CPSC 121: Models of Computation Assignment ... - Ugrad.cs.ubc.ca

CPSC 121: Models of Computation
Assignment #2, due Thursday February 2 nd , 2011 at 17:00
[14] 1. Most programming languages provide ways to manipulate the bits of signed integers
represented using two’s complement. For instance, if x and y are two such integers,
then
• x & y denotes the integer whose i th bit is the i th bit of x and the i th bit of y.
For instance, assuming that we have eight bit integers, 109 10 = 01101101 2 and
51 10 = 00110011 2 , and so 109 & 51 = 00100001 2 = 33 10 .
• x | y denotes the integer whose i th bit is the i th bit of x or the i th bit of y. For
instance, 109 | 51 = 01111111 2 = 127 10 .
• x ˆ y denotes the integer whose i th bit is the i th bit of x xor the i th bit of y.
For instance, 109 ˆ 51 = 01011110 2 = 94 10 .
• x ≫ i denotes the integer obtained by “deleting” the rightmost i bits of x,
and adding i 0 bits to the left of the remaining bits. For instance, 109 ≫ 3 =
00001101 (the bits 101 were removed).
We can perform several operations simply by manipulating the sequence of bits that
represents an integer using these operators. In this problem, we look at some of
them.
[2] a. There are several situations where we would like to only retain some of the
bits of an integer x. For instance, the last 3 bits of 109 equal 101 2 = 5 10 .
Given x, what expression would give you an integer that corresponds to the
last 3 bits of x
[2] b. Continuing from the previous question, what expression would give an integer
that corresponds to the next three bits of x (those corresponding to the
positions 2 5 , 2 4 and 2 3 ) For instance, for 51, this would be 110 2 = 6 10 .
[4] c. The expression x & −x gives the largest power of 2 that divides x.
i. Verify that this is the case for the two 8-bit integers 109 and 51.
ii. Explain why this is always true for every integer x. Hints: think of how
you compute −x using two’s complement; this expression also helps you
find the rightmost bit that is a 1.
[6] d. Finding the leftmost 1 bit in a binary integer seems more complicated than
finding the rightmost 1 bit, and there is no simple expression like the one from
part (c). However, given two unsigned integers x and y, we can determine
whether or not their leftmost 1 bits are in the same position using the following
fact:
The leftmost 1 bits of x and y are in the same position if and only if
x ˆ y ≤ x & y.

Explain why this statement holds. Note that, since it is a biconditional, you
will need two explanations: one that explains why if the leftmost 1 bits of x
and y are in the same position, then xˆy ≤ x&y, and one that explains why if
the leftmost 1 bits of x and y are not in the same position, then x ˆ y > x & y.
[10] 2. Computers represent characters by associating with each character a specific sequence
of 0’s and 1’s. In this question, you will be dealing with the ASCII and
Unicode encodings discussed in class. You can find a table listing the decimal (labeled
Dec) and hexadecimal (labeled Hx) codes for each ASCII character (labeled
Chr) at
http://www.lookuptables.com
by clicking on the “ASCII table” button near the top of the page. This site also
contains tables for Unicode accessible by clicking on the “Unicode v4” button.
[2] a. Unix and Linux systems terminate each line of a text file with a single linefeed
(LF) character. MacOS terminates each line with a single carriage return
(CR) character. Windows terminates each line with two characters: a carriage
return character (CR) followed by a linefeed character (LF). What are the
ASCII codes for these two characters
[2] b. In one sentence, explain how given an integer between 0 and 9, you would
obtain the ASCII code for the character representing that digit. For instance,
the integer 7 would be converted to the character ’7’.
[2] c. In one sentence, explain how given its ASCII code, you would convert an
uppercase letter into the corresponding lowercase letter.
[2] d. Is the operation from part (c) simpler to describe in decimal or in binary
Why
[2] e. What is the hexadecimal Unicode character for the picture of an airplane
(hint: see “Dingbats”)
[8] 3. Following on the previous question, design a circuit that has 8 inputs b 7 , b 6 , . . . b 0 ,
and two outputs d and l:
• d will be true if the character whose ASCII code is b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 represents
a digit.
• l will be true if the character whose ASCII code is b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 represents
an uppercase or lowercase letter.
Hint: do not write a truth table with 256 lines! Instead, try to find a pattern for the
sequences of bits for which one or the other should be true. A brute-force working
solution will be worth 6/8. To get full marks, your solution needs to be both correct
and elegant (it doesn’t need to use the smallest possible number of gates, but it
shouldn’t be too ugly).

[16] 4. Determine the validity of the following arguments using rules of inference. You may
use inference rules from class or one of the texts (Epp or Rosen), or you may use a
logical equivalence from class or the texts as an inference rule. You must state what
inference rule or logical equivalence you use at each step.
[8] a. 1. (∼p ∨ q) → r
2. r → (s ∨ t)
3. ∼s ∧ ∼u
4. ∼u → ∼t
∴ p
[8] b. Given the following premises:
(1) If the Toronto Maple Leafs do not win in overtime, then Donald Cerise
will not be happy.
(2) If the clouds do not cross Vancouver Island or an arctic wind is blowing,
then it will be cold.
(3) If it is cold, then the Toronto Maple Leafs will win in overtime, or the Los
Angeles Kings will lose in overtime.
(4) The clouds will not cross Vancouver island, or it will snow Saturday.
(5) If Donald Cerise is unhappy, then the Los Angeles Kings will not lose in
overtime.
(6) The Toronto Maple Leafs did not win in overtime.
we can conclude that it will snow Saturday.