1 We'll assume the "Version 3" implementation of lists and numbers throughout. So:
3 <pre><code>zero ≡ \s z. z
4 succ ≡ \n. \s z. s (n s z)
5 iszero ≡ \n. n (\x. false) true
6 add ≡ \m \n. m succ n
7 mul ≡ \m \n. \s. m (n s)</code></pre>
11 <pre><code>empty ≡ \f z. z
12 make-list ≡ \hd tl. \f z. f hd (tl f z)
13 isempty ≡ \lst. lst (\hd sofar. false) true
14 extract-head ≡ \lst. lst (\hd sofar. hd) junk</code></pre>
16 The `junk` in `extract-head` is what you get back if you evaluate:
20 As we said, the predecessor and the extract-tail functions are harder to define. We'll just give you one implementation of these, so that you'll be able to test and evaluate lambda-expressions using them in Scheme or OCaml.
22 <pre><code>predecesor ≡ (\shift n. n shift (make-pair zero junk) get-second) (\pair. pair (\fst snd. make-pair (successor fst) fst))
23 extract-tail ≡ (\shift lst. lst shift (make-pair empty junk) get-second) (\hd pair. pair (\fst snd. make-pair (make-list hd fst) fst))</code></pre>
25 The `junk` is what you get back if you evaluate:
31 Alternatively, we might reasonably declare the predecessor of zero to be zero (this is a common construal of the predecessor function in discrete math), and the tail of the empty list to be the empty list.
34 For these exercises, assume that `LIST` is the result of evaluating:
36 (make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
39 1. What would be the result of evaluating:
43 2. Based on your answer to question 1, how might you implement the **map** function? Expected behavior:
45 <pre><code>map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))</code></pre>
47 3. Based on your answer to question 1, how might you implement the **filter** function? The expected behavior is that:
51 should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
53 4. How would you implement map using the either the version 1 or the version 2 implementation of lists?
55 5. Our version 3 implementation of the numbers are usually called "Church numerals". If `m` is a Church numeral, then `m s z` applies the function `s` to the result of applying `s` to ... to `z`, for a total of *m* applications of `s`, where *m* is the number that `m` represents or encodes.
57 Given the primitive arithmetic functions above, how would you implement the less-than-or-equal function? Here is the expected behavior, where `one` abbreviates `succ zero`, and `two` abbreviates `succ (succ zero)`.
59 less-than-or-equal zero zero ~~> true
60 less-than-or-equal zero one ~~> true
61 less-than-or-equal zero two ~~> true
62 less-than-or-equal one zero ~~> false
63 less-than-or-equal one one ~~> true
64 less-than-or-equal one two ~~> true
65 less-than-or-equal two zero ~~> false
66 less-than-or-equal two one ~~> false
67 less-than-or-equal two two ~~> true
69 You'll need to make use of the predecessor function, but it's not important to understand how the implementation we gave above works. You can treat it as a black box.