Bitwise

Problems and Solutions
for
Bit and String Manipulations
by
Willi-Hans Steeb
International School for Scientiﬁc Computing
at
University of Johannesburg, South Africa
Yorick Hardy
Department of Mathematical Sciences
at
University of South Africa, South Africa

Preface
The purpose of this book is to supply a collection of problems in bitwise
operations and string manipulations.
The material was tested in our lectures given around the world.
Any useful suggestions and comments are welcome.
The International School for Scientiﬁc Computing (ISSC) provides certiﬁ-
cate courses for this subject. Please contact the authors if you want to do
this course.
e-mail addresses of the authors:
steebwilli@gmail.com
Home pages of the authors:
http://issc.uj.ac.za
v

Contents
1 Basic Bitwise Operations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quickies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Explain the Output of the C++ Program . . . . . . . . . . 12
1.3.1 Bitwise Operations . . . . . . . . . . . . . . . . . . . 12
1.3.2 Shift Operations . . . . . . . . . . . . . . . . . . . . 16
1.4 bitset class . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Advanced Bitwise Manipulations 31
2.1 Write a C++ Program . . . . . . . . . . . . . . . . . . . . . 31
2.2 Gray Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Binary and Arithmetic Operations . . . . . . . . . . . . . . 38
2.4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Binary Matrices 57
4 Reversible Logic Gates 64
5 Floating Point Numbers 70
6 Cellular Automata 75
7 String Manipulations 77
Bibliography 81
Index 82
vi

Chapter 1
Basic Bitwise Operations
1.1 Introduction
Bit is short for binary digit with either of the two digits 0 and 1 in the
binary number system. The bit is the smallest unit of storage in a binary
system. Binary refers to base 2 arithmetic using the digits 0 and 1. Thus
a bit is a binary digit (i.e. a digit in the binary number system). It is the
most basic unit of information in digital computing.
A boolean function is a function f with domain {0, 1}n
and range {0, 1}, for
some positive integer n. Here {0, 1}n
denotes the n-fold Cartesian product
of the set {0, 1} with itself, that is, the set of binary n-tuples. Thus we also
write
f : {0, 1}n
→ {0, 1}.
There are 22n
boolean function of n variables.
We also have a set of m boolean functions f1, f2, . . . , fm, where fj :
{0, 1}n
→ {0, 1} and j = 1, 2, . . . , m. We also write
f : {0, 1}n
→ {0, 1}m
.
There are 2m2n
functions f : {0, 1}n
→ {0, 1}m
.
A truth table is a tabular description of a combinational circuit (such as
an AND-gate, OR-gate, NAND-gate) listing all possible states of the input
1

2 Problems and Solutions
variables together with a statement of the output variable(s) for each of
those possible states. The truth table for the AND-gate, OR-gate, XOR-
gate and NOT-gate are
AND
0 0 0
0 1 0
1 0 0
1 1 1
OR
0 0 0
0 1 1
1 0 1
1 1 1
XOR
0 0 0
0 1 1
1 0 1
1 1 0
NOT
0 1
1 0
The NAND-gate is an AND-gate followed by a NOT-gate. The NOR-gate
is an OR-gate followed by a NOT-gate. Both are universal gates, i.e. all
other gates can be built from these gates.
Notation. In the text we use for the AND-operation ·, we use for the
OR-operation +, we use for the XOR-operation ⊕ and we use for the NOT-
operation x.
Programming Languages. In C++, C, Java, C# and Perl the bitwise
AND is the ampersand operator &. The bitwise OR is |. The bitwise XOR
is ^. The bitwise NOT (one complement) is ~. In C and C++ the order of
precedence of the bitwise operators is &, ^, |.
The shift operations allow bits to be moved to the left or right in a word.
There are three types of shift operations: logical, rotate and arithmetic.
The logical shift moves bits to the left or right. The bits which fall off
the end of the word are discarded and the word is filled with 0’s from the
opposite end. For example a logical right shift of the 8 bit binary number
1000 1011 provides 0100 0101. Shift instructions include a repeat value,
which is the number of times the single bit shift operation is repeated. A
rotate operation (not implemented in C++, C, Java, C# and Perl) is a
circular shift in which no bits are discarded. An arithmetic right shift is
similar to a logical right shift except that the leftmost bits are filled with
the sign bit of the original number instead of 0’s. For example, an arith-
metic right shift of the 8 bit number 1000 1011 provides 1100 0101. >>
is the right shift operation in C++, C, Java, C# and Perl. << is the left
shift operation in these languages. In C++ and C they can be applied to
integral operands, that is char, short, int, and long, whether signed or
unsigned. The size of char is 8 bits, the size of short is 16 bits, and the
size of int and long is 32 bits. The sign bit is at the left most position, for
example for int it is at position 31 (counting from 0).
Hex notation is indicated in C, C++, Java and Perl as 0x. For example
0xFF is the number 255 (base 10).

Basic Bitwise Operations 3
The AND-operation performs a bit-by-bit logical AND of two bitstrings of
the same length.
The OR-operation performs a bit-by-bit logical OR of two bitstrings of the
same length.
The XOR-operation performs a bit-by-bit logical exclusive OR of two bit-
strings of the same length.
The MOD-operation returns the remainder (modulus) from dividing two
integer numbers.
The NOT-operation performs a bit-by-bit complement of a bitstring.
SHL shifts the value of the bitstring to the left count bits. A negative count
causes the data to be shifted the opposite way.
SHR shifts the value of the bitstring to the right count bits. A negative
count causes the data to be shifted the opposite way.

1.2 Quickies
Problem 1. Let x ∈ {0, 1}.
(i) Find x · x, x ⊕ x, x + x.
(ii) Find x · ¯x, x ⊕ ¯x, x + ¯x.
(iii) Find x · 0, x · 1, x ⊕ 0, x ⊕ 1, x + 0, x + 1.
Problem 2. Let x, y ∈ {0, 1}.
(i) Solve the boolean equation
x · y = x + y.
(ii) Solve the boolean equation
x ⊕ y = x · y.
(iii) Solve the boolean equation
x ⊕ y = x + y.
Problem 3. Let x1, x2, x3, x4 ∈ {0, 1}.
(i) Is
(x1 + x2) · (x3 + x4) = x1 · x3 + x1 · x4 + x2 · x3 + x2 · x4 ?
(ii) Is
(x1 + x2) ⊕ (x3 + x4) = x1 ⊕ x3 + x1 ⊕ x4 + x2 ⊕ x3 + x2 ⊕ x4 ?
Problem 4. Let x1, x2, x3 ∈ { 0, 1 }. Let ⊕ be the XOR-operation and ·
be the AND-operation. Is
(x1 · x2) ⊕ x3 = x1 · (x2 ⊕ x3) ?
Problem 5. Let a, b, c, d ∈ { 0, 1 }. Consider the boolean function f :
{0, 1}4
→ {0, 1}
f(a, b, c, d) = a + b ⊕ c · d.
In what order of precedence are the operations AND ·, XOR ⊕, and OR +
applied? Find f(1, 1, 1, 1).
Problem 6. (i) Can the boolean function f : {0, 1}2
→ {0, 1}
f(x, y) = x + (x · y)

be simpliﬁed? Here + denotes the XOR-operation and · the AND-operation.
(ii) Can be boolean function f : {0, 1}2
→ {0, 1}
f(x, y) = x · (x + y)
be simpliﬁed? Here + denotes the XOR-operation and · the AND-operation.
Problem 7. Let x, y, z ∈ {0, 1}. Is
(x · y) + (y · z) + (z · x) = (x · y) ⊕ (y · z) ⊕ (z · x)
where · denotes the AND-operation, + the OR-operation and ⊕ the XOR-
operation?
Problem 8. Let a, b, c ∈ { 0, 1 }. Is
(a ⊕ b) · c = a ⊕ (b · c) ?
Problem 9. Let a, b, x, y ∈ {0, 1}. Solve the bitwise equation
a ⊕ b = x · y.
Problem 10. (i) Let x, y, z be bitstrings of the same length. Is the
XOR-operation associative, i.e.
(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) ?
(ii) Let x and y be arbitrary bitstrings of the same length. Let ⊕ be the
XOR operation. Calculate
(x ⊕ y) ⊕ y.
Problem 11. Let x, y, z ∈ {0, 1}. Is
(x ⊕ y) · z = (x · z) ⊕ (y · z) ?
Problem 12. Let be the exclusive-NOR operation, i.e. x y = 1 if
and only if x = y for the boolean variables x and y. Find the solutions of
x + y = x y, x · y = x y, x ⊕ y = x y.
Problem 13. Show that the bitwise expression
b ⊕ (a · c)

