X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=5d75a855a5a09ef187dcdd9b6e4e22bb11b6ee7a;hp=4f80620631159be4834260dabe32e119cb6cef1a;hb=9a23d7fc448e48268a7b60ce9ff3e55222cd833a;hpb=7bb54cf783a20dd6f8ff464f5be97e72b87f4045 diff --git a/assignment2.mdwn b/assignment2.mdwn index 4f806206..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: @@ -73,7 +107,7 @@ For these exercises, assume that `LIST` is the result of evaluating:
      -
    1. What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint): +
    2. What would be the result of evaluating (see [[hints/Assignment 2 hint]] for a hint): LIST make-list empty @@ -87,7 +121,7 @@ For these exercises, assume that `LIST` is the result of evaluating: should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`. -
    3. 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. @@ -103,6 +137,6 @@ Given the primitive arithmetic functions above, how would you implement the less 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.