+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
--------------------
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.
<pre><code>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))</code></pre>
The `junk` is what you get back if you evaluate:
(make-list a (make-list b (make-list c (make-list d (make-list e empty)))))
-(16). What would be the result of evaluating:
+<OL start=16>
+<LI>What would be the result of evaluating (see [[Assignment 2 hint 1]] for a hint):
- LIST make-list empty
+ LIST make-list empty
- [[Assignment 2 hint 1]]
+<LI>Based on your answer to question 16, how might you implement the **map** function? Expected behavior:
-(17). Based on your answer to question 1, 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)))))
- <pre><code>map f LIST <~~> (make-list (f a) (make-list (f b) (make-list (f c) (make-list (f d) (make-list (f e) empty)))))</code></pre>
+<LI>Based on your answer to question 16, how might you implement the **filter** function? The expected behavior is that:
-18. Based on your answer to question 1, how might you implement the **filter** function? The expected behavior is that:
+ filter f LIST
- 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`.
- should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
+<LI>What goes wrong when we try to apply these techniques using the version 1 or version 2 implementation of lists?
-4. How would you implement map using the either the version 1 or the version 2 implementation of lists?
+<LI>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.
-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)`.
- 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
- 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 important to understand how the implementation we gave above works. You can treat it as a black box.
+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.
+</OL>