contains the AND-gate, XOR-gate, NOT-gate and FANOUT. First write
down the truth table.
Problem 14. Given the truth table
row x2 x1 x0 z
0 0 0 0 0
1 0 0 1 0
2 0 1 0 1
3 0 1 1 0
4 1 0 0 0
5 1 0 1 1
6 1 1 0 0
7 1 1 1 1
Find the boolean expression (sum of products) for this truth table. Can the
expression be simplified?
Problem 15. Given k-bit inputs and m-bit outputs. How many boolean
function are there?
Problem 16. The bitwise NOT-operation is defined by 0 -> 1 and
1 -> 0. Thus in a bitstring the NOT-operation replaces 0’s by 1’s and
1’s by 0’s. How can the bitwise NOT-operation be implemented using the
XOR-operation?
Problem 17. Can the expression
E = x · z + x · y + x · y + y · z
be simplified?
Problem 18. The truth table of a typical encoder with inputs a, b, c, d
and outputs c0, c1 is
a b c d c0 c1
1 0 0 0 0 0
0 1 0 0 0 1
0 0 1 0 1 0
0 0 0 1 1 1
Find the Boolean expression for this truth table.
Problem 19. The truth table of a full adder with inputs a, b, cin and
outputs y, cout is

a b cin y cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Find the Boolean expression for this truth table.
Problem 20. A 1-bit full adder is a circuit with three 1-bit inputs (the
bits to be added) and two 1-bit outputs (the sum and the carry). It is given
by
sum = (a ⊕ b) ⊕ c
carry = a · b + a · c + b · c = a · b + (a + b) · c = a · b + (a ⊕ b) · c.
Find the thruth table.
Problem 21. We deﬁne the binary dot product of two bitstrings x and y
of the same length n as
x · y := (x1y1 + x2y2 + · · · + xnyn) mod 2.
Let x, y, z be bitstrings of the same length. Verify that
(x ⊕ y) · z ≡ (x · z) ⊕ (y · z).
Problem 22. Let a, b, c ∈ {0, 1}. Find all solutions of
a ⊕ b ⊕ c = a · b · c.
Problem 23. Prove the following statements by (i) using a truth table,
(ii) using the properties of boolean algebra (algebraic proof).
(a) 1 + a ≡ 1.
(b) a ⊕ b ≡ a ⊕ b
(c) (a · b) ⊕ (a · b) ≡ a · (b ⊕ b).
Problem 24. Let x ∈ { 0, 1 }2n
and
Y (x) := { y ∈ {0, 1}2n
: h(x, y) = n }

C(x, y) := { (x, y) : x ∈ { 0, 1}2n
, y ∈ Y (x) }
where h(x, y) denotes the Hamming distance between x and y.
(a) Determine |Y (x)|, where | | denotes the cardinality, i.e. the number of
elements.
(b) Determine |C(x, y)|, where | | denotes the cardinality, i.e. the number
of elements.
(c) Let (x, y). How many unique pairs are (x, y) in C(x, y)?
(d) Which group properties hold for Y (x0) where x0 = (0, 0, . . . , 0)?
Problem 25. The NAND-gate is a universal gate. The XOR-gate can
be represented with four NAND-gates. Use multi expression programming
to implement the XOR-gate using NAND-gates.
Problem 26. Show that the XOR-gate can be built from 4 NAND-gates.
Show that the AND-gate can be built from 2 NAND-gates. Show that the
OR-gate can be built from 3 NAND-gates. Show that the NOR-gate can
be built from 4 NAND-gates.
Problem 27. Consider the one-dimensional map f : [0, 1] → [0, 1]
f(x) = 4x(1 − x).
A computational analysis using a finite state machine with base 2 arithmetic
in fixed point operation provides one-dimensional maps with a lattice of 2N
sites labeled by numbers
x =
N
j=1
j
2j
, j ∈ { 0, 1 }
and N defines the machine’s precision. Consider N = 6 bits and x =
1/8. Calculate the orbit f(x), f(f(x)), f(f(f(x))), . . . with this precision.
Discuss.
Problem 28. Find the truth table for the boolean function
f(a, a , b, b ) = (a · b ) ⊕ (a · b).
Problem 29. Consider the boolean function
f(x1, x2, x3) = x1 ⊕ x2 ⊕ x3.
Find the disjunctive normal form.

Problem 30. Show that every boolean function f : {0, 1}n
→ {0, 1} can
be expanded as follows
f(x1, x2, . . . , xn) = x1 · f(1, x2, . . . , xn) + x1 · f(0, x2, . . . , xn).
Problem 31. Apply the expansion theorem given at the previous exercise
repeatedly to each variable of
f(x1, x2, x3) = x1 · x2 + x2 · x3
to obtain its disjunctive normal form.
Problem 32. Consider the first-order discrete time dynamical system
xk+1 = 2xk mod 1 k = 0, 1, 2, . . .
and
sk =
1 if xk ≥ 0.5
0 if xk < 0.5
where x0 ∈ [0, 1]. We call s = s0s1s2 . . . the output symbol. Show that if
x0 ∈ [0.78125, 0.8125] then the output coincide for the first three bits.
Problem 33. Let n be the number of discrete symbols s1, s2, . . . , sn
that can be used. Let m be the length of the message string. Find the
number M of messages. Then consider the special case n = m = 2.
Problem 34. Consider a bitstring of length m which has exactly m1 ones
and m2 zeros (m1 + m2 = m).
(i) Find the number of different possible bitstrings.
(ii) Consider the special case m = 4, m1 = m2 = 2.
Problem 35. The D-type latch works as follows. When the control signal
latch-enable (LEN) is high, the latch is in the transparent mode and the
input signal ¯D is available at the output. When the LEN signal is low, the
input data is latched to the output and is retained until LEN goes back to
high.
(i) Give the truth table from this description.
(ii) Give the boolean equation for this latch.
(iii) Give the circuit.
Problem 36. Consider two unsigned int (32 bits) n and n + 1. What
is the condition that the parity of n and n + 1 are the same? For example
1 and 2 have the same parity but 2 and 3 not.

Problem 37. Given the boolean function
f(a, b, c, d) = a · d + (a · c + b) · (c · d + e).
Find the truth table. Discuss. Write a C++ program that generates the
truth table.
Problem 38. Write the boolean function
f(x, y, z) = (x · ¯y) + x · z) + ¯x
in disjunctive normal form.
Problem 39. Consider the boolean function f : {0, 1}3
→ {0, 1}
f(a, b, c) = a · b · ¯c + ¯a · c + ¯a · ¯b.
Find the truth table.
Problem 40. Simplify the boolean expression
f(x, y, z) = x · (¯x + y) + y · (y + z) + y.
Problem 41. Given the boolean function
f(x1, x2, x3, x4) = (x1 + x2) · (x3 + x4).
Give the truth table.
Problem 42. Consider the truth table
I_1 I_2 I_3 O
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Find a boolean expression.
Problem 43. Let a, b ∈ {0, 1}. Find all solutions of the equation a ⊕ b =
a · b, where ⊕ denotes the XOR-operation and · the AND-operation.

Problem 44. Let a, b, c, d ∈ {0, 1}. Find all solutions of the equation
a·b = c⊕d, where ⊕ denotes the XOR-operation and · the AND-operation.
Problem 45. Let i1, i2, i3 be the input bits and o1, o2, o3 be the output
bits. We say that the input bits and the output bits pass the GHZ test if
they satisfy the system of boolean equation
i1 ⊕ i2 ⊕ i3 = 0
o1 ⊕ o2 ⊕ o3 ⊕ (i1 ∨ i2 ∨ i3) = 1.
Find all solutions of this system of boolean equation. Here ⊕ denotes the
XOR-operation and ∨ denotes the OR-operation.
Problem 46. After the ASCII table the capital A is identiﬁed with the
integer 65 (base 10) and the small a is identiﬁed with the integer 97 (base
10). Write down these two numbers in binary (8 bits) and apply the XOR-
operation. Discuss.
Problem 47. Prove (i) using a truth table, (ii) using the properties of
boolean algebra (algebraic proof)
(a) 1 + x = 1
(b) ¯x ⊕ ¯y = x ⊕ y
(c) (x · ¯y) ⊕ x = x · y
(d) that ⊕ is associative.
Problem 48. Let I, J, K, M be unsigend int. Write a C++ program
that implements
((I XOR J) AND (K XOR M)) OR ((I XOR K) AND (J XOR M)).
Discuss.
Problem 49. Let a, b, c, x, y, z ∈ {0, 1}. Solve the boolean equation
a · (b · c) = x + (y + z).

