`λ`

s 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:
+
+(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)
+
+5. `(\x y. x y) x y`
+6. `(\x y. x y) (x y)`
+7. `\x y. x y (x y)`
+
Reduction
---------
Find "normal forms" for the following---that is, reduce them until no more reductions are possible. As mentioned in the notes, we'll write `λx`

as `\x`. If we ever say "reduce" without qualifications, we mean just "beta-reduce" (as opposed to "(beta + eta)-reduce").
-1. `(\x \y. y x) z`
-2. `(\x (x x)) z`
-3. `(\x (\x x)) z`
-4. `(\x (\z x)) z`
-5. `(\x (x (\y y))) (\z (z z))`
-6. `(\x (x x)) (\x (x x))`
-7. `(\x (x x x)) (\x (x x x))`
+8. `(\x \y. y x) z`
+9. `(\x (x x)) z`
+10. `(\x (\x x)) z`
+11. `(\x (\z x)) z`
+12. `(\x (x (\y y))) (\z (z z))`
+13. `(\x (x x)) (\x (x x))`
+14. `(\x (x x x)) (\x (x x x))`
+
Booleans
@@ -27,9 +46,8 @@ In Racket, these functions can be defined like this:
(Note that they are different from Racket's *primitive* boolean values `#t` and `#f`.)
-(I know the numbering of the homework problems will restart instead of continuing with 8, 9, ... 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.)
-8. Define a `neg` operator that negates `true` and `false`.
+15. Define a `neg` operator that negates `true` and `false`.
Expected behavior:
@@ -41,9 +59,9 @@ In Racket, these functions can be defined like this:
evaluates to 10.
-9. Define an `or` operator.
+16. Define an `or` operator.
-10. Define an `xor` operator. If you haven't seen this term before, here's a truth table:
+17. Define an `xor` operator. If you haven't seen this term before, here's a truth table:
true xor true == false
true xor false == true
@@ -65,7 +83,7 @@ To extract the first element of a triple t, you write:
Here are some definitions in Racket:
- (define make-triple (lambda (fst) (lambda (snd) (lambda (trd) (lambda (f) (((f fst) snd) trd))))))
+ (define make-triple (lambda (fst) (lambda (snd) (lambda (trd) (lambda (f) (((f fst) snd) trd))))))
(define fst_of_three (lambda (fst) (lambda (snd) (lambda (trd) fst))))
(define snd_of_three (lambda (fst) (lambda (snd) (lambda (trd) snd))))
@@ -82,7 +100,7 @@ argument*, and returns *the result of* operating on its elements with that
function. In other words, the triple is a higher-order function.
-11. Define the `swap12` function that permutes the elements of a triple. Expected behavior:
+18. Define the `swap12` function that permutes the elements of a triple. Expected behavior:
(define t (((make-triple 10) 20) 30))
((t swap12) fst_of_three) ; evaluates to 20
@@ -91,14 +109,61 @@ function. In other words, the triple is a higher-order function.
Write out the definition of `swap12` in Racket.
-12. Define a `dup3` function that duplicates its argument to form a triple
+19. Define a `dup3` function that duplicates its argument to form a triple
whose elements are the same. Expected behavior:
((dup3 10) fst_of_three) ; evaluates to 10
((dup3 10) snd_of_three) ; evaluates to 10
-13. Define a `dup27` function that makes
+20. Define a `dup27` function that makes
twenty-seven copies of its argument (and stores them in a data structure of
your choice).
+Folds and Lists
+---------------
+
+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.
+
+
+
+Numbers
+-------
+
+26. Given that we've agreed to Church's encoding of the numbers:
+
+ BLAH
+
+ How would you express the `succ` function in the Lambda Calculus?
+