+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
--------------------
<LI>`(\x y z. x z (y z)) (\u v. u)`
</OL>
+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
-----------------
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:
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>How would you implement map using the either the version 1 or the version 2 implementation of lists?
+<LI>What goes wrong when we try to apply these techniques using the version 1 or 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.
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>