Syntax
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-Insert all the implicit `( )`s and <code>λ</code>s into the following abbreviated expressions.
+Insert all the implicit `( )`s and <code>λ</code>s into the following abbreviated expressions. Don't just insert them *freely*; rather, provide the official expression, without any notational shortcuts, that is syntactically identical to the form presented.
1. `x x (x x x) x`
2. `v w (\x y. v x)`
Booleans
--------
+For these questions, and the ones on triples below, we're setting them up so as to encourage you to experiment with Racket and to formulate your answer in Scheme/Racket syntax. But you can answer in Lambda Calculus syntax if you prefer.
+
Recall our definitions of true and false.
> **true** is defined to be `\t f. t`
15. Define a `neg` operator that negates `true` and `false`.
- Expected behavior:
+ Expected behavior:
(((neg true) 10) 20)
- evaluates to 20, and
+ evaluates to `20`, and
(((neg false) 10) 20)
- evaluates to 10.
+ evaluates to `10`.
16. Define an `or` operator.
Recall our definitions of ordered triples.
-> the triple **(**a**,**b**, **c**)** is defined to be `\f. f a b c`
+> the triple **(**a**, **b**, **c**)** is defined to be `\f. f a b c`
-To extract the first element of a triple t, you write:
+To extract the first element of a triple `t`, you write:
t (\fst snd trd. fst)
(t snd_of_three) ; will evaluate to 20
If you're puzzled by having the triple to the left and the function that
-operates on it come second, think about why it's being done this way: the pair
+operates on it come second, think about why it's being done this way: the triple
is a package that takes a function for operating on its elements *as an
argument*, and returns *the result of* operating on its elements with that
function. In other words, the triple is a higher-order function.
21. Using Kapulet syntax, define `fold_left`.
-22. Using Kapulet syntax, define `filter` (problem 7 in last week's homework) in terms of `fold_right`.
-
-23. Using Kapulet syntax, define `&&` in terms of `fold_right`. (To avoid trickiness about the infix syntax, just call it `append`.)
-
-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`.
-
-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`?
-
- 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]]. 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.
-
- 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.
-
-<!--
-Hint. Suppose the list we want to reverse is [10, 20, 30]. Applying `fold_right` to this will begin by computing `f 30 z` for some `f` and `z` that we specify. If we made the result of that be something like `30 & blah`, or any larger structure that contained something of that form, it's not clear how we could, using just the resources of `fold_right`, reach down into that structure and replace the `blah` with some other element, as we'd evidently need to, since after the next step we should get `30 & (20 & blah)`. What we'd like instead is something like this:
-
- 30 & < >
-
-Where `< >` isn't some *value* but rather a *hole*. Then with the next step, we want to plug into that hole `20 & < >`, which contains its own hole. Getting:
+22. Using Kapulet syntax, define `filter` (problem 7 in last week's homework) in terms of `fold_right` and other primitive syntax like `lambda`, `&`, and `[]`. Don't use `letrec`! All the `letrec`-ing that happens should come from the one inside the definition of `fold_right`.
- 30 & (20 & < >)
+23. Using Kapulet syntax, define `&&` in terms of `fold_right`. (To avoid trickiness about the infix syntax, just call it `append`.) As with problem 22 (the previous problem), don't use `letrec`!
-And so on. That is the key to the solution. The questions you need to answer, to turn this into something executable, are:
+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 `'err`. As with problem 22, don't use `letrec`!
-1. What is a hole? How can we implement it?
+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. So that's how we can express `reverse` in terms of `fold_left`. How would you express `reverse` in terms of `fold_right`? As with problem 22, don't use `letrec`!
-2. What should `f` be, so that the result of `f (20, 30 & < >)`, is `30 & (20 & < >)`?
+ 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.
-3. What should `z` be, so that the result of `f (30, z)` is `30 & < >`?
+ There are also other, less cool answers. Perhaps you'll find one of them first.
-4. At the end of the `fold_right`, we're going to end with something like `30 & (20 & (10 & < >))`. But what we want is `[30, 20, 10]`. How can we turn what we've gotten into what we want?
+ Even if you don't get any answer, we think the experience of working on the problem, and then understanding the answer when we reveal it, will be satisfying and worthwhile. It also fits our pedagogical purposes for some of the recurring themes of the class.
--->
Numbers
-------