1.3 Explain the Output of the C++ Program
1.3.1 Bitwise Operations
Problem 50. In the following C++ program that bitwise AND & is
applied. What is the output?
// bitand.cpp
#include <iostream>
using namespace std;
int main(void)
{
int x = 17;
int r = x & (-x);
cout << "r = " << r << endl;
x = 101;
r = x & (-x);
cout << "r = " << r << endl;
x = -5;
r = x & (-x);
cout << "r = " << r << endl;
return 0;
}
Problem 51. What is the output of the following C++ program?
// XORANDOR.cpp
#include <iostream>
int main(void)
{
unsigned int i = 10; unsigned int j = 11;
unsigned int k = 12; unsigned int l = 13;
unsigned r = ((i^j) & (k*l)) | ((i^k) & (j^l));
cout << "r = " << r << endl;
return 0;
}
Problem 52. In the following C++ program the bitwise XOR ^ is used.
What is the output?
// branch.cpp
#include <iostream>

int main(void)
{
unsigned int a = 10; unsigned int b = 9;
unsigned int x = 10;
x = a^b^x;
cout << "x = " << x << endl;
a = 7; b = 9; x = 9;
x = a^b^x;
cout << "x = " << x << endl;
return 0;
}
Problem 53. In the following C++ program we apply the NOT operation
~ and the OR operation |. What is the output?
// deMorgan.cpp
#include <iostream>
int main(void)
{
unsigned int x = 8345; unsigned int y = 34512;
unsigned int r1 = x & y;
unsigned int r2 = ~(~x | ~y);
if(r1==r2) cout << "true"; else cout << "false";
return 0;
}
Problem 54. What is output of the following C++ program using the
AND and NOT operations
// clearbit.cpp
#include <iostream>
int main(void)
{
unsigned int i = 476; unsigned int j = 677;
unsigned int r = i & ~j;
cout << "r = " << r << endl;
return 0;
}
// output.cpp

#include <iostream>
int main(void)
{
unsigned int i=1; unsigned int j=2;
unsigned int k=3;
unsigned int f1 = (((i) & (j)) | ((~i) & (k)));
unsigned int f2 = (((i) & (k)) | ((j) & (~k)));
unsigned int f3 = ((i) ^ (j) ^ (k));
unsigned int f4 = ((j) ^ ((i) | (~k)));
cout << "f1 = " << f1 << endl;
cout << "f2 = " << f2 << endl;
cout << "f3 = " << f3 << endl;
cout << "f4 = " << f4 << endl;
return 0;
}
Problem 56. The following C++ program uses the AND operation.
What is the output?
// maxmin.cpp
#include <iostream>
int main(void)
{
int x = 22; int y = 18;
int t = ((x-y) & -(x < y));
int r1 = y + t;
cout << "r1 = " << r1 << endl;
int r2 = x-t;
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 57. The following C++ uses the AND operation. Note that !
is the logical NOT. What is the output?
// parity1.cpp
#include <iostream>
int main(void)
{
unsigned int v = 19;
bool p = false;

while(v != 0) // short-cut while(v)
{ p = !p; v = v & (v-1); }
cout << "p = " << p << endl;
return 0;
}
Problem 58. The following C++ code uses the AND-operation. Here !
is the logical NOT. What is the output?
// power.cpp
#include <iostream>
int main(void)
{
unsigned int i1 = 2345678;
bool b1 = !(i1 & (i1-1)) && (i1 > 0);
cout << "b1 = " << b1 << endl;
bool b2 = !(i2 & (i2-1)) && (i2 > 0);
cout << "b2 = " << b2 << endl;
bool b3 = !(i3 & (i3-1)) && (i3 > 0);
cout << "b3 = " << b3 << endl;
return 0;
}
Problem 59. The following C++ code use the NOT and AND-operation.
What is the output?
// bitstozero.cpp
#include <iostream>
int main(void)
{
unsigned int j = 4236571;
unsigned int r = j & ~0xFF;
cout << "r = " << r << endl;
return 0;
}
Problem 60. In the following C++ program we use the XOR and NOT-
operation. What is the output?
// XORNOT.cpp

#include <iostream>
int main(void)
{
unsigned int j;
unsigned int r = j ^ (~j);
cout << "r = " << r << endl;
return 0;
}
Problem 61. The following program uses the OR operation. What is
the output?
// turnonbit.cpp
#include <iostream>
int main(void)
{
unsigned int r = v | (v + 1);
cout << "r = " << r << endl;
return 0;
}
1.3.2 Shift Operations
Problem 62. The following C++ program uses the shift operation and
the XOR operation. What is the output?
// parity.cpp
#include <iostream>
int main(void)
{
unsigned int x = 7;
unsigned int y;
y = x^(x >> 1);
y = y^(y >> 2);
y = y^(y >> 4);
y = y^(y >> 8);
y = y^(y >> 16);
cout << "y = " << y << endl;
unsigned int r = y & 1;
cout << "r = " << r << endl;

return 0;
}
// pparity.cpp
#include <iostream>
unsigned int p(unsigned int v)
{
v ^= v >> 16;
v ^= v >> 8;
v ^= v >> 4;
v &= 0xf;
return (0x6996 >> v) & 1;
}
int main(void)
{
unsigned int v1 = 8;
unsigned int r1 = p(v1);
cout << "r1 = " << r1 << endl;
cout << "r2 = " << r2 << endl;
cout << "r3 = " << r3 << endl;
return 0;
}
Problem 64. The following C++ program uses the shift-operation.
What is the output?
// leadingzeros.cpp
#include <iostream>
unsigned int leading(unsigned int x)
{
if(x==0) return 32;
unsigned int n = 0;
if(x <= 0x0000FFFF) { n += 16; x = x << 16; }
if(x <= 0x00FFFFFF) { n += 8; x = x << 8; }
if(x <= 0x0FFFFFFF) { n += 4; x = x << 4; }
if(x <= 0x3FFFFFFF) { n += 2; x = x << 2; }

if(x <= 0x7FFFFFFF) { n += 1; }
return n;
}
int main(void)
{
unsigned int x = 4;
unsigned int r = leading(x);
cout << "r = " << r << endl;
x = 100;
r = leading(x);
cout << "r = " << r << endl;
x = 255;
r = leading(x);
cout << "r = " << r << endl;
return 0;
}
Problem 65. The following C++ program utilizes the AND, OR and
shift operation. What is the output?
// reversing.cpp
#include <iostream>
unsigned int reversing(unsigned int x)
{
x = (x & 0x55555555) << 1 | (x & 0xAAAAAAAA) >> 1;
x = (x & 0x33333333) << 2 | (x & 0xCCCCCCCC) >> 2;
x = (x & 0x0F0F0F0F) << 4 | (x & 0xF0F0F0F0) >> 4;
x = (x & 0x00FF00FF) << 8 | (x & 0xFF00FF00) >> 8;
x = (x & 0x0000FFFF) << 16 | (x & 0xFFFF0000) >> 16;
return x;
}
int main(void)
{
unsigned int x1 = 0;
unsigned int r1 = reversing(x1);
cout << "r1 = " << r1 << endl;
unsigned int r2 = reversing(x2);
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 66. The following C++ program uses the shift and OR opera-

tion. What is the output of the following C++ program?
// isqrt.cpp
#include <iostream>
unsigned int isqrt(unsigned int x)
{
unsigned int m, y, b;
m = 0x40000000;
y = 0;
while(m != 0)
{
b = y | m;
y = y >> 1;
if(x >= b) { x = x-b; y = y | m; }
m = m >> 2;
}
return y;
}
int main(void)
{
unsigned int r1 = isqrt(x1);
cout << "r1 = " << r1 << endl;
unsigned int r2 = isqrt(x2);
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 67. The following program uses the shift operation. What is
the output?
// cuberoot.cpp
#include <iostream>
unsigned int cr(unsigned int x)
{
int s = 30;
unsigned int y, b;
y = 0;
while(s >= 0)
{
y = y << 1;

b = (3*y*(y+1) + 1) << s;
s -= 3;
if(x >= b) { x -= b; y += 1; } // end if
} // end while
return y;
}
int main(void)
{
unsigned int r1 = cr(x);
cout << "r1 = " << r1 << endl; // =>
x = 200;
unsigned int r2 = cr(x);
cout << "r2 = " << r2 << endl; // =>
return 0;
}
// shiftOR.cpp
#include <iostream>
unsigned int shiftOR(unsigned int v)
{
v = v | (v >> 1);
v = v | (v >> 2);
v = v | (v >> 4);
v = v | (v >> 8);
v = v | (v >> 16);
return v - (v >> 1);
}
int main(void)
{
unsigned int j = 66;
unsigned int result = shiftOR(j);
cout << "result = " << result << endl;
return 0;
}
Problem 69. The function int f(int) in the following C++ program
uses the XOR operation and shift operation. What is the output?
// absolute.cpp
#include <iostream>

int f(int i)
{
int t = sizeof(int);
int r;
r = (i^(i >> 31))-(i >> 31);
return r;
}
int main(void)
{
int n1 = -87;
int r1 = f(n1);
cout << "r1 = " << r1 << endl;
int n2 = 99;
int r2 = f(n2);
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 70. The following C++ program uses the AND operation and
shift operation. What is the output?
// countingbits.cpp
#include <iostream>
const unsigned char Table[] =
{ 0,1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4,
1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5,
1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7,
1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7,
2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,
3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7,
3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7,
4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8 };
int main(void)

