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ece473 fn sp2013.pdf
ece473_fn_sp2013.pdf
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ece473 fn sp2013.pdf-EE473 Final Exam Purdue Unive...
ece473_fn_sp2013.pdf-EE473 Final Exam Purdue University Spring
ece473 fn sp2013.pdf-EE473 Final Ex...
ece473_fn_sp2013.pdf-EE473 Final Exam Purdue University Spring
Page 1
EE473 Final Exam
Purdue University
Spring 2013
There are 6 questions in this final exam. You are to answer 3 out of the 6 questions. Indicate below which 3
questions you wish to have graded. If you do not indicate exactly 3 questions, we will not grade your exam.
The 3 questions that you choose to have graded are equally weighted. All of your solutions must be written
on the exam handout and no other handins will be accepted or graded. We have a supply of extra exams,
thus if you want more space to write your solution, raise your hand and ask for another copy of the exam.
We will only grade one solution per problem per student.
If in the process of writing your solution you
explore alternate approaches, you must clearly cross out the portion of your solution that you do not wish
to be graded. All of the copies of this exam that you received should be returned to the proctor at the end
of this exam. No copies of this exam may be removed from the exam room by students. All exams handed
in must have the student’s name, PUID, and ECN userid indicated below or they will not be graded. Good
luck and have fun!.
Circle the
three
questions you wish to have graded: 1 2 3 4 5 6.
Name:
PUID:
ECN userid:
1
Page 2
Problem 1:
Part A:
Write an
inductive
definition of a function
all-subsets
that takes a set of numbers as input, represented
as a list, and returns the set of all subsets of that list of numbers, as output, represented as a list of lists.
Part B:
Write an
inductive
definition of a function
all-permutations
that takes a list of numbers as input, and
returns the set of all permutations of that list of numbers, as output, represented as a list of lists.
You are to formulate the definitions for both parts A and B in the style that we have been using throughout
the semester, that is as a
functional
definition without assignment statements or any form of variable or
data structure mutation. Write your answers in a combination of
English
and mathematical notation. Do
not write your answers in
Scheme
.
2
Page 7
Problem 2:
In digital circuit design, we often wish to find the smallest circuit that computes a given
Boolean function.
Assume you are given a Boolean function, specified by a truth table, that computes a
single Boolean output from
n
Boolean inputs. You wish to find the circuit, composed solely out of two-input
NAND gates, that is built out of the minimal number of such gates, that implements the specified Boolean
function. Describe a nondeterministic algorithm that searches that space of circuits to find the smallest one
that implements a specified target Boolean function. Write your answer in a combination of
English
and
mathematical notation. Do not write your answer in
Scheme
. Note that we do not restrict the circuit to be
in sum-of-products or product-of-sums form thus the methods that you have learned, like Karnaugh Maps,
are not applicable.
Hint: one way to approach this problem is to consider digital circuits to be represented as expressions in
an extremely small
Scheme
-like language containing only Boolean-valued variables, a procedure
nand
, and
the syntax
let
to handle fanout. For example, the following contains fanout of the output of
(nand a b)
:
(let ((x (nand a b))) (nand x x))
.
7
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