Assignment 1: Recursive Functions and Lists solution


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In this assignment, you will write two programs that plot ASCII text approximations of the
Mandelbrot set: one in Haskell, and one in C.
Figure 1: Illustration of the Mandelbrot set.
The portion of your solution in Haskell should be defined within a module named Mandelbrot,
and you should submit the file Mandelbrot.hs (or Mandelbrot.lhs). File names are
case sensitive. The portion of your solution in C should be defined within a file named
mandelbrot.c. Verbal responses to non-programming questions should be in the form of
comments in your code.
Note: The solution to each part of the Haskell problems is meant to be extremely short
(one to four lines). You may (and should) use functions from earlier problems to solve parts
of later problems. Unless otherwise specified, your solutions may utilize functions from the
standard prelude as well as material from the lecture notes and textbook.
Problem 1. (25 pts)
(a) Define a recursive function prefix that accepts a positive integer n and a list xs as
input, and returns a list containing only the first n elements in the input list. This
function must have a recursive definition, and may not use any library functions.
(b) Define a recursive function suffix that accepts a positive integer n and a list xs as
input, and returns the list of elements that remain after the first n elements are dropped
from the front of the list. This function must have a recursive definition, and
may not use any library functions.
(c) Define a function split that takes a positive integer n, an element y, and a list xs.
The function should insert the specified element y after every n elements in the list.
This function should work even when applied to infinite lists. The following
example illustrates how this function should work:
split 2 ’X’ [’a’, ’b’, ’c’, ’d’] = [’a’, ’b’, ’X’, ’c’, ’d’, ’X’, ’e’]
Problem 2. (45 pts)
(a) Define a function plane that takes a single argument r and returns the list of all points
on the cartesian plane of the form (x/r, y/r) where x/r is between −2 and 1 while
y/r is between −1 and 1. The list of points should be ordered from left to right, from
bottom to top. For example, plane 1 should return:
[(-2.0, -1.0), (-1.0, -1.0), ( 0.0, -1.0), ( 1.0, -1.0),
(-2.0, 0.0), (-1.0, 0.0), ( 0.0, 0.0), ( 1.0, 0.0),
(-2.0, 1.0), (-1.0, 1.0), ( 0.0, 1.0), ( 1.0, 1.0)]
Note that the length of the list should grow as r grows. You are allowed to use list
(b) Consider the function 푃(푥,푦) defined as follows:
푃(푥,푦)(푢, 푣) = (푢
2 − 푣
2 + 푥, 2푢푣 + 푦)
We define the orbit 푂(푥, 푦) of a point (푥, 푦) to be an infinite list of items:
푂(푥, 푦) = {(0, 0), 푃(푥,푦)(0, 0), 푃(푥,푦)(푃(푥,푦)(0, 0)), 푃(푥,푦)(푃(푥,푦)(푃(푥,푦)(0, 0))), . . .}
In other words, the 푛th entry of the list 푂(푥, 푦) is the 푃(푥,푦)
function composed with
itself 푛 times and then applied to (0, 0). Define a Haskell function orbit that takes a
single point (x,y) as an argument and returns an infinite list corresponding to 푂(푥, 푦).
You may want to define a helper function corresponding to 푃(푥,푦)
(c) Define a recursive function disp that takes two arguments: a number d and a list of
pairs. Every pair in this input list consists of a number followed by a character, and
you can assume the input list is always in ascending order. For example, a possible
input list might be:
[(0.15, ’#’), (0.5, ’x’), (1, ’.’)]
The function disp should return the character from the list that corresponds to the
smallest number on the list that is greater than the input d, and if d is larger than all
the number in the list, disp should return a space character, ’ ’. For example,
disp 0.01 [(0.15, ’#’), (0.5, ’x’), (1, ’.’)] = ’#’
disp 0.4 [(0.15, ’#’), (0.5, ’x’), (1, ’.’)] = ’x’
disp 100 [(0.15, ’#’), (0.5, ’x’), (1, ’.’)] = ’ ’
The Mandelbrot set is defined to be the set of points (x,y) for which the orbit sequence of
points does not escape to infinity but stays within a fixed distance from the origin.
One way to approximate the Mandelbrot set is to consider a certain element within the orbit
of every point on the plane (such as the 12th element) and to check whether that element
is within a certain fixed distance from the origin. You should use the following function to
calculate distances of points from the origin:1
norm (x,y) = x*x + y*y
These distance values can then be used with disp and orbit to turn points on the plane
into appropriate ASCII characters within an ASCII plot of the Mandelbrot set.
(d) Define a function mandelbrot that takes three arguments: r represents the resolution
of the approximation (to be used with the plane function), i represents the index of
the elements to check in the orbit lists of the points, and l represents the formatting
list (to be used with the disp function). This function should return a list of characters
(which is equivalent in Haskell to a string) that corresponds to a picture approximating
the shape of the Mandelbrot set on the plane. You will need to combine the split,
plane, disp, and orbit functions appropriately; list comprehensions are allowed.
Once you’ve defined the function mandelbrot, you can generate an ASCII version of
an approximation of the Mandelbrot set by evaluating the expression:
putStr (mandelbrot 20 20 [(0.15, ’#’), (0.5, ’x’), (1, ’.’)])
If your solution is correct, the output should unmistakably resemble Figure 1.
Problem 3. (30 pts)
(a) Implement a program in C that takes two positive integers as command line arguments
and displays an ASCII text approximation of the Mandelbrot set using the resolution
and orbit index specified by the integers. For each positive integer inputs r and i, the
output of your C program should match the results of
putStr (mandelbrot r i [(0.15, ’#’), (0.5, ’x’), (100, ’.’)])
(b) Briefly describe three points of contrast or comparison between the two solutions (in
Haskell and in C). These could be about the kinds of issues you had to think about
while writing the solutions or how language features and characteristics affected your
solutions (e.g. in terms of quality, flexibility, or efficiency).
1Note for those who are curious: this function is preferable over the Euclidean metric because the built-in
sqrt function does not behave well on extremely small double precision values. Ideally, this assignment
would use the Rational type to represent numbers, but this is a more advanced topic.