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The purpose of this assignment is to begin your introduction to Standard ML.
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Note that in some of the following functions it will be useful to first define auxilliary functions
that do much of the work. In some cases I have explicitly told you when this is so; in other cases you
will need to recognize it for yourself.
In the descriptions of the functions, I have included the types (and proposed names) of the
functions arguments. This was done just to make totally clear what I expected the functions
to do. As we will/have discussed in class, it is not actually necessary for you to include
the types when you implement the functions.
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- Define the function
power (x : real) (y : int) which will raise any real to any
integer power (positive or negative). The function should be recursive. You should assume that
any number raised to the power 0 is 1.
- In this problem you will write a recursive versions of the fibonnaci function, which is defined
by the recurrence:
fib(0) = 0
fib(1) = 1
fib(n+2) = fib(n+1) + fib(n)
Define the function naiveFib (n : int) which computes the fibonacci function directly
according to the definition using a branching recursion. The time this function takes to compute the
value of the fibonacci number of n will grow exponentially as n grows.
- Define the function
prime (n : int) which returns true
if its argument is a prime number and false otherwise.
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- Define the function
mySize (s : string) which returns the number of characters in the
given string. This should not rely on any built-in functions other than explode.
- Define the function
removeVowels (s : string) which returns a copy of the string s
with all its vowels removed. To simplify the function you should first define the function vowel
(c : char) which returns true if c is a vowel, and
false otherwise, and use that function in constructing removeVowels.
- Define the function
sieve (n : integer) which returns a list of all the numbers from 2 up
to n that are primes. You should not call the function prime written earlier, but rather use the
technique known as the Sieve of Eratosthenes. The sieve works by first building a list of all
the numbers from 2 to n. Then it removes all the numbers bigger than 2 that are divisible by 2. Then
it removes all the numbers bigger than 3 that are divisible by 3. Then all the numbers bigger than 5
divisible by 5 (after all we have already removed the numbers divisible by 2 so there won't be
anything left divisible by 4). This process is repeated, removing all the multiples of each prime
bigger than that prime.
- Define the function
diffChars (s : string) which returns the list containing
one copy of each of the characters in the given string s. The order of the returned list
is not important, it may vary depending on the approach you take to the problem. Thus if the
input string were "Hello World" then the output should be some
permutation of the list
[#"H",#"e",#"l",#"o",#" ",#"W",#"r",#"d"]
Note that upper and lowere case versions of the same character may be
considered to be different. (That is, you need not do anything special to
deal with the case of characters.)
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