# Homework #4 Solution

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In Lab problems a-c:

Problem a:

Consider the following definition:

&gt; mm f g = (.) (map f) (map g)

What is the type of

1. ((-) 20) ((+) 9) ? What are the values of

mm ((-) 20) ((+) 9) [3,-2, -9, 50] mm ((-) 20) ((+) 9) [-3,2, -9, 50]

Describe in word what mm does.

Define mm2 to be equivalent to mm but

only using only one ‘map’ application, and compose, (.).

Problem b:

Consider the following definition:

&gt; lc f p = \lst -&gt; [f x | x&lt;-lst, p x]

What is the type of

lc (\n -&gt; n -1) odd ?

What are the values of :

lc (\n -&gt; n -1) odd [2,33,50,11,23,4]

lc (\n -&gt; n -1) ((.) not odd) [2,33,50,11,23,4]

Define hf to be equivalent to lc but uses map and filter instead of list comprehension.

Problem c:

‘prodPRL nums’, defined below, produces a list of partial products of the numbers in nums. The first value in the resulting list will be the prod of all the items, the second will be the prod of all the items except the first, etc. For example:

…&gt; prodPRL [3,-2,10]

[-60,-6,3]

 > prodPRL (l:ls) = prods [l] ls > where (n:ns) = prods ((a*n):a:as) ns > prods (a:as) > prods as [] = as

Define prodPRLHOF to be equivalent to prodPRL but using a fold. I have started the prodPRLHOF definition, you only need to complete the definition for f.

&gt; prodPRLHOF :: (Num t1, Foldable t) =&gt; t t1 -&gt; [t1]

 > prodPRLHOF = foldr f [] > where [] = > f y > f y (x:xs) =

Remember foldr can be defined as follows:

&gt; foldr :: (a -&gt; b -&gt; b) -&gt; b -&gt; [a] -&gt; b &gt; foldr f accumulator [] = accumulator

&gt; foldr f accumulator (x:xs) = f x (foldr f accumulator xs)

———————- HW ———————-

Problem 1: Find operators (that can be partially applied), sec1 and sec2, so that map sec1 . filter sec2

has the same effect as

filter (&gt;0) . map (+1)

Problem 1 Answer:

Problem 2: Remember in the second Haskell assignment problem 6 you used primitive recursion to define a function composeList.

composeList took a list of functions and composed them into a single function.

… &gt;composeList [ (-) 3 , (*) 2, (+)5 ] 7 -21

One possible solution is

&gt; composeList :: [(a -&gt; a) -&gt; a -&gt; a] -&gt; a -&gt; a &gt; composeList [] = id

&gt; composeList (h:t) = (.) h composeList t

Define composeHOF to be equivalent to composeList but use a ‘fold?’ You will need to specify the type composeHof

&gt; composeHOF :: Foldable t =&gt; t (b -&gt; b) -&gt; b -&gt; b

Problem 2 Answer:

Problem 3: The harmonic series is the following infinite series:

 1 + 1 1 1 1 1 – + – + – + – + … + … – .. 2 3 4 5 i

(http://en.wikipedia.org/wiki/Harmonic_series_(mathematics))

Write a function sumHarmonic such that sumHarmonic i is the sum of the first in terms of this series. For example, sumHarmonic 4 ~&gt; 1 + 1 + 1 + 1 ~&gt; 2.08333…

2 3 4

In the last assignment you defined sumHarmonic using a simple recursive style. One possible solution is

&gt; sumHarmonic :: (Eq a, Fractional a) =&gt; a -&gt; a &gt; sumHarmonic 1 = 1

&gt; sumHarmonic n = 1 / n + sumHarmonic (n-1)

Rewrite the above definition using either foldr or foldl. ??? should be a lamdba expression that accumulates the values

sumH n = foldr ??? 0 [1..n]

or

sumH n = foldl ??? 0 [1..n]

…&gt; sumHarmonic 20

3.59773965714368

Problem 3 Answer:

Problem 4: In the last homework, problem 6, you wrote a function to create the next thue value. Below is one possible solution.

&gt; thue (s:sx) = s : (mod (s+1) 2): thue sx &gt; thue [ ] = [ ]

Using the circular list idea demostrated in fibSeq (http://bingweb.binghamton.edu/%7Ehead/CS471/NOTES/HASKELL/4hF02.html)

Define thueSeq to be a sequence of thue values. You should use list comprehension. You may use the ‘thue’ definition from your homework or the solution provided above. You may assume the sequence starts with “0”.

For example:

&gt; take 4 thueSeq

&gt; [[0],[0,1],[0,1,1,0],[0,1,1,0,1,0,0,1]]

Problem 4 Answer:

&gt; thueSeq :: [[Integer]]

Problem 5: thueSeqMap is another definition of thueSeq using map instead of list comprehension.