{
unsigned int c1 = Table[v1 & 0xff] + Table[(v1 >> 8) & 0xff]
+ Table[(v1 >> 16) & 0xff] + Table[v1 >> 24];
cout << "c1 = " << c1 << endl;
unsigned int c2 = Table[v2 & 0xff] + Table[(v2 >> 8) & 0xff]
+ Table[(v2 >> 16) & 0xff] + Table[v2 >> 24];
cout << "c2 = " << c2 << endl;
return 0;
}
Problem 71. What is the output of the following C++ program
// output1.cpp
#include <iostream>
unsigned int add(unsigned int x,unsigned int y)
{
unsigned int low = (x & 0xffff) + (y & 0xfff);
unsigned int high = (x >> 16) + (y >> 16) + (low >> 16);
return (high << 16) | (low & 0xffff);
}
int main(void)
{
unsigned int x = 17; unsigned int y = 23;
unsigned int result = add(x,y);
return 0;
}
Problem 72. The following C++ program uses the XOR, AND and shift
operation. What is the output?
// parity2.cpp
#include <iostream>
unsigned int parity(unsigned char v)
{
v ^= v >> 4;
v &= 0xf; // short cut for v = v & 0xf
return ((0x6996 >> int(v)) & 1);
}

int main(void)
{
unsigned char v1 = 19;
unsigned int r1 = parity(v1);
cout << "r1 = " << r1 << endl;
unsigned char v2 = 18;
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 73. The following C++ program uses the shift and AND op-
eration. What is the output?
// count.cpp
#include <iostream>
int main(void)
{
unsigned int i = 234;
unsigned int c;
for(c=0;i>0;i>>=1) { c += i & 1; }
cout << "c = " << c << endl;
return 0;
}
Problem 74. The following C++ operation applies the shift operation.
What is the output?
// signbit.cpp
#include <iostream>
int main(void)
{
int i1 = 345271;
int r1 = (i1 >> 31);
cout << "r1 = " << r1 << endl;
int i2 = -471273;
int r2 = (i2 >> 31);
cout << "r2 = " << r2 << endl;
return 0;
}
Problem 75. The following C++ program uses the shift operation. What
is the output?

// multiply.cpp
#include <iostream>
int main(void)
{
unsigned int r1 = (i1 << 3) - i1;
cout << "r1 = " << r1 << endl;
unsigned int r2 = (i2 << 4) + i2;
cout << "r2 = " << r2 << endl;
unsigned int r3 = (i3 << 6) + i3;
cout << "r3 = " << r3 << endl;
unsigned int r4 = (i4 << 3) + (i4 << 2) + i4;
cout << "r4 = " << r4 << endl;
return 0;
}
Problem 76. The following C++ program uses the shift operation and
AND-operation. What is the output?
// parity3.cpp
#include <iostream>
unsigned int parity(unsigned int v)
{
v ^= v >> 16;
v ^= v >> 8;
v ^= v >> 4;
v &= 0xf;
return ((0x6996 >> v) & 1);
}
int main(void)
{
cout << "r1 = " << r1 << endl;
cout << "r2 = " << r2 << endl;
return 0;
}

Problem 77. What is the output of the following C++ program
// Hammingunsigned1.cpp
#include <iostream>
int main(void)
{
unsigned int x = 4;
unsigned int y = 4294967295;
unsigned int r = x^y;
int j = 31;
int count = 0;
while(j >= 0)
{
if((r & 1)==1) count++;
r = r >> 1;
j--;
}
cout << "count = " << count;
return 0;
}
Problem 78. The following C++ program uses the XOR, AND, OR and
shift operations. What is the output?
// bitsswapping.cpp
#include <iostream>
unsigned int swap(unsigned int v,unsigned int i,unsigned int j,
unsigned int n)
{
unsigned int temp = ((v >> i)^(v >> j)) & ((1 << n)-1);
return (v^((temp << i)|(temp << j)));
}
int main(void)
{
unsigned int i = 2, j = 5;
unsigned int n = 2;
unsigned int result = swap(v,i,j,n);
return 0;
}

// overflow.cpp
#include <iostream>
int main()
{
unsigned int j = 1;
unsigned int r = i+j;
cout << "r = " << r << endl;
return 0;
}
Use the binary representation of the number 4294967295 and the number
1.
// divide.cpp
#include <iostream>
int main(void)
{
unsigned int i0 = i & 0xFFFF;
unsigned int i1 = i >> 0x10;
unsigned int r0 = i0*0xAAAB;
unsigned int t = i1*0xAAAB + (r0 >> 0x10);
unsigned int r1 = t & 0xFFFF;
unsigned int r2 = t >> 0x10;
r1 += i0*0xAAAA;
unsigned p = i1*0xAAAA + r2 + (r1 >> 0x10);
unsigned result = p >> 1;
cout << "result = " << result;
return 0;
}
Problem 81. In C and C++ the precedence of the bitwise operators is
as follows:
0 ~ one’s complement
1 & bitwise AND
2 ^ bitwise XOR
3 | bitwise OR
What is the output of the following C++ program?

// precedence.cpp
#include <iostream>
int main(void)
{
unsigned int a = 10;
unsigned int b = 11;
unsigned int c = 12;
unsigned int r1 = a & b | c;
cout << "r1 = " << r1 << endl;
unsigned int r2 = a & (b | c);
cout << "r2 = " << r2 << endl;
return 0;
}

1.4 bitset class
Problem 82. The following program uses the bitset class of C++ and
utilizes pointers. What is the output of the program?
// bitset1.cpp
#include <iostream>
#include <bitset>
const unsigned int n = 32;
void swap(bitset<n>* b1,bitset<n>* b2)
{ *b1 = (*b1)^(*b2); *b2 = (*b2)^(*b1); *b1 = (*b1)^(*b2); }
int main()
{
bitset<n> bs1;
bs1.flip(4);
cout << "bs1 = " << bs1 << endl;
bitset<n>* pbs1 = new bitset<n>;
pbs1 = &bs1;
cout << "pbs1 = " << pbs1 << endl;
bitset<n> bs2;
bs2.flip(7);
cout << "bs2 = " << bs2 << endl;
bitset<n>* pbs2 = new bitset<n>;
pbs2 = &bs2;
swap(pbs1,pbs2);
cout << "after swapping:" << endl;
cout << "bs1 = " << bs1 << endl;
cout << "bs2 = " << bs2 << endl;
return 0;
}
Problem 83. The following program uses the bitset class of C++. A
pointer to bitset is declared. What is the output of the following program?
// bitsetpointer.cpp
#include <iostream>
#include <bitset>
int main(void)
{
bitset<n>* bp = new bitset<n>;
(*bp).set();

(*bp).flip(4);
cout << *bp << endl;
delete bp;
return 0;
}
Problem 84. The following program uses the bitset class of C++.
What is the output of the program?
// uinttobitset1.cpp
#include <iostream>
#include <bitset>
void convert(bitset<32>& s,unsigned int i,int n)
{
int count = 0;
while(count < n)
{
int t = 1 & i;
if(t != 0) s.set(count,1);
else s.set(count,0);
i = i >> 1;
count++;
}
} // end convert
int main(void)
{
bitset<n> s;
convert(s,i,n);
cout << "s = " << s << endl;
return 0;
}
Problem 85. (i) Given an unsigned int number (32 bits). Write a C++
program that converts it to a bitstring of the bitset class.
(ii) Given an signed int number (32 bits). Write a C++ program that
converts it to a bitstring of the bitset class.
(iii) Given a ﬂoating point number float (32 bits). Write a C++ program
that converts it to a bitstring of the bitset class.
(iv) Given a ﬂoating point number double (64 bits). Write a C++ program
that converts it to a bitstring of the bitset class.

Problem 86. (i) Given a bitstring of the bitset class of length 32. Write
a C++ program that converts the bitstring into an unsigned int. First one
has to check whether the bitstring refers to NaN.
(ii) Given a bitstring of the bitset class of length 32. Write a C++ program
that converts the bitstring into a signed int. First one has to check whether
the bitstring refers to NaN.
(iii) Given a bitstring of the bitset class of length 32. Write a C++
program that converts the bitstring into a ﬂoat. First one has to check
whether the bitstring refers to NaN.
(iv) Given a bitstring of the bitset class of length 64. Write a C++
program that converts the bitstring into a double. First one has to check
whether the bitstring refers to NaN.
Problem 87. Show that
6710 = 749 = 1038 = 1247 = 1516 = 2325 = 10034 = 21113 = 10000112
and 6710 = 4316.

Chapter 2
Advanced Bitwise
Manipulations
2.1 Write a C++ Program
Problem 1. Let a, b, c, x, y, z ∈ { 0, 1 }. Let + be the bitwise OR and ·
be the bitwise AND. Write a C++ program that ﬁnds all solutions of the
equation
a + b + c = x · y · z.
Count the number of solutions.
Problem 2. Let a, b, c, x, y, z ∈ { 0, 1 }. Let ⊕ be the bitwise XOR and ·
be the bitwise AND. Write a C++ program that ﬁnds all solutions of the
equation
a ⊕ b ⊕ c = x · y · z.
Count the number of solutions.
Problem 3. Let x and y be unsigned int’s. Write a function
unsigned int setbits(x,p,n,y)
that returns x with the n bits that begin at position p set to the rightmost
n bits of y, leaving the other bits unchanged.
31

Problem 4. Given two unsigned int x and y. Write a C++ program
that finds the Hamming distance between x and y. Apply the XOR opera-
tion.
Problem 5. Given a bitstring with an even number n of bits. Half (n/2)
of the bits are 0’s and therefore the other half are 1’s.
(i) In how many ways can we form such a bitstring for a given n.
(ii) Write a C++ program using bitset<N> that finds all the possible bit-
strings for a given n and stores them in lexicographical order. For example,
for n = 2 we have 01, 10. For n = 4 we have
0011 0101 0110 1001 1010 1100
Problem 6. Let x, y be bitstrings of the same length n. We define a
scalar product
x y := (x0 · y0) ⊕ (x1 · y1) ⊕ · · · ⊕ (xn−1 · yn−1).
Write a C++ program that implements this scalar product.
Problem 7. Given two unsigned int m and n. Write a C++ function
bool check(unsigned int m,unsigned int n)
which returns true if m and n are both even and false otherwise. Apply
bitwise operations.
Problem 8. Given an unsigned int (32 bits) in C++. Write a C++
program that reverses the bits.
Problem 9. Let x be an unsigned int. Write a function invert(x,p,n)
that returns x with the n bits that begin at position p inverted (i.e. 1 is
changed into 0 and vice versa), leaving others unchanged.
Problem 10. Let x be an unsigned int. Write a C++ function
unsigned int rightrot(x,n)
that returns the value of the integer x rotated to the right by n bit positions.
Apply the shift operation.
Problem 11. Rewrite the following C++ program without the if ... else
using bitwise operations. What is the program doing?

