1 For these assignments, you'll probably want to use our [[lambda evaluator]] to check your work. This accepts any grammatical lambda expression and reduces it to normal form, when possible.
7 Insert all the implicit `( )`s and <code>λ</code>s into the following abbreviated expressions:
12 4. `w (\x y z. x z (y z)) u v`
14 Mark all occurrences of `x y` in the following terms:
18 <LI>`(\x y. x y) (x y)`
19 <LI> `\x y. x y (x y)`
22 Reduce to beta-normal forms:
25 <LI>`(\x. x (\y. y x)) (v w)`
26 <LI>`(\x. x (\x. y x)) (v w)`
27 <LI>`(\x. x (\y. y x)) (v x)`
28 <LI>`(\x. x (\y. y x)) (v y)`
30 <LI>`(\x y. x y y) u v`
31 <LI>`(\x y. y x) (u v) z w`
32 <LI>`(\x y. x) (\u u)`
33 <LI>`(\x y z. x z (y z)) (\u v. u)`
39 Reduce the following forms, if possible:
49 <LI> Give Combinatory Logic combinators that behave like our boolean functions.
50 You'll need combinators for `true`, `false`, `neg`, `and`, `or`, and `xor`.
53 Using the mapping specified in the lecture notes,
54 translate the following lambda terms into combinatory logic:
63 <LI> For each translation, how many I's are there? Give a rule for
64 describing what each I corresponds to in the original lambda term.
70 We'll assume the "Version 3" implementation of lists and numbers throughout. So:
72 <pre><code>zero ≡ \s z. z
73 succ ≡ \n. \s z. s (n s z)
74 iszero ≡ \n. n (\x. false) true
75 add ≡ \m \n. m succ n
76 mul ≡ \m \n. \s. m (n s)</code></pre>
80 <pre><code>empty ≡ \f z. z
81 make-list ≡ \hd tl. \f z. f hd (tl f z)
82 isempty ≡ \lst. lst (\hd sofar. false) true
83 extract-head ≡ \lst. lst (\hd sofar. hd) junk</code></pre>
85 The `junk` in `extract-head` is what you get back if you evaluate:
89 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.
91 <pre><code>predecesor ≡ (\shift n. n shift (make-pair zero junk) get-second) (\pair. pair (\fst snd. make-pair (successor fst) fst))
93 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>
95 The `junk` is what you get back if you evaluate:
101 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.
104 For these exercises, assume that `LIST` is the result of evaluating:
106 (make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
110 <LI>What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint):
114 <LI>Based on your answer to question 16, how might you implement the **map** function? Expected behavior:
116 map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))
118 <LI>Based on your answer to question 16, how might you implement the **filter** function? The expected behavior is that:
122 should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
124 <LI>What goes wrong when we try to apply these techniques using the version 1 or version 2 implementation of lists?
126 <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.
128 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)`.
130 less-than-or-equal zero zero ~~> true
131 less-than-or-equal zero one ~~> true
132 less-than-or-equal zero two ~~> true
133 less-than-or-equal one zero ~~> false
134 less-than-or-equal one one ~~> true
135 less-than-or-equal one two ~~> true
136 less-than-or-equal two zero ~~> false
137 less-than-or-equal two one ~~> false
138 less-than-or-equal two two ~~> true
140 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.