X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment4.mdwn;h=9b7ec2c028622dc92e17dced54403aa6043b8896;hp=34fb044d6197fd875e56a4f15ecc26bc2a233e37;hb=e5f582330945c265a211cbebb30bc06a94f3ed91;hpb=260a35e134bfd7d8105208f13d4333cc9e50470a diff --git a/assignment4.mdwn b/assignment4.mdwn index 34fb044d..9b7ec2c0 100644 --- a/assignment4.mdwn +++ b/assignment4.mdwn @@ -1,43 +1,143 @@ -Assignment 4 ------------- - #Reversing a list# -How would you define an operation to reverse a list? (Don't peek at the +
+
1. 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# - + +
+
1. 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# -[[Implementing trees]] +First, read this: [[Implementing trees]] + +
+
1. 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.) + +
2. 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# + +
+
1. (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? + + +
2. (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.) +