4 Insert all the implicit `( )`s and <code>λ</code>s into the following abbreviated expressions.
9 4. `w (\x y z. x z (y z)) u v`
11 Mark all occurrences of `(x y)` in the following terms:
13 <small>(I know the numbering of the homework problems will restart instead of continuing with 5, 6, 7, ... It's too much of a pain to fix it right now. We'll put in a better rendering engine later that will make this work right without laborious work-arounds on our part. Please just renumber the problems appropriately)</small>
16 6. `(\x y. x y) (x y)`
22 Find "normal forms" for the following---that is, reduce them until no more reductions are possible. As mentioned in the notes, we'll write <code>λx</code> as `\x`. If we ever say "reduce" without qualifications, we mean just "beta-reduce" (as opposed to "(beta + eta)-reduce").
28 12. `(\x (x (\y y))) (\z (z z))`
29 13. `(\x (x x)) (\x (x x))`
30 14. `(\x (x x x)) (\x (x x x))`
37 Recall our definitions of true and false.
39 > **true** is defined to be `\t f. t`
40 > **false** is defined to be `\t f. f`
42 In Racket, these functions can be defined like this:
44 (define true (lambda (t) (lambda (f) t)))
45 (define false (lambda (t) (lambda (f) f)))
47 (Note that they are different from Racket's *primitive* boolean values `#t` and `#f`.)
50 15. Define a `neg` operator that negates `true` and `false`.
62 16. Define an `or` operator.
64 17. Define an `xor` operator. If you haven't seen this term before, here's a truth table:
66 true xor true == false
67 true xor false == true
68 false xor true == true
69 false xor false == false
76 Recall our definitions of ordered triples.
78 > the triple **(**a**,**b**, **c**)** is defined to be `\f. f a b c`
80 To extract the first element of a triple t, you write:
84 Here are some definitions in Racket:
86 (define make-triple (lambda (fst) (lambda (snd) (lambda (trd) (lambda (f) (((f fst) snd) trd))))))
87 (define fst_of_three (lambda (fst) (lambda (snd) (lambda (trd) fst))))
88 (define snd_of_three (lambda (fst) (lambda (snd) (lambda (trd) snd))))
92 (define t (((make-triple 10) 20) 30))
93 (t fst_of_three) ; will evaluate to 10
94 (t snd_of_three) ; will evaluate to 20
96 If you're puzzled by having the triple to the left and the function that
97 operates on it come second, think about why it's being done this way: the pair
98 is a package that takes a function for operating on its elements *as an
99 argument*, and returns *the result of* operating on its elements with that
100 function. In other words, the triple is a higher-order function.
103 18. Define the `swap12` function that permutes the elements of a triple. Expected behavior:
105 (define t (((make-triple 10) 20) 30))
106 ((t swap12) fst_of_three) ; evaluates to 20
107 ((t swap12) snd_of_three) ; evaluates to 10
109 Write out the definition of `swap12` in Racket.
112 19. Define a `dup3` function that duplicates its argument to form a triple
113 whose elements are the same. Expected behavior:
115 ((dup3 10) fst_of_three) ; evaluates to 10
116 ((dup3 10) snd_of_three) ; evaluates to 10
118 20. Define a `dup27` function that makes
119 twenty-seven copies of its argument (and stores them in a data structure of
126 21. Using Kapulet syntax, define `fold_left`.
128 22. Using Kapulet syntax, define `filter` (problem 7 in last week's homework) in terms of `fold_right`.
130 23. Using Kapulet syntax, define `&&` in terms of `fold_right`. (To avoid trickiness about the infix syntax, just call it `append`.)
132 24. Using Kapulet syntax, define `head` in terms of `fold_right`. When applied to a non-empty list, it should give us the first element of that list. When applied to an empty list, let's say it should give us `'error`.
134 25. We mentioned in the Encoding notes that `fold_left (flipped_cons, []) xs` would give us the elements of `xs` but in the reverse order. That is, this is how we can express `reverse` in terms of `fold_left`. How would you express `reverse` in terms of `fold_right`?
136 This problem does have an elegant and concise solution, but it may be hard for you to figure it out. We think it will a useful exercise for you to try, anyway. We'll give a [[hint|assignment2 hint]]. Don't look at the hint until you've gotten really worked up about the problem. Before that, it probably will just be baffling. If your mind has really gotten its talons into the problem, though, the hint might be just what you need to break it open.
138 Even if you don't get the answer, we think the experience of working on the problem, and then understanding the answer when we reveal it, will be satisfying and worhtwhile. It also fits our pedagogical purposes for one of the recurring themes of the class.
144 26. Given that we've agreed to Church's encoding of the numbers:
146 <code>0 ≡ \f z. z</code>
147 <code>1 ≡ \f z. f z</code>
148 <code>2 ≡ \f z. f (f z)</code>
149 <code>3 ≡ \f z. f (f (f z))</code>
152 How would you express the `succ` function in the Lambda Calculus?