X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=5d75a855a5a09ef187dcdd9b6e4e22bb11b6ee7a;hp=04c3a282de2d35b47efc56d066d6a1291a49acc8;hb=HEAD;hpb=f6ffcb0a876f3e27d478217390ff2874c6b12dfb;ds=sidebyside diff --git a/assignment2.mdwn b/assignment2.mdwn deleted file mode 100644 index 04c3a282..00000000 --- a/assignment2.mdwn +++ /dev/null @@ -1,70 +0,0 @@ -We'll assume the "Version 3" implementation of lists and numbers throughout. So: - -
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 - -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. -