Advanced Bitwise Manipulations 33
// iftoxor0.cpp
#include <iostream>
#include <ctime>
#include <cstdlib>
int main(void)
{
unsigned int a, b, x;
srand((unsigned int) time(NULL));
x = rand()%2;
cout << "x = " << x << endl;
a = 0; b = 1;
if(x==a) x = b; else x = a;
cout << "x = " << x << endl;
return 0;
}
Problem 12. Let i, j be integers. The doz function is “difference or
zero” and is defined as
doz(i, j) :=
i − j if i ≥ j
0 if i < j
Write a C++ program using bitwise operations that implements this func-
tion.
Problem 13. Given an unsigned int (32 bits). Write a C++ program
with a function unsigned int lasttwo(unsigned int) that return the
integer number represented by the last two bits, i.e. if "00" return 0, if
"01" return 1, if "10" return 2 and if "11" return 3.
Problem 14. Let n = 0, 1, 2, . . .. We define the sequence
an :=



0 if the number of 1’s in the binary representation
of n is even
1 if the number of 1’s in the binary representation
of n is odd
Write a C++ program that finds this sequence.
Problem 15. Let v, d be unsigned integers. To find the remainder of
integer division v/d one use v%d. Write a C++ program that uses bitwise
operations to do this operation. We assume that d is of the form 1, 2, 4, 8,
... i.e. of the form 2n
.

Problem 16. Given an unsigned int j. Write a C++ program with a
function
bool test5(unsigned int j)
that tests whether j can be divided by 5 without remainder. The bitwise
AND must be used.
Problem 17. Given an unsigned int j. Write a C++ program with a
function
bool test3(unsigned int j)
that tests whether j can be divided by 3 without remainder. The bitwise
AND must be used.
Problem 18. Write a C++ program that ﬁnd the integer part log2 of an
unsigned int using bitwise operations.
Problem 19. In the following C++ program we calculate the (ﬂoor) of
the average of two unsigned integers x and y as (x+y)/2. The result is
352516352 which is obviously wrong. Explain why. Fix the problem using
bitwise operations.
// average.cpp
#include <iostream>
int main(void)
{
unsigned int y = 1000000001;
unsigned int r1 = (x + y)/2;
cout << "r1 = " << r1 << endl;
return 0;
}
Problem 20. The bitwise XOR operation ^ has a special property and
this can be used directly on encryption. Given two arbitrary bitstrings a
and b of the same length, then the following expression for bitwise XOR
holds
(a ^ b) ^ b = a .
In encryption the b would be called the key. Consider a character string, for
example ”Willi-Hans”. Write a C++ program that uses the XOR operation
that encrypts and decrypts this character string byte by byte. As a key use

unsigned char key = 50;
Problem 21. Given a bitstring of length n, where n is even and n ≥ 2.
Test whether the bitstring is alternating or not. For example
10101010
01010101
are the two alternating bitstrings of length 8.
2.2 Gray Code
Problem 22. An n-bit Gray code is a sequence of all the n-bit binary
numbers, ordered in such a way that each number differs from its predeces-
sor and its successor by exactly 1 bit and the first and last differ by 1 bit
too. Give a 2-bit Grey sequence.
Problem 23. Give two different Gray codes for three bits.
Problem 24. Given a Gray code sequence for n − 1 bits. Construct a
Gray code sequence for n bits using the Gray code sequence for n − 1 bits.
Problem 25. (i) Consider the traveling saleswoman problem. Given
eight cities with the space coordinates
(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1),
(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1).
Consider the coordinates as a bitstring. We identify the bitstrings with
base 10 numbers, i.e. 000 → 0, 001 → 1, 010 → 2, 011 → 3, 100 → 4,
101 → 5, 110 → 6, 111 → 7. Thus we set
x0 = (0, 0, 0), x1 = (0, 0, 1), x2 = (0, 1, 0), x3 = (0, 1, 1)
x4 = (1, 0, 0), x5 = (1, 0, 1), x6 = (1, 1, 0), x7 = (1, 1, 1).
The traveling saleswoman does not like her husband. Thus she wants to
find the longest route starting at (0, 0, 0) and returning to (0, 0, 0) after
visiting each city once. Is there more than one solution?
(ii) Consider the traveling salesman problem and the eight cities given in
(i). The traveling salesman likes his girlfriend. Thus he wants to find the
shortest route starting at (0, 0, 0) and returning to (0, 0, 0) after visiting
each city once. What is the connection with the Gray code? Is there more
than one solution?

Problem 26. Extend the previous problem to 16 cities in hyperspace R4
with the coordinates
(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1),
(0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1),
(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1),
(1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1).
Problem 27. Give a 4-bit Gray code.
Problem 28. Find a Gray code for 5 bits. Start of with
0 0 0 0 0
0 0 0 0 1
0 0 0 1 1
0 0 0 1 0
0 0 1 1 0
0 0 1 1 1
0 0 1 0 1
0 0 1 0 0
0 1 1 0 0
Problem 29. The fundamental quantum-dot cellular automata logic de-
vice is a three-input majority logic gate. The truth table containing all
possible input combinations (a, b, c) in Gray code is given by (O is the
output)
a b c O
0 0 0 0
0 0 1 0
0 1 1 1
0 1 0 0
1 1 0 1
1 1 1 1
1 0 1 1
1 0 0 0
(i) Find the boolean function (sum of products). Can the expression be
simpliﬁed?
(ii) Show that the majority gate can implement an AND-gate and an OR-
gate.
(iii) Can it implement a NOT-gate?

Problem 30. What is the output of the following C++ code?
// gray.cpp
#include <iostream>
unsigned int gray(unsigned int g)
{
g ^= (g >> 16);
g ^= (g >> 8);
g ^= (g >> 4);
g ^= (g >> 2);
g ^= (g >> 1);
return g;
}
int main(void)
{
unsigned int g1 = 0;
unsigned int r1 = gray(g1);
cout << "r1 = " << r1 << endl;
cout << "r2 = " << r2 << endl;
cout << "r3 = " << r3 << endl;
cout << "r4 = " << r4 << endl;
cout << "r5 = " << r5 << endl;
return 0;
}

2.3 Binary and Arithmetic Operations
Problem 31. Usually we describe arithmetic operation in terms of bi-
nary operations, but in some cases it is interesting to consider the reverse
problem. Let a, b ∈ N0 and a, b < 2n
. Thus we can write the binary
representation
a =
n−1
j=0
aj2j
, a0, a1, . . . , an−1 ∈ {0, 1}
b =
n−1
j=0
bj2j
, b0, b1, . . . , bn−1 ∈ {0, 1}.
Now we can deﬁne the bitwise operations as functions on whole numbers:
NOT(a) : N0 → N0, NOT(a) =
n−1
j=0 NOT2(aj)2j
,
AND(a, b) : N0 × N0 → N0, AND(a, b) =
n−1
j=0 AND2(aj, bj)2j
,
OR(a, b) : N0 × N0 → N0, OR(a, b) =
n−1
j=0 OR2(aj, bj)2j
,
XOR(a, b) : N0 × N0 → N0, XOR(a, b) =
n−1
j=0 XOR2(aj, bj)2j
.
The boolean functions NOT2, AND2, OR2 and XOR2 are given by
aj bj NOT2(aj AND2(aj, bj) OR2(aj, bj) XOR2(aj, bj)
0 0 1 0 0 0
0 1 1 0 1 1
1 0 0 0 1 1
1 1 0 1 1 0
In the following use only the arithmetic operations for addition, subtraction,
multiplication, quotient and remainder.
1. Give the formula for NOT(a) without referring to the binary repre-
sentation.
2. Given an algorithm to calculate AND(a, b) without referring to the
binary representation.
3. Give a formula for XOR(a, b) in terms of a, b and the AND function.
4. Give a formula for OR(a, b) in terms of a, b and the AND function.
5. Implement bitwise operations on integers using the answers above.
Problem 32. (i) Let xj ∈ {+1, −1} with j = 1, 2, . . . , 9. Find all solu-
tions of the system of six equations
x1x2x3 = 1, x4x5x6 = 1, x7x8x9 = 1,

