+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 our [lambda-let page](/lambda-let.html), based on Chris Barker's JavaScript lambda calculator and [Oleg Kiselyov's Haskell lambda calculator](http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell).
+
+
More Lambda Practice
--------------------
<LI>`(\x y z. x z (y z)) (\u v. u)`
</OL>
+Combinatory Logic
+-----------------
+
+Reduce the following forms, if possible:
+
+<OL start=16>
+<LI> `Kxy`
+<LI> `KKxy`
+<LI> `KKKxy`
+<LI> `SKKxy`
+<LI> `SIII`
+<LI> `SII(SII)`
+
+<LI> Give Combinatory Logic combinators that behave like our boolean functions.
+ You'll need combinators for `true`, `false`, `neg`, `and`, `or`, and `xor`.
+</OL>
+
+Using the mapping specified in the lecture notes,
+translate the following lambda terms into combinatory logic:
+
+<OL start=23>
+<LI> `\x.x`
+<LI> `\xy.x`
+<LI> `\xy.y`
+<LI> `\xy.yx`
+<LI> `\x.xx`
+<LI> `\xyz.x(yz)`
+<LI> For each translation, how many I's are there? Give a rule for
+ describing what each I corresponds to in the original lambda term.
+</OL>
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>