Due by 11:59pm on Monday, 2/2
Download hw02.zip. Inside the archive, you will find a file called hw02.py, along with a copy of the OK autograder.
Submission: When you are done, submit with
python3 ok --submit
. You may submit more than once before
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Readings: You might find the following references useful:
Several doctests refer to these one-argument functions:
def square(x):
return x * x
def triple(x):
return 3 * x
def identity(x):
return x
def increment(x):
return x + 1
Implement piecewise
, which takes two one-argument functions, f
and g
,
along with a number b
. It returns a new function that takes a number x
and
returns either f(x)
if x
is less than b
, or g(x)
if x
is greater than
or equal to b
.
def piecewise(f, g, b):
"""Returns the piecewise function h where:
h(x) = f(x) if x < b,
g(x) otherwise
>>> def negate(x):
... return -x
>>> abs_value = piecewise(negate, identity, 0)
>>> abs_value(6)
6
>>> abs_value(-1)
1
"""
"*** YOUR CODE HERE ***"
If f
is a numerical function and n
is a positive integer, then we
can form the nth repeated application of f
, which is defined to be
the function whose value at x
is f(f(...(f(x))...))
. For example,
if f
adds 1
to its argument, then the nth repeated application of
f
adds n
. Write a function that takes as inputs a function f
and
a positive integer n
and returns the function that computes the nth
repeated application of f:
def repeated(f, n):
"""Return the function that computes the nth application of f.
>>> add_three = repeated(increment, 3)
>>> add_three(5)
8
>>> repeated(triple, 5)(1) # 3 * 3 * 3 * 3 * 3 * 1
243
>>> repeated(square, 2)(5) # square(square(5))
625
>>> repeated(square, 4)(5) # square(square(square(square(5))))
152587890625
"""
"*** YOUR CODE HERE ***"
Hint: You may find it convenient to use compose1
from the textbook:
def compose1(f, g):
"""Return a function h, such that h(x) = f(g(x))."""
def h(x):
return f(g(x))
return h
The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. The purpose was to show that functions are sufficient to describe all of number theory: if we have functions, we do not need to assume that numbers exist, but instead we can invent them.
Your goal in this problem is to rediscover this representation known as Church
numerals. Here are the definitions of zero
, as well as a function that
returns one more than its argument:
def zero(f):
return lambda x: x
def successor(n):
return lambda f: lambda x: f(n(f)(x))
First, define functions one
and two
such that they have the same behavior
as successor(zero)
and successsor(successor(zero))
respectively, but do
not call successor
in your implementation.
Next, implement a function church_to_int
that converts a church numeral
argument to a regular Python integer.
Finally, implement functions add_church
, mul_church
, and pow_church
that
perform addition, multiplication, and exponentiation on church numerals.
def one(f):
"""Church numeral 1: same as successor(zero)"""
"*** YOUR CODE HERE ***"
def two(f):
"""Church numeral 2: same as successor(successor(zero))"""
"*** YOUR CODE HERE ***"
three = successor(two)
def church_to_int(n):
"""Convert the Church numeral n to a Python integer.
>>> church_to_int(zero)
0
>>> church_to_int(one)
1
>>> church_to_int(two)
2
>>> church_to_int(three)
3
"""
"*** YOUR CODE HERE ***"
def add_church(m, n):
"""Return the Church numeral for m + n, for Church numerals m and n.
>>> church_to_int(add_church(two, three))
5
"""
"*** YOUR CODE HERE ***"
def mul_church(m, n):
"""Return the Church numeral for m * n, for Church numerals m and n.
>>> four = successor(three)
>>> church_to_int(mul_church(two, three))
6
>>> church_to_int(mul_church(three, four))
12
"""
"*** YOUR CODE HERE ***"
def pow_church(m, n):
"""Return the Church numeral m ** n, for Church numerals m and n.
>>> church_to_int(pow_church(two, three))
8
>>> church_to_int(pow_church(three, two))
9
"""
"*** YOUR CODE HERE ***"