x1x4x7 = 1, x2x5x8 = 1, x3x6x9 = −1.
Write a C++ program that runs of all 29
combinations to find the solutions.
(ii) Let yj ∈ {0, 1}. Find all solutions of the system of six equations
y1 ⊕ y2 ⊕ y3 = 1, y4 ⊕ y5 ⊕ y6 = 1, y7 ⊕ y8 ⊕ y9 = 1,
y1 ⊕ y4 ⊕ y7 = 1, y2 ⊕ y5 ⊕ y8 = 1, y3 ⊕ y6 ⊕ y9 = −1
where ⊕ denotes the XOR-operation.
2.4 Theory
Problem 33. Let n ≥ 2 and even. How many bitstrings of length n can
one form with n/2 0’s and n/2 1’s. For n = 4 write down all of them in
lexicographical order.
Problem 34. Consider a binary string S of length n ≥ 1 with symbols
1 and 0. We define ∆(S) as the number of 1’s of S minus the number
of 0’s. For example, ∆(“1001001”) = −1. We call a string S balanced if
every substring T of consecutive symbols of S has −2 ≤ ∆(T) ≤ 2. Thus
“1001001” is not balanced, since it contains the substring “00100”. The
string 01010101 is balanced. Let bn be the number of balanced strings of
length n. Obviously we have b1 = 2 with the strings “0” and “1” and b2 = 4
with the strings “00”, “01”, “10” and “11”. Find a recursion relation for
bn.
Problem 35. Let
α = 1 + a1x + a2x2
+ · · ·
be a formal power series with coefficients in the field of two elements (char-
acteristic 2). We define
an :=



1 if every block of zeros in the binary expansion
of n has an even number of zeros in the block
0 otherwise
For example, a11 = 0 since 11 = 10112 and a36 = 1 since 36 = 1001002.
Show that α3
+ xα + 1 = 0.
Problem 36. Consider two bitstrings of length N,
b = b0b1b2 . . . bN−1, c = c0c1c2 . . . cN−1
and the map
c0 = b0

c1 = b0 ⊕ b1
c2 = b0 ⊕ b1 ⊕ b2
...
cN−1 = b0 ⊕ b1 ⊕ · · · ⊕ bN−1
(i) Let b = b0b1b2b3 = 1011, i.e. N = 4. Calculate c.
(ii) Find the inverse map if it exists.
Problem 37. We consider multiplication of two two-binary numbers a1a0
and b1b0. For example, if a1a0 = 10 (2 in decimal) and b1b0 = 11 three
in decimal, then we find for the product (output) O3O2O1O0 = 0110 (6 in
decimal). Find all possible products and put them into a function table.
Problem 38. The algebraic Reed-Muller expansion of boolean functions
is one of the fundamental approaches in the design of digital systems, es-
pecially when dealing with the VLSI technology. This algebraic repre-
sentation is useful in error detection and error correction models. Let
f(x1, x2, . . . , xn) be a boolean function. We define with respect to xj
(j = 1, 2, . . . , n)
fxj (x) := f(x1, . . . , xj−1, 1, xj+1, . . . , xn)
f¯xj (x) := f(x1, . . . , xj−1, 0, xj+1, . . . , xn)
∂f
∂xj
:= fxj
(x) ⊕ f¯xj
(x)
as the positive cofactors of f, negative cofactor of f, and the boolean deriva-
tive of f. Then the Reed-Muller expansion (also called Davio expansion) is
given by
f = f¯xj
⊕ xj ·
∂f
∂xj
where ⊕ is the XOR operation and · is the AND operation. The complement
of the variable is denoted by an overbar, that is ¯x = 1 − x. Note that
(a ⊕ b) · c = a ⊕ (b · c) in general, for example for a = 1, b = 0, c = 0.
Consider the sum-of-product form of the boolean function
f(x1, x2, x3, x4) = ¯x1 · x3 + ¯x1 · x2 · x4 + x1 · ¯x2 · ¯x3 + x1 · ¯x3 · ¯x4.
Apply the Reed-Muller expansion to find a simpler expression.
Problem 39. Let f(x1, x2, . . . , xn) be a boolean function of n variables.
Shannon’s expansion is given by
f(x1, x2, . . . , xn) = x1 · f1(1, x2, . . . , xn) + ¯x1 · f(0, x2, . . . , xn) (1)

and
f(x1, x2, . . . , xn) = ¯x1 · f(0, x2, . . . , xn) ⊕ x1 · f(1, x2, . . . , xn). (2)
Apply these two expansions to the boolean function
f(x1, x2, x3) = (x1 + x2) · (x1 + x3).
Discuss.
Problem 40. A J-K flip-flop can memorize a single bit of information.
The next state Q(t + 1) of a J-K flip lop is characterized as a function of
both the present state Q(t) and the present two inputs J(t) and K(t). The
truth table for the J-K flip flop is
J(t) K(t) Q(t) Q(t + 1)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0
Find the minterm expression and simplify it.
Problem 41. Consider the set {+1, −1} and multiplication. Then we
have the group table
· +1 -1
+1 +1 -1
-1 -1 +1
The neutral element of the group is +1. Consider now the set {0, 1} and
the XOR operation. Then we have the group table
⊕ 0 1
0 0 1
1 1 0
The neutral element of the group is 0. Show that the two group are iso-
morphic.
Problem 42. The Cantor sequence is constructed as follows. Given the
natural numbers
n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

We write them in ternary notation as
0, 1, 2, 10, 11, 12, 20, 21, 22, 100, . . .
Then the Cantor sequence bn (n = 0, 1, 2, . . .) is defined as: if n in ternary
has only 0s and 2s, then bn = 1, otherwise we set bn = 0. The first 9 terms
of the sequence are given by
1, 0, 1, 0, 0, 0, 1, 0, 1, . . .
Write a C++ program that generates this sequence. Is the sequence chaotic?
Problem 43. The Thue-Morse sequence is defined as follows. Let V =
{ 0, 1 }, s0 = 0 (intial value of the sequence) and
p1 : 0 → 01, p2 : 1 → 10.
Find the first five elements in the construction of the Thue-Morse sequence.
Write a C++ program that generates the sequence. Use the bitset class.
Problem 44. Consider the map
f(x) = 2x mod 1.
We can associate with each point in [0, 1] an itinerary (an infinite sequence
of 0’s and 1’s) on which the shift map represents the action of f. Show that
all points in the subinterval
[(k − 1)2−n
, k2−n
], 1 ≤ k ≤ 2n
are represented by a finite sequence of n symbols.
Problem 45. A boolean function f : { 0, 1 }⊗n
→ { 0, 1 }⊗m
(m ≥ n) is
periodic with period p with respect to bitwise modulo 2 addition, i.e. for
all x we have
f(x) = f(x + p).
Give an example of such a periodic boolean function.
Problem 46. Let x = x1x2 . . . xn, y = y1y2 . . . yn, z = z1z2 . . . zn be
three n-bit binary strings. Let ⊕ denote bitwise addition modulo 2, i.e.
x ⊕ y = z ⇒ for all k, zk = xk + yk mod 2.
We define a scalar product of x and y by
x • y := (x1 · y1) + (x2 · y2) + · · · + (xn · yn) mod 2.

Show that this binary scalar product • is distributive over bitwise modulo
2 addition ⊕, i.e.
(x ⊕ y • z = (x • z) ⊕ (y • z).
Problem 47. Consider the full adder given by the truth table
A B C SUM CARRY
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Implement the full adder using only NAND-gates.
Problem 48. The Cantor series approximation is deﬁned as follows.
For arbitrary chosen integers n1, n2, . . . (equal or larger than 2), we can
approximate any real number r0 as follows
xj = integer part(rj), j = 0, 1, 2, . . .
rj+1 = (rj − xj)nj
and
r0 ≈ x0 +
N
j=1
xj
n1n2 · · · nj
.
The approximation error is
En =
1
n1n2 · · · nN
.
Apply this approximation to r0 = 2/3 and the golden mean number with
nj = 2 for all j and N = 4. Thus the xj form a bit sequence.
Problem 49. Let
Σ2 := { s := (s0s1s2 . . .) : sj = 0 or 1 }.
Σ2 is known as sequence space on the symbols 0 and 1. If we deﬁne the
distance between the two sequences s and t by
d[s, t] =
∞
j=0
|sj − tj|
2j

then Σ2 is a metric space.
(i) Let s = (1111 . . .), t = (0000 . . .). Find the distance.
(ii) Consider the periodic sequences
s = (010101 . . .), t = (101010 . . .).
Find the distance.
(iii) A dynamics is given by the shift map σ : Σ2 → Σ2 defined by
σ(s0s1s2 . . .) := (s1s2s3 . . .).
Let s = (1000 · · ·) and t = σ(s). Find the distance.
Problem 50. Periodic points are identified with exactly repeating se-
quences
s = (s0s1 . . . sn−1, s0s1 . . . sn−1, s0s1 . . . sn−1, . . .).
How many periodic points of period n there are?
Problem 51. Show that
a · b = a + b
using (i) truth tables and (ii) properties of boolean algebra (with a+1 = 1).
Problem 52. (i) Given a digital circuit with 4 input lines and four
output lines. The output is the two’s complement of the input. Give the
truth table.
(ii) Find the disjunctive normal form.
(iii) Construct a logic circuit using a PAL.
(iv) Write a VHDL program tha simulates this circuit.
Problem 53. The genetic code vector space is presented by the Ga-
lois field of four bases (GF(4)). We consider the invertible map f :
{ A, C, G, T} → { (aj, aj+1) } from the base set to the set of binary du-
plets aj, aj+1, where f(X) = (aj, aj+1) and aj ∈ {0, 1}. Since A maps to
T, C maps to G we introduce the map
f(A) = (0, 0), f(C) = (1, 0), f(G) = (0, 1), f(T) = (1, 1).
We also write (0,0) etc as bitstrings "00". Using the NOT-operation we
have 00 = 11, 01 = 10. Thus the sum of binary digits corresponding to
DNA complementary basis is always (1, 1). Therefore we have eight ordered
base sets, namely
{ G, A, T, C }, { G, T, A, C }, { C, A, T, G }, { C, T, A, G }

