X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=f0e8a0989cfaf28919d7b03ead3ce51529a969e7;hp=82957888e1c773cb007feb9b7ec7fd9ef55ac735;hb=4c0fb46d0cb9dcbfa5687140afeca2fdb48f668c;hpb=ac5b18441c0c498fa20ccd2db601d895d665929b diff --git a/assignment2.mdwn b/assignment2.mdwn index 82957888..f0e8a098 100644 --- a/assignment2.mdwn +++ b/assignment2.mdwn @@ -1,3 +1,6 @@ +For these assignments, you'll probably want to use a "lambda calculator" to check your work. This accepts any grammatical lambda expression and reduces it to normal form, when possible. See the page on [[using the programming languages]] for instructions and links about setting this up. + + More Lambda Practice -------------------- @@ -30,6 +33,33 @@ 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) + +* 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) + +* 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 +86,7 @@ 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: @@ -72,37 +103,37 @@ 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))))) -1. What would be the result of evaluating: - - LIST make-list empty +
      +
    1. What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint): - [[Assignment 2 hint 1]] + LIST make-list empty -2. 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))))) -3. 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. +