From e56ba2579b115d5061d06d129b5e4ca90914983c Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Fri, 1 Oct 2010 20:33:24 -0400 Subject: [PATCH] edits --- temp.mdwn | 206 +++++++++++++++++++++++++++++++++++++++----------------------- 1 file changed, 131 insertions(+), 75 deletions(-) diff --git a/temp.mdwn b/temp.mdwn index 585e666c..e91eeee5 100644 --- a/temp.mdwn +++ b/temp.mdwn @@ -1,78 +1,134 @@ - - -do eta-reductions too - - - - - -
```+let and = \l r. l (r true false) false in
+
+; version 1 lists
+let makePair = \f s g. g f s in
+let fst = true in
+let snd = false in
+let nil = makePair true meh in
+let isNil = \x. x fst in
+let makeList = \h t. makePair false (makePair h t) in
+let head = \l. isNil l err (l snd fst) in
+let tail = \l. isNil l err (l snd snd) in
+
+; a list of numbers to experiment on
+let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
+
+; a fixed-point combinator for defining recursive functions
+let Y = \f. (\h. f (h h)) (\h. f (h h)) in
+
+; church numerals
+let isZero = \n. n (\x. false) true in
+let succ = \n s z. s (n s z) in
+let mult = \m n s. m (n s) in
+let length = Y (\length l. isNil l 0 (succ (length (tail l)))) in
+let pred = \n. isZero n 0 (length (tail (n (\p. makeList meh p) nil))) in
+let leq = \m n. isZero(n pred m) in
+let eq = \m n. and (leq m n)(leq n m) in
+
+eq 2 2 yes no
+```
+ + +Then `length mylist` evaluates to 3. + +1. What does `head (tail (tail mylist))` evaluate to? + +2. Using the `length` function as a model, and using the predecessor +function, write a function that computes factorials. (Recall that n!, +the factorial of n, is n times the factorial of n-1.) + +Warning: my browser isn't able to compute factorials of numbers +greater than 2 (it does't provide enough resources for the JavaScript +interpreter; web pages are not supposed to be that computationally +intensive). + +3. (Easy) Write a function `listLenEq` that returns true just in case two lists have the +same length. That is, + + listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true + + listLenEq mylist (makeList meh (makeList meh nil))) ~~> false + + +4. (Still easy) Now write the same function, but don't use the length function. + +5. In assignment 2, we discovered that version 3-type lists (the ones that +work like Church numerals) made it much easier to define operations +like `map` and `filter`. But now that we have recursion in our toolbox, +reasonable map and filter functions for version 1 lists are within our +reach. Give definitions for `map` and a `filter` for verson 1 type lists. + +#Computing with trees# + +Linguists analyze natural language expressions into trees. +We'll need trees in future weeks, and tree structures provide good +opportunities for learning how to write recursive functions. +Making use of the resources we have at the moment, we can approximate +trees as follows: instead of words, we'll use Church numerals. +Then a tree will be a (version 1 type) list in which each element is +itself a tree. For simplicity, we'll adopt the convention that +a tree of length 1 must contain a number as its only element. +Then we have the following representations: + +
```+   (a)           (b)             (c)
+    .
+   /|\            /\              /\
+  / | \          /\ 3            1 /\
+  1 2  3        1  2               2 3
+
+[[1];[2];[3]]  [[[1];[2]];[3]]   [[1];[[2];[3]]]
```
- + +Limitations of this scheme include the following: there is no easy way +to label a constituent with a syntactic category (S or NP or VP, +etc.), and there is no way to represent a tree in which a mother has a +single daughter. + +When processing a tree, you can test for whether the tree contains +only a numeral (in which case the tree is leaf node) by testing for +whether the length of the list is less than or equal to 1. This will +be your base case for your recursive functions that operate on these +trees. + +1. Write a function that sums the number of leaves in a tree. + +Expected behavior: + +
```+let t1 = (makeList 1 nil) in
+let t2 = (makeList 2 nil) in
+let t3 = (makeList 3 nil) in
+let t12 = (makeList t1 (makeList t2 nil)) in
+let t23 = (makeList t2 (makeList t3 nil)) in
+let ta = (makeList t1 t23) in
+let tb = (makeList t12 t3) in
+let tc = (makeList t1 (makeList t23 nil)) in
+
+sum-leaves t1 ~~> 1
+sum-leaves t2 ~~> 2
+sum-leaves t3 ~~> 3
+sum-leaves t12 ~~> 3
+sum-leaves t23 ~~> 5
+sum-leaves ta ~~> 6
+sum-leaves tb ~~> 6
+sum-leaves tc ~~> 6
+```
+ +2. Write a function that counts the number of leaves. + -- 2.11.0