{ A, C, G, T }, { A, G, C, T }, { T, C, G, A }, { T, G, C, A }
Write a C++ program using the bitset class that converts a given DNA
sequence, for example ”ATGCAATTCTCGCTA”, into the corresponding
bitstring.
Problem 54. Solve the following Max-SAT problem with ﬁve clauses and
four variables x1, x2, x3, x4
f1(x1, x2.x3, x4) = x1 ∨ ¬x2
f2(x1, x2, x3, x4) = ¬x1 ∨ x3 ∨ ¬x4
f3(x1, x2, x3, x4) = ¬x1 ∨ ¬x2
f4(x1, x2, x3, x4) = x1 ∨ ¬x3 ∨ x4
f5(x1, x2, x3, x4) = x2 ∨ x3 ∨ ¬x4
using backtracking branch-and-bound, where ¬ is the NOT and ∨ is the
OR.
Problem 55. Let and m be positive integers satisfying 1 ≤ < m. Let
X := X−m+1, . . . , X−1, X0, X1, X2, . . .
be a binary sequence satisfying the recursion relation
Xk = Xk− ⊕ Xk−m
for k = 1, 2, . . .. Here ⊕ denotes addition modulo 2. Any such sequence
will be called a linear-feedback shift sequence. For k ≥ 0 the m-tuple
[X] := (Xk−m+1, . . . , Xk−1, Xk)
will be called the state of X at time k. Let = 1, m = 2, X−1 = 1, X0 = 1.
Find the sequence X1, X2, . . . . Write a C++ pogram that implements this
sequence.
Problem 56. A Boolean function f : {0, 1}2
→ {0, 1} is constant if for
every x ∈ {0, 1}2
we have f(x) = c, where c ∈ {0, 1}. A Boolean function
is balanced if
|{x ∈ {0, 1}2
: f(x) = 0 }| = |{x ∈ {0, 1}2
: f(x) = 1 }|
i.e. f maps to an equal number of zeros and ones. Find all Boolean func-
tions from {0, 1}2
to {0, 1} which are either constant or balanced. Deter-
mine which of these functions are separable, i.e. for which function can we
ﬁnd a hyperplane which separates the inputs x which give f(x) = 1 and
the inputs y which give f(y) = 0.

Problem 57. The average information gain when one of a mutually
exclusive set of events with probabilities p0, . . . , pn−1 occurs is defined as
the Shannon entropy
H(p0, p1, . . . , pn−1) := −
n−1
j=0
pj log(pj)
with the convention 0 log(0) = 0. Consider an information source that
transmits words composed out of letters of an alphabet of eight elements.
If all letters have equal probability, pj = 1/8, j = 0, 1, . . . , 7, to appear
per transmitted letter, a compact way of representing the letters in binary
notation is the 3-bit code
0th → 000,
1st → 001,
...
7th → 111.
Find the Shannon entropy.
Problem 58. Show that the XOR-gate can be built from 4 NAND-gates.
Let A1, A2 be the inputs and O the output.
Problem 59. Consider the function f : [−1, 1] × [−1, 1] → R
f(x1, x2) = x1x2.
We want to find the maxima of the (fitness) function f in the given domain
[−1, 1] × [−1, 1]. Consider the two bitstrings
b1 = "0000000011111111" b2 = "1010101010101010"
where the 8 bits on the right-hand side belong to x1 and the 8 bits on the
left-hand side belong to x2. Find the value of the fitness function for the
two bitstring. Apply the NOT-operation to the two bitstrings to obtain the
new bitstrings b3 = NOT(b1) and b4 = NOT(b2). Select the two fittest of
the four bitstrings for survival.
Problem 60. Consider the domain [−1, 1]×[−1, 1] in R2
and the bitstring
(16 bits)
0101010100001111
The 8 bits on the right-hand side belong to x and the 8 bits on the left-hand
side belong to y (counting from right to left starting at 0). Find the real

values x∗
and y∗
for this bitstring. Let f : [−1, 1] × [−1, 1] → R be given
by
f(x, y) = x4
+ y4
.
Find the value f(x∗
, y∗
). How far is this value from the minimum of f in
the given domain?
Problem 61. Show that the full-adder can be built with nine NAND-
gates. Give the circuit. Describe the circuit using multiexpression program-
ming.
Problem 62. Let a, b ∈ {0, 1} and
0: a
1: b
2: NOT 0
3: OR 1,2
Give the output.
Problem 63. Let a, b, c, d ∈ {0, 1} and
0: a
1: b
2: c
3: d
4: XOR 0,2
5: XOR 1,3
6: 0R 4,5
Give the output.
Problem 64. Let a, b ∈ {0, 1}. Consider the sequence of expressions in
multiexpression programming
0: a
1: b
2: NOT 0
3: OR 1,2
Give the output at 3.
Problem 65. Let a, b, c, d ∈ {0, 1}. Consider the sequence of expressions
in multiexpression programming

0: a
1: b
2: c
3: d
4: XOR 0,2
5: XOR 1,3
6: OR 4,5
Give the output at 6.
Problem 66. Express the majority gate
I_1 I_2 I_3 O
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
with multiexpression programming.
Problem 67. Build the 4 bit magnitude comparator using only NAND
gates.
Problem 68. Build the 8 bit priority encoder using only NAND gates.
Problem 69. (i) Consider a bit-string of length n counting the bit-string
from right to left and starting counting from 0. Write a C++ program
using the bitset class that selects at random a bit and then the bits on
the left and on the right are switched around and the selected bit stays the
same. This means all bits except the selected bit are negated. For example
consider the bitstring 10101110 and we select bit number 2, then we have
01010101.
(ii) Assume that all bits are set in the bitstring. Can one ﬁnd a sequence
of this operation so that all bits are oﬀ, i.e. set to 0.
Problem 70. In C++ the data type double consists of 64 bits. Convert
this bitstring of a double into a character array char a[8] counting from
left to right in the bitstring of the double.

Problem 71. The data type float consists of 32 bits. Thus the number
of possible bitstrings is
232
= 68719476736
Apply the bitset class of C++ to generate all these bitstrings and convert
them into the corresponding float or NaN (not a number).
Problem 72. Let
σ0,0, σ1,0, σ−1,0, σ0,1, σ0,−1 ∈ {0, 1}.
Consider the Hamiltonian
H = σ0,1 ⊕ σ0,0 ⊕ σ−1,0 + σ−1,0 ⊕ σ0,0 ⊕ σ1,0
where ⊕ denotes the XOR-operation and + is the arithmetic addition. Find
the energy levels of H by running through all possible bistrings. There are
25
bitstrings of length 5. Write a C++ program using the bitset class
that finds all the levels.
Problem 73. Show that the NAND-operation is not associative in gen-
eral, i.e. N(N(a, b), c) = N(a, N(b, c)). Consider the case a = 0, b = 0,
c = 1.
Problem 74. Consider the map
xt+1 = 2xt mod 1
with x0 ∈ [0, 1) to find the binary expansion of the rational number x0 =
3/13.
Problem 75. Let F2 be the field with the elements 0 and 1. Consider
the vector v := (v0, v1, . . . , vm−1) from the vector space Fm
2 . Then the
complement of the vector v is defined as
¯v := (1 − v0, 1 − v1, . . . , 1 − vm−1).
We define a subset of vectors Wn ⊆ F2n
2 as follows: We set W0 := {(0)} and
for j ≥ 0 the set Wj+1 consists of all vectors given by (w, w) and (w, ¯w),
where w is any vector from Wj. Therefore Wj+1 has twice as many vectors
as the set Wj. One calls Wn the set of Hadamard vectors. The length of
the Hadamard vectors in Wn is 2n
.
(i) Let W1 = {(0, 0), (0, 1)}. Find W2.
(ii) Write a C++ program using the bitset class to generate the Hadamard
vectors.

