+++ /dev/null
-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.
-
-
-More Lambda Practice
---------------------
-
-Insert all the implicit `( )`s and <code>λ</code>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:
-
-<OL start=5>
-<LI>`(\x y. x y) x y`
-<LI>`(\x y. x y) (x y)`
-<LI> `\x y. x y (x y)`
-</OL>
-
-Reduce to beta-normal forms:
-
-<OL start=8>
-<LI>`(\x. x (\y. y x)) (v w)`
-<LI>`(\x. x (\x. y x)) (v w)`
-<LI>`(\x. x (\y. y x)) (v x)`
-<LI>`(\x. x (\y. y x)) (v y)`
-
-<LI>`(\x y. x y y) u v`
-<LI>`(\x y. y x) (u v) z w`
-<LI>`(\x y. x) (\u u)`
-<LI>`(\x y z. x z (y z)) (\u v. u)`
-</OL>
-
-Combinatory Logic
------------------
-
-Reduce the following forms, if possible:
-
-<OL start=16>
-<LI> `Kxy`
-<LI> `KKxy`
-<LI> `KKKxy`
-<LI> `SKKxy`
-<LI> `SIII`
-<LI> `SII(SII)`
-
-<LI> Give Combinatory Logic combinators that behave like our boolean functions.
- You'll need combinators for `true`, `false`, `neg`, `and`, `or`, and `xor`.
-</OL>
-
-Using the mapping specified in the lecture notes,
-translate the following lambda terms into combinatory logic:
-
-<OL start=23>
-<LI> `\x.x`
-<LI> `\xy.x`
-<LI> `\xy.y`
-<LI> `\xy.yx`
-<LI> `\x.xx`
-<LI> `\xyz.x(yz)`
-<LI> For each translation, how many I's are there? Give a rule for
- describing what each I corresponds to in the original lambda term.
-</OL>
-
-Lists and Numbers
------------------
-
-We'll assume the "Version 3" implementation of lists and numbers throughout. So:
-
-<pre><code>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)</code></pre>
-
-And:
-
-<pre><code>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</code></pre>
-
-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.
-
-<pre><code>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))</code></pre>
-
-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)))))
-
-
-<OL start=16>
-<LI>What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint):
-
- LIST make-list empty
-
-<LI>Based on your answer to question 16, 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)))))
-
-<LI>Based on your answer to question 16, 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`.
-
-<LI>What goes wrong when we try to apply these techniques using the version 1 or version 2 implementation of lists?
-
-<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.
-
-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 essential to understand how the implementation we gave above works. You can treat it as a black box.
-</OL>
-