X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=5d75a855a5a09ef187dcdd9b6e4e22bb11b6ee7a;hp=bcfba912fc6dfd28a4a37df1590370a9b764f4b6;hb=641e2e0035a5e3225f719c0ef201f93362b0f8ad;hpb=c7978bc281c36093f85f5c478f2844fc124919a5 diff --git a/assignment2.mdwn b/assignment2.mdwn index bcfba912..5d75a855 100644 --- a/assignment2.mdwn +++ b/assignment2.mdwn @@ -1,3 +1,6 @@ +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 -------------------- @@ -30,6 +33,36 @@ Reduce to beta-normal forms:
  • `(\x y z. x z (y z)) (\u v. u)` +Combinatory Logic +----------------- + +Reduce the following forms, if possible: + +
      +
    1. `Kxy` +
    2. `KKxy` +
    3. `KKKxy` +
    4. `SKKxy` +
    5. `SIII` +
    6. `SII(SII)` + +
    7. Give Combinatory Logic combinators that behave like our boolean functions. + You'll need combinators for `true`, `false`, `neg`, `and`, `or`, and `xor`. +
    + +Using the mapping specified in the lecture notes, +translate the following lambda terms into combinatory logic: + +
      +
    1. `\x.x` +
    2. `\xy.x` +
    3. `\xy.y` +
    4. `\xy.yx` +
    5. `\x.xx` +
    6. `\xyz.x(yz)` +
    7. For each translation, how many I's are there? Give a rule for + describing what each I corresponds to in the original lambda term. +
    Lists and Numbers ----------------- @@ -56,6 +89,7 @@ The `junk` in `extract-head` is what you get back if you evaluate: 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: @@ -72,35 +106,37 @@ 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))))) -16. What would be the result of evaluating [[Assignment 2 hint 1]]: +
      +
    1. What would be the result of evaluating (see [[hints/Assignment 2 hint]] for a hint): - LIST make-list empty + LIST make-list empty -17. Based on your answer to question 1, how might you implement the **map** function? Expected behavior: +
    2. 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)))))
      + map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty))))) -18. Based on your answer to question 1, how might you implement the **filter** function? The expected behavior is that: +
    3. Based on your answer to question 16, how might you implement the **filter** function? The expected behavior is that: - filter f LIST + 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`. +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? +
    4. What goes wrong when we try to apply these techniques using the version 1 or 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. +
    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)`. +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 + 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. +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. +