Problem 76. Let n be an odd number and n ≥ 3. Write a C++ program
using the bitset class that parses trough all bitstrings of length n. If the
number of 0’s is larger then the number of 1’s the output should be 0 and
the number of 0’s is smaller then the number of 1’s the output should be
1. Then write down the truth table (majority gate).
Problem 77. Consider the lattice
(0,1)
(-1,0) (0,0) (1,0)
(-1,0)
and the Hamiltonian
H = σ0,1 ⊕ σ0,0 ⊕ σ−1,0 + σ−1,0 ⊕ σ0,0 ⊕ σ1,0
where σ0,0, σ1,0, σ−1,0, σ0,1, σ0,−1 ∈ {0, 1}, ⊕ is the XOR-operation and +
the arithmetic addition. Find the enery levels of ˆH by running through all
possible bitstrings. There are 25
= 32 bitstrings of length 5. Write a C++
programs that finds the energy level using the bitset class of C++.
Problem 78. Let x ∈ {0, 1}. Consider the boolean functions (fj :
{0, 1} → {0, 1})
f1(x) = x ⊕ x, f2(x) = x · x, f3(x) = x + x
where ⊕ is the XOR-operation, · is the AND-operation and + is the OR-
operation. Find the fixed points of f1, f2 and f3.
→ {0, 1}2
f1(x, y) = y ⊕ y, f2(x, y) = x ⊕ x.
(i) Find the fixed points of the boolean function.
(ii) Start with (x, y) = (1, 1) and iterate the boolean function. Discuss.
(iii) Start with (x, y) = (1, 0) and iterate the boolean function. Discuss.
(iv) Start with (x, y) = (0, 1) and iterate the boolean function. Discuss.
→ {0, 1}2
f1(x, y) = x ⊕ y, f2(x, y) = x ⊕ y.
(ii) Start with (x, y) = (1, 0) and iterate the boolean function at least twice.
Discuss.

(iii) Start with (x, y) = (0, 1) and iterate the boolean function at least
twice. Discuss.
(iv) Start with (x, y) = (1, 1) and iterate the boolean function. Discuss.
Problem 81. Given the boolean function f : {0, 1}3
→ {0, 1} as truth
table
x_1 x_2 x_3 f(x_1,x_2,x_3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Consider the unit cube in R3
with vertices (corner points) (0, 0, 0), (0, 0, 1),
(0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1). Find the plane in R3
that separates the 0 from the 1’s.
→ {0, 1}3
f1(x, y, z) = y⊕y⊕y, f2(x, y, z) = z⊕z⊕z, f3(x, y, z) = x⊕x⊕x.
(ii) Start with (x, y, z) = (1, 0, 1) and iterate the boolean function. Discuss.
(iii) Start with (x, y, z) = (1, 0, 0) and iterate the boolean function. Discuss.
(iv) Start with (x, y, z) = (0, 1, 0) and iterate the boolean function. Discuss.
Problem 83. The NAND-gate is given by
i_1 i_2 o
0 0 1
0 1 1
1 0 1
1 1 1
where i_1, i_2 are the inputs and o is the output. The XNOR-gate is
given by
i_1 i_2 o
0 0 1
0 1 0
1 0 0
1 1 1

where i_1, i_2 are the inputs and o is the output. Show that the XNOR-
gate can be build with five NAND-gates.
Problem 84. Let n ≥ 1. Consider the boolean function f : {0, 1}n
→
{0, 1}. Then the boolean derivatives are defined as
∂f
∂xj x
:= f(x1, . . . , xj, . . . , xn) ⊕ f(x1, . . . , ¯xj, . . . , xn)
for j = 1, . . . , n.
(i) Consider the boolen function f : {0, 1}2
→ {0, 1} given by
f(x1, x2) = x1 + x2.
Find the boolean derivatives ∂f/∂x1, ∂f/∂x2.
(ii) Consider the boolen function f : {0, 1}2
→ {0, 1} given by
f(x1, x2) = x1 · x2.
Find the boolean derivatives ∂f/∂x1, ∂f/∂x2.
(iii) Consider the boolean function f : {0, 1}4
→ {0, 1}
f(x1, x2, x3, x4) = x1 + x2 ⊕ x3 · x4.
Find the boolean derivatives ∂f/∂x1, ∂f/∂x2, ∂f/∂x3, ∂f/∂x4.
Problem 85. (i) The binary Thue-Morse sequence is generated by the
substitutions
0 → 01, 1 → 10.
Start with 0 and give the sequence for the first four steps. Give a C++
implementation.
(ii) Study the binary sequence given by the substitutions
0 → 11, 1 → 10
for the first four steps starting with 1. Give a C++ implementation.
(iii) The Rudin-Shapiro sequence is generated by the two-digit substitutions
00 → 0001, 01 → 0010, 10 → 1101, 11 → 1110.
Start with 00 and give the sequence after three substitutions. Give a C++
implementation.
Problem 86. Consider the NAND gate. Show that the NOT gate can
be build with one NAND gate. Show that the AND gate can be build with

two NAND gates. Show that the OR gate can be build with three NAND
gates. Show that the NOR gate can be build with four NAND gates. Show
that the XOR gate can be build with four NAND gates. Show that the
XNOR gate can be build with five NAND gates.
Problem 87. Consider the unit cube with the 8 corner points jk (j, k, ∈
{0, 1})
000, 001, 010, 011, 100, 101, 110, 111.
Let xjk ∈ {0, 1}. Each corner point has three nearest neighbours. Find all
the energy levels for
E = x000 ⊕ x001 ⊕ x010 ⊕ x100 + x001 ⊕ x000 ⊕ x011 ⊕ x101
+x010 ⊕ x000 ⊕ x011 ⊕ x110 + x011 ⊕ x001 ⊕ x010 ⊕ x111
+x100 ⊕ x000 ⊕ x110 ⊕ x101 + x101 ⊕ x001 ⊕ x100 ⊕ x111
+x110 ⊕ x111 ⊕ x100 ⊕ x010 + x111 ⊕ x011 ⊕ x110 ⊕ x101
where ⊕ is the XOR-operation and + is the arithmetic plus. Find the
energy level for all possible 28
= 256 configurations.
Problem 88. (i) Let s1(0), s2(0), s3(0) ∈ {+1, −1}. Study the time-
evolution (t = 01, 2, . . .) of the coupled system of equations
s1(t + 1) = s2(t)s3(t)
s2(t + 1) = s1(t)s3(t)
s3(t + 1) = s1(t)s2(t)
for the eight possible initial conditions, i.e. (i) s1(0) = s2(0) = s3(0) = 1,
(ii) s1(0) = 1, s2(0) = 1, s3(0) = −1, (iii) s1(0) = 1, s2(0) = −1, s3(0) = 1,
(iv) s1(0) = −1, s2(0) = 1, s3(0) = 1, (v) s1(0) = 1, s2(0) = −1, s3(0) =
−1, (vi) s1(0) = −1, s2(0) = 1, s3(0) = −1, (vii) s1(0) = −1, s2(0) = −1,
s3(0) = 1, (viii) s1(0) = −1, s2(0) = −1, s3(0) = −1. Which of these initial
conditions are fixed points?
(ii) Let s1(0), s2(0), s3(0) ∈ {+1, −1}. Study the time-evolution (t =
01, 2, . . .) of the coupled system of equations
s1(t + 1) = s2(t)s3(t)
s2(t + 1) = s1(t)s2(t)s3(t)
s3(t + 1) = s1(t)s2(t)
for the eight possible initial conditions, i.e. (i) s1(0) = s2(0) = s3(0) = 1,
(ii) s1(0) = 1, s2(0) = 1, s3(0) = −1, (iii) s1(0) = 1, s2(0) = −1, s3(0) = 1,
(iv) s1(0) = −1, s2(0) = 1, s3(0) = 1, (v) s1(0) = 1, s2(0) = −1, s3(0) =
−1, (vi) s1(0) = −1, s2(0) = 1, s3(0) = −1, (vii) s1(0) = −1, s2(0) = −1,

s3(0) = 1, (viii) s1(0) = −1, s2(0) = −1, s3(0) = −1. Which of these initial
conditions are fixed points?
Problem 89. Let x1(0), x2(0), x3(0) ∈ {0, 1} and let ⊕ be the XOR-
operation. Study the time-evolution (t = 01, 2, . . .) of the coupled system
of equations
x1(t + 1) = x2(t) ⊕ x3(t)
x2(t + 1) = x1(t) ⊕ x3(t)
x3(t + 1) = x1(t) ⊕ x2(t)
for the eight possible initial conditions, i.e. (i) x1(0) = x2(0) = x3(0) = 0,
(ii) x1(0) = 0, x2(0) = 0, x3(0) = 1, (iii) x1(0) = 0, x2(0) = 1, x3(0) = 0,
(iv) x1(0) = 1, x2(0) = 0, x3(0) = 0, (v) x1(0) = 0, x2(0) = 1, x3(0) = 1,
(vi) x1(0) = 1, x2(0) = 0, x3(0) = 1, (vii) x1(0) = 1, x2(0) = 1, x3(0) = 0,
(viii) x1(0) = 1, x2(0) = 1, x3(0) = 1. Which of these initial conditions are
fixed points?
Problem 90. The Cantor series approximation is defined as follows.
For arbitrary chosen integers n1, n2, . . . (equal or larger than 2), we can
approximate any