X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment2.mdwn;h=3e932561960bbc7792c34d868ea008aee1b2fb25;hp=4f80620631159be4834260dabe32e119cb6cef1a;hb=fee1f7831003a7edee103dd7999b390abc7583c8;hpb=7bb54cf783a20dd6f8ff464f5be97e72b87f4045
diff --git a/assignment2.mdwn b/assignment2.mdwn
index 4f806206..3e932561 100644
--- a/assignment2.mdwn
+++ b/assignment2.mdwn
@@ -1,3 +1,6 @@
+For these assignments, you'll probably want to use our [[lambda evaluator]] to check your work. This accepts any grammatical lambda expression and reduces it to normal form, when possible.
+
+
More Lambda Practice
--------------------
@@ -30,6 +33,36 @@ Reduce to beta-normal forms:
`(\x y z. x z (y z)) (\u v. u)`
+Combinatory Logic
+-----------------
+
+Reduce the following forms, if possible:
+
+
+- `Kxy`
+
- `KKxy`
+
- `KKKxy`
+
- `SKKxy`
+
- `SIII`
+
- `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:
+
+
+- `\x.x`
+
- `\xy.x`
+
- `\xy.y`
+
- `\xy.yx`
+
- `\x.xx`
+
- `\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
-----------------
@@ -56,6 +89,7 @@ The `junk` in `extract-head` is what you get back if you evaluate:
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.
```
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))
```

The `junk` is what you get back if you evaluate:
@@ -87,7 +121,7 @@ For these exercises, assume that `LIST` is the result of evaluating:
should evaluate to a list containing just those of `a`, `b`, `c`, `d`, and `e` such that `f` applied to them evaluates to `true`.
-How would you implement map using the either the version 1 or the version 2 implementation of lists?
+What goes wrong when we try to apply these techniques using the version 1 or version 2 implementation of lists?
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.
@@ -103,6 +137,6 @@ Given the primitive arithmetic functions above, how would you implement the less
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.