X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=5d75a855a5a09ef187dcdd9b6e4e22bb11b6ee7a;hp=85ec6ce71385d856231a482a0bdc74a4aaaa1e16;hb=HEAD;hpb=70fdac4d0a4db28dc391e3ea13b9c590b6ef9760 diff --git a/assignment2.mdwn b/assignment2.mdwn deleted file mode 100644 index 85ec6ce7..00000000 --- a/assignment2.mdwn +++ /dev/null @@ -1,142 +0,0 @@ -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 --------------------- - -Insert all the implicit `( )`s and λs 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: - -
    -
  1. `(\x y. x y) x y` -
  2. `(\x y. x y) (x y)` -
  3. `\x y. x y (x y)` -
- -Reduce to beta-normal forms: - -
    -
  1. `(\x. x (\y. y x)) (v w)` -
  2. `(\x. x (\x. y x)) (v w)` -
  3. `(\x. x (\y. y x)) (v x)` -
  4. `(\x. x (\y. y x)) (v y)` - -
  5. `(\x y. x y y) u v` -
  6. `(\x y. y x) (u v) z w` -
  7. `(\x y. x) (\u u)` -
  8. `(\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 ------------------ - -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 (see [[Assignment 2 hint 1]] for a hint): - - LIST make-list empty - -
  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))))) - -
  3. 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`. - -
  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. - -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. -
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