4 Insert all the implicit `( )`s and <code>λ</code>s into the following abbreviated expressions:
9 4. `w (\x y z. x z (y z)) u v`
11 Mark all occurrences of `x y` in the following terms:
15 <LI>`(\x y. x y) (x y)`
16 <LI> `\x y. x y (x y)`
19 Reduce to beta-normal forms:
22 <LI>`(\x. x (\y. y x)) (v w)`
23 <LI>`(\x. x (\x. y x)) (v w)`
24 <LI>`(\x. x (\y. y x)) (v x)`
25 <LI>`(\x. x (\y. y x)) (v y)`
27 <LI>`(\x y. x y y) u v`
28 <LI>`(\x y. y x) (u v) z w`
29 <LI>`(\x y. x) (\u u)`
30 <LI>`(\x y z. x z (y z)) (\u v. u)`
37 We'll assume the "Version 3" implementation of lists and numbers throughout. So:
39 <pre><code>zero ≡ \s z. z
40 succ ≡ \n. \s z. s (n s z)
41 iszero ≡ \n. n (\x. false) true
42 add ≡ \m \n. m succ n
43 mul ≡ \m \n. \s. m (n s)</code></pre>
47 <pre><code>empty ≡ \f z. z
48 make-list ≡ \hd tl. \f z. f hd (tl f z)
49 isempty ≡ \lst. lst (\hd sofar. false) true
50 extract-head ≡ \lst. lst (\hd sofar. hd) junk</code></pre>
52 The `junk` in `extract-head` is what you get back if you evaluate:
56 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.
58 <pre><code>predecesor ≡ (\shift n. n shift (make-pair zero junk) get-second) (\pair. pair (\fst snd. make-pair (successor fst) fst))
60 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>
62 The `junk` is what you get back if you evaluate:
68 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.
71 For these exercises, assume that `LIST` is the result of evaluating:
73 (make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
77 <LI>What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint):
81 <LI>Based on your answer to question 16, how might you implement the **map** function? Expected behavior:
83 map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))
85 <LI>Based on your answer to question 16, how might you implement the **filter** function? The expected behavior is that:
89 should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
91 <LI>How would you implement map using the either the version 1 or the version 2 implementation of lists?
93 <LI>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.
95 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)`.
97 less-than-or-equal zero zero ~~> true
98 less-than-or-equal zero one ~~> true
99 less-than-or-equal zero two ~~> true
100 less-than-or-equal one zero ~~> false
101 less-than-or-equal one one ~~> true
102 less-than-or-equal one two ~~> true
103 less-than-or-equal two zero ~~> false
104 less-than-or-equal two one ~~> false
105 less-than-or-equal two two ~~> true
107 You'll need to make use of the predecessor function, but it's not essential to understand how the implementation we gave above works. You can treat it as a black box.