This is a course web page of David Casperson 

Equations for gcd^{′}
In Scheme, write a function that
counts the cons
es in a
Scheme expression.
Make your function tailrecursive if possible.
Haskell has builtin functions
for gcd
and
lcm
. Nonetheless the goal of this exercise is to
build your own gcd'
function, using the
definition above.
Use Haskell guards to write the function.
In Haskell write a function
isSorted :: Ord a => [a] > Bool
that returns true
if and only if the list is sorted.
In Haskell write a function
insert :: Ord a => a > [a] > [a]
that inserts an element
into an ordered list.
Write a function insertionSort :: Ord a => [a] > [a]
that uses insert
and foldr
.
data JoinList a = Empty  Elt a  Join (JoinList a) (JoinList a)write the following functions:
fromList:: [a] > JoinList a
JoinList
, preferably in a
“balanced” way.
toList:: JoinList a > [a]
JoinList
to a list.
Show
for JoinList
a
so that you can print JoinList
s.
!!!
operator for JoinList
s
that finds the ith element of
a JoinList
.
uncurry
that
takes a function of type a → b → c
and returns a function of type (a,b) → c
.
Recode the toList
function from Question 12 by using
continuations. If your original solution used continuations,
find another tailrecursive approach.
nth
so that
nth(2,[a,b,c,d],c).
returns true.
Make your rules cope with as many
situations as possible. For instance, it would be nice if your
rules find both
N=0
and
N=2
when given the query.
nth(N,[a,b,a,d],a)
. Comment on the combinations of
variables that rules handle.
N.B., this is builtin as predicate nth0
, and there
are answers
on StackOverflow
and elsewhere for how
to do this. Write your own! (first.)
N.B., you may want to check out the
predicates var
and ground
.
split_list(+List,Left,Right)
given by
split_list(List,Left,Right) : split_list(List,List,Left,Right). split_list([WWalker], [_,_Runner], [WLeft], Right) : split_list(Walker, Runner, Left, Right). split_list(Walker, [], [], Walker). split_list(Walker, [X], [], Walker).splits a list into two pieces.
What happens with odd sized lists? Use guitracer.
and trace, split_list([1,2,3,4,5],L,R).
to watch how
this code works. Explain the code.
split_list/4
predicate leaves extraneous choice points. Show how to use
“!
” to remove the extraneous choice
points.
split_list
or code of your own design,
write Prolog code to merge sort a
list of numbers. (The recursive part of merge sort looks
something like:
m_sort(In,Out) : split_list(In, L, R), m_sort(L,L1), m_sort(R,R1), merge(L1,R1,Out).
c(
n)
, where
c(
n)
is defined by
modulo
in the Racket Help desk for
information on helpful arithmetic functions.
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