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-More Lambda Practice
---------------------
-
-Insert all the implicit `( )`s and λ
s into the following abbreviated expressions:
-
-1. `x x (x x x) x`
-2. `v w (\x y. v x)`
-3. `(\x y. x) u v`
-4. `w (\x y z. x z (y z)) u v`
-
-Mark all occurrences of `x y` in the following terms:
-
-
zero ≡ \s z. z
-succ ≡ \n. \s z. s (n s z)
-iszero ≡ \n. n (\x. false) true
-add ≡ \m \n. m succ n
-mul ≡ \m \n. \s. m (n s)
-
-And:
-
-empty ≡ \f z. z
-make-list ≡ \hd tl. \f z. f hd (tl f z)
-isempty ≡ \lst. lst (\hd sofar. false) true
-extract-head ≡ \lst. lst (\hd sofar. hd) junk
-
-The `junk` in `extract-head` is what you get back if you evaluate:
-
- extract-head empty
-
-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.
-
-predecesor ≡ (\shift n. n shift (make-pair zero junk) get-second) (\pair. pair (\fst snd. make-pair (successor fst) fst))
-extract-tail ≡ (\shift lst. lst shift (make-pair empty junk) get-second) (\hd pair. pair (\fst snd. make-pair (make-list hd fst) fst))
-
-The `junk` is what you get back if you evaluate:
-
- predecessor zero
-
- extract-tail empty
-
-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.
-
-
-For these exercises, assume that `LIST` is the result of evaluating:
-
- (make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
-
-
-1. What would be the result of evaluating:
-
- LIST make-list empty
-
- [[Assignment 2 hint 1]]
-
-2. Based on your answer to question 1, how might you implement the **map** function? Expected behavior:
-
- map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))
-
-3. Based on your answer to question 1, how might you implement the **filter** function? The expected behavior is that:
-
- filter f LIST
-
- should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
-
-4. How would you implement map using the either the version 1 or the version 2 implementation of lists?
-
-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.
-
- 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)`.
-
- less-than-or-equal zero zero ~~> true
- less-than-or-equal zero one ~~> true
- less-than-or-equal zero two ~~> true
- less-than-or-equal one zero ~~> false
- less-than-or-equal one one ~~> true
- less-than-or-equal one two ~~> true
- less-than-or-equal two zero ~~> false
- less-than-or-equal two one ~~> false
- less-than-or-equal two two ~~> true
-
- 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.
-