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) * 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 ----------------- 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.