Lists are a fundamental Racket data type.
A list is defined as either:
the empty list (
a pair (
cons cell) of anything and a list.
As simple as this may seem, it gives us precise formal rules to prove that something is a list.
List operations — predicates:
We can derive
list? from the above rules:
But why can’t we define
list? more simply as
The difference between the above definition and the proper one can be observed in the full Racket language, not in the student languages (where there are no pairs with non-list values in their tails).
List operations — destructors for pairs (
c<x>r combination for
<x> that is made of up to four
ds — we will probably not use much more than
Example for recursive function involving lists:
Use different tools, esp:
How come we could use
list as an argument — use the syntax checker
Main idea: lists are a recursive structure, so functions that operate on lists should be recursive functions that follow the recursive definition of lists.
Another example for list function — summing a list of numbers
Also show how to implement
rcons, using this guideline.
reverse — solve the problem using
rcons can be generalized into something very useful:
How would we use
append instead of
How much time will this take? Does it matter if we use
reverse using tail recursion.
When you have some common value that you need to use in several places, it is bad to duplicate it. For example:
What’s bad about it?
It’s longer than necessary, which will eventually make your code less readable.
It’s slower — by the time you reach the last case, you have evaluated the two sequences three times.
It’s more prone to bugs — the above code is short enough, but what if it was longer so you don’t see the three occurrences on the same page? Will you remember to fix all places when you debug the code months after it was written?
In general, the ability to use names is probably the most fundamental concept in computer science — the fact that makes computer programs what they are.
We already have a facility to name values: function arguments. We could split the above function into two like this:
But instead of the awkward solution of coming up with a new function
just for its names, we have a facility to bind local names —
In general, the syntax for a
let special form is
But note that the bindings are done “in parallel”, for example, try this:
(Note that “in parallel” is quoted here because it’s not really parallelism, but just a matter of scopes: the RHSs are all evaluated in the surrounding scope!)
Using this for the above problem:
Some notes on writing code (also see the style-guide in the handouts section)
Code quality will be graded to in this course!
Use abstractions whenever possible, as said above. This is bad:
But don’t over abstract:
(define one 1) or
(define two "two")
Always do test cases (show coverage tool), you might want to comment them, but you should always make sure your code works.
Do not under-document, but also don’t over-document.
INDENTATION! (Let DrRacket decide; get used to its rules) –> This is part of the culture that was mentioned last time, but it’s done this way for good reason: decades of programming experience have shown this to be the most readable format. It’s also extremely important to keep good indentation since programmers in all Lisps don’t count parens — they look at the structure.
As a general rule,
if should be either all on one line, or the
condition on the first and each consequent on a separate line.
define — either all on one line or a newline after
the object that is being define (either an identifier or a an
identifier with arguments).
Another general rule: you should never have white space after an open-paren, or before a close paren (white space includes newlines). Also, before an open paren there should be either another open paren or white space, and the same goes for after a closing paren.
Use the tools that are available to you: for example, use
instead of nested
ifs (definitely do not force the indentation to
make a nested
if look like its C counterpart — remember to let
DrRacket indent for you).
Another example — do not use
(+ 1 (+ 2 3)) instead of
(+ 1 2 3)
(this might be needed in extremely rare situations, only when you
know your calculus and have extensive knowledge about round-off
Another example — do not use
(cons 1 (cons 2 (cons 3 null)))
(list 1 2 3).
Also — don’t write things like:
since it’s the same as just
A few more of these:
(Actually the first two are almost the same, for example,
(and 1 2)
Use these as examples for many of these issues:
The fact that in Racket we can use functions as values is very useful
— for example,
foldr, many more.
You should generally know what tail calls are, but here’s a quick review of the subject. A function call is said to be in tail position if there is no context to “remember” when you’re calling it. Very roughly, this means that function calls that are not nested in argument expressions of another call are tail calls. This definition is something that depends on a context, for example, in an expression like
both calls to
foo are tail calls, but they’re tail calls of this
expression and therefore apply to this context. It might be that this
code is inside another call, as in
foo calls are now not in tail position. The main feature of
all Scheme implementations including Racket wrt tail calls is that calls
that are in tail position of a function are said to be “eliminated”.
That means that if we’re in an
f function, and we’re about to call
in tail position and therefore whatever
g returns would be the result
f too, then when Racket does the call to
g it doesn’t bother
f context — it won’t remember that it needs to “return”
f and will instead return straight to its caller. In other words,
when you think about a conventional implementation of function calls as
frames on a stack, Racket will get rid of a stack frame when it can.
Another way to see this is to use DrRacket’s stepper to step through a function call. The stepper is generally an alternative debugger, where instead of visualizing stack frames it assembles an expression that represents these frames. Now, in the case of tail calls, there is no room in such a representation to keep the call — and the thing is that in Racket that’s perfectly fine since these calls are not kept on the call stack.
Note that there are several names for this feature:
“Tail recursion”. This is a common way to refer to the more limited optimization of only tail-recursive functions into loops. In languages that have tail calls as a feature, this is too limited, since they also optimize cases of mutual recursion, or any case of a tail call.
“Tail call optimization”. In some languages, or more specifically in
some compilers, you’ll hear this term. This is fine when tail calls
are considered only an “optimization” — but in Racket’s case (as
well as Scheme), it’s more than just an optimization: it’s a language
feature that you can rely on. For example, a tail-recursive function
(define (loop) (loop)) must run as an infinite loop, not just
optimized to one when the compiler feels like it.
“Tail call elimination”. This is the so far the most common proper name for the feature: it’s not just recursion, and it’s not an optimization.
Often, people who are aware of tail calls will try to use them always. That’s not always a good idea. You should generally be aware of the tradeoffs when you consider what style to use. The main thing to remember is that tail-call elimination is a property that helps reducing space use (stack space) — often reducing it from linear space to constant space. This can obviously make things faster, but usually the speedup is just a constant factor since you need to do the same number of iterations anyway, so you just reduce the time spent on space allocation.
Here is one such example that we’ve seen:
In this case the first (recursive) version version consumes space linear to the length of the list, whereas the second version needs only constant space. But if you consider only the asymptotic runtime, they are both O(length(l)).
A second example is a simple implementation of
In this case, both the asymptotic space and the runtime consumption are the same. In the recursive case we have a constant factor for the stack space, and in the iterative one (the tail-call version) we also have a similar factor for accumulating the reversed list. In this case, it is probably better to keep the first version since the code is simpler. In fact, Racket’s stack space management can make the first version run faster than the second — so optimizing it into the second version is useless.
Types can become interesting when dealing with higher-order functions.
map receives a function and a list of some type, and
applies the function over this list to accumulate its output, so its
map can use more than a single list, it will apply the
function on the first element in all lists, then the second and so on.
So the type of
map with two lists can be described as:
Here’s a hairy example — what is the type of this function:
Begin by what we know — both
maps, call them
the double- and single-list types of
map respectively, here they are,
with different names for types:
Now, we know that
map2 is the first argument to
map1, so the type of
map1s first argument should be the type of
From here we can conclude that
If we use these equations in
map1’s type, we get:
foo’s two arguments are the 2nd and 3rd arguments of
its result is
map1s result, so we can now write the type of
This should help you understand why, for example, this will cause a type error:
and why this is valid:
An important “discovery” in computer science is that we don’t need names for every intermediate sub-expression — for example, in almost any language we can write the equivalent of:
Such languages are put in contrast to assembly languages, and were all put under the generic label of “high level languages”.
(Here’s an interesting idea — why not do the same for function values?)