Problem 5 Answer:

Problem 6.

Using an HOF (map, fold and/or filter ) define flattenT that returns a list of value in each tuple.

&gt; flattenT :: Foldable t =&gt; t (a, a) -&gt; [a]

6a) The output should be in the same order as the values appear in the original list.

e.g.

…&gt; flattenT [(1,2), (3,4), (11,21),(-5,45)] [1,2,3,4,11,21,-5,45]

6b) flattenR is the same as 7a, however, the values appear in the reverse order from the original list. Do not use the builtin reverse.

e.g.

…&gt; flattenR [(1,2), (3,4), (11,21),(-5,45)] [45,-5,21,11,4,3,2,1]

Problem 6 Answer:

&gt; flattenT :: Foldable t =&gt; t (a, a) -&gt; [a]

&gt; flattenR :: Foldable t =&gt; t (a, a) -&gt; [a]

Problem 7: An inductive definition of a binary tree and a pretty print function was provided in the last assignment (repeated below).

 > data Tree a = Nil (Tree a) deriving Eq > | Node a (Tree a) > instance Show a => Show (Tree a) where > show t = show’ t 0 > where ind = replicate ind ‘ ‘ ++ “Nil” > show’ Nil > show’ (Node v l r) ind = > replicate ind ‘ ‘ ++ “(Node ” ++ show v ++ “\n” ++ > show’ l (ind+1) ++ “\n” ++ > show’ r (ind+1) ++ “\n” ++ > replicate ind ‘ ‘ ++ “)”

If ‘Tree a’ is a BST (binary search tree) then ‘tree_search’ returns True if v (value) is a member of the “Tree a”

 > tree_search :: Ord a => Tree a -> a -> Bool > tree_search Nil v = False > tree_search (Node e l r) v > | v == e = True > | v < e = tree_search l v > | v > e = tree_search r v

‘tree_insert’, defined below, inserts a value into a BST and maintances the BST invariant. Below you see a new operator, “@”.

Using “@”, pronounced ‘as’, it is possible to label all or part of an argument.

So we can use ‘nd’ on line 226 (5 lines below) instead of ‘Node e l r’.

 > tree_insert :: Ord a => Tree a -> a -> Tree a > tree_insert Nil v = (Node v Nil Nil) > tree_insert nd@(Node e l r) v > | v == e = nd > | v < e = (Node e (tree_insert l v) r) > | v > e = (Node e l (tree_insert r v))

Provided below are a few default binary search trees so you do not need to keep entering them into ghci.

&gt; tree1 = (Node 5 (Node 2 (Node 1 Nil Nil) (Node 3 Nil Nil)) (Node 7 Nil (Node 9 Nil Nil)))

&gt; tree2 = (Node 5 (Node 2 (Node 1 Nil Nil) Nil) (Node 7 Nil (Node 9 Nil Nil)))

Using our inductive definition of a Tree and define a function called ‘tree_fold f acc tree” that combines all the values of a tree inorder.

In reality, this is no different than performing an inorder traversal of the tree, except that you are accumulating (combining) values along the way.

Here is a rough sketch of the algorithm to help you get started:

1. tree_fold f acc Nil = acc (returns the combination thus far)

1. tree_fold f acc (Node v l r) =

1. Recurse on the left, getting a new accumulated value

1. Perform the accumulating operation (f) on the current value (v) and the returned value from (a)

1. Recurse on the right, passing the new accumulated value from (b)

Problem 7 Answer:

&gt; tree_fold :: (a -&gt; b -&gt; a) -&gt; a -&gt; Tree b -&gt; a

Problem 8: To demonstrate the computational power of tree_fold, write a function ‘inorder_list tree’ that uses tree_fold to build a list of all the elements

of tree in order. Since the tree_fold recurses inorder, all you simply need to think about is the lambda that should build the list

Problem 8 Answer:

&gt; inorder_list :: Tree a -&gt; [a]

Problem 9: Given two lists where all the elements are unique and are in increasing order. Define the function ‘merge’ that returns a list where all the elements are unique and are in increasing order.

Problem 9 Answer:

Problem 10: (Thompson 17.24/30 )

Define the list of numbers whose only prime factors are 2, 3, and 5, the so-called Hamming numbers:

…&gt; hamming

1,2,3,4,5,6,8,9,10,12,15,16,18,20,24 …

You may consider using any combinition of the following techiques to express your solution list comprehension notation, and/or explicit recursion, and/or local definitions .

You may ‘merge’ defined in the last homework useful.

(Hint: Apply the circular list idea demostrated in fibSeq (http://bingweb.binghamton.edu/%7Ehead/CS471/NOTES/HASKELL/4hF02.html))

Problem 10 Answer:

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