#Reversing a list#
- How would you define an operation to reverse a list? (Don't peek at the
[[lambda_library]]! Try to figure it out on your own.) Choose whichever
implementation of list you like. Even then, there are various strategies you
can use.
(See [[hints/Assignment 4 hint 1]] if you need some hints.)
#Comparing lists for equality#
- Suppose you have two lists of integers, `left` and `right`. You want to
determine whether those lists are equal: that is, whether they have all the
same members in the same order. (Equality for the lists we're working with is
*extensional*, or parasitic on the equality of their members, and the list
structure. Later in the course we'll see lists which aren't extensional in this
way.)
How would you implement such a list comparison?
(See [[hints/Assignment 4 hint 2]] if you need some hints.)
#Enumerating the fringe of a leaf-labeled tree#
First, read this: [[Implementing trees]]
- Write an implementation of leaf-labeled trees. You can do something v3-like, or use the Y combinator, as you prefer.
You'll need an operation `make_leaf` that turns a label into a new leaf. You'll
need an operation `make_node` that takes two subtrees (perhaps leaves, perhaps
other nodes) and joins them into a new tree. You'll need an operation `isleaf`
that tells you whether a given tree is a leaf. And an operation `extract_label`
that tells you what value is associated with a given leaf. And an operation
`extract_left` that tells you what the left subtree is of a tree that isn't a
leaf. (Presumably, `extract_right` will work similarly.)
- The **fringe** of a leaf-labeled tree is the list of values at its leaves,
ordered from left to right. For example, the fringe of this tree:
.
/ \
. 3
/ \
1 2
is `[1;2;3]`. And that is also the fringe of this tree:
.
/ \
1 .
/ \
2 3
The two trees are different, but they have the same fringe. We're going to
return later in the term to the problem of determining when two trees have the
same fringe. For now, one straightforward way to determine this would be:
enumerate the fringe of the first tree. That gives you a list. Enumerate the
fringe of the second tree. That also gives you a list. Then compare the two
lists to see if they're equal.
Write the fringe-enumeration function. It should work on the
implementation of trees you designed in the previous step.
Then combine this with the list comparison function you wrote for question 2,
to yield a same-fringe detector. (To use your list comparison function, you'll
have to make sure you only use Church numerals as the labels of your leaves,
though nothing enforces this self-discipline.)
#Mutually-recursive functions#
- (Challenging.) One way to define the function `even` is to have it hand off
part of the work to another function `odd`:
let even = \x. iszero x
; if x == 0 then result is
true
; else result turns on whether x's pred is odd
(odd (pred x))
At the same tme, though, it's natural to define `odd` in such a way that it
hands off part of the work to `even`:
let odd = \x. iszero x
; if x == 0 then result is
false
; else result turns on whether x's pred is even
(even (pred x))
Such a definition of `even` and `odd` is called **mutually recursive**. If you
trace through the evaluation of some sample numerical arguments, you can see
that eventually we'll always reach a base step. So the recursion should be
perfectly well-grounded:
even 3
~~> iszero 3 true (odd (pred 3))
~~> odd 2
~~> iszero 2 false (even (pred 2))
~~> even 1
~~> iszero 1 true (odd (pred 1))
~~> odd 0
~~> iszero 0 false (even (pred 0))
~~> false
But we don't yet know how to implement this kind of recursion in the lambda
calculus.
The fixed point operators we've been working with so far worked like this:
let X = Y T in
X <~~> T X
Suppose we had a pair of fixed point operators, `Y1` and `Y2`, that operated on
a *pair* of functions `T1` and `T2`, as follows:
let X1 = Y1 T1 T2 in
let X2 = Y2 T1 T2 in
X1 <~~> T1 X1 X2 and
X2 <~~> T2 X1 X2
If we gave you such a `Y1` and `Y2`, how would you implement the above
definitions of `even` and `odd`?
- (More challenging.) Using our derivation of Y from the [Week3
notes](/week3/#index4h2) as a model, construct a pair `Y1` and `Y2` that behave
in the way described.
(See [[hints/Assignment 4 hint 3]] if you need some hints.)