the derivative.

The first step is, therefore, to make `epsilon`

some kind of
parameter rather than a global constant. That leaves open what kind of
parameter it should be (number or stream?) as well as when it should
be supplied.

`epsilon`

values may depend
on both. Thus, we get:
```
fun d-dx(f :: (Number -> Number)) ->
(Number -> (Number -> Number)):
lam(x :: Number) -> (Number -> Number):
lam(epsilon :: Number) -> Number:
(f(x + epsilon) - f(x)) / epsilon
end
end
end
```

`square`

example:
`d-dx-square = d-dx(square)`

`d-dx`

without
any reference to streams: we have merely made a constant into a
parameter.```
tenths = block:
fun by-ten(d):
new-denom = d / 10
lz-link(new-denom, lam(): by-ten(new-denom) end)
end
by-ten(1)
end
```

```
check:
take(3, tenths) is [list: 1/10, 1/100, 1/1000]
end
```

`square`

—`10`

:
`d-dx-square-at-10 = d-dx-square(10)`

`(Number -> Number)`

: given a value for `epsilon`

, it computes
the derivative using that value. We know, analytically, that the
value of this derivative should be `20`

. We can now (lazily) map
`tenths`

to provide increasingly better approximations for
`epsilon`

and see what happens:
`lz-map(d-dx-square-at-10, tenths)`

`20.1`

, `20.01`

,
`20.001`

, and so on: progressively better numerical
approximations to `20`

.Exercise

Extend the above program to take a tolerance, and draw as many values from the

`epsilon`

stream as necessary until the difference between successive approximations of the derivative fall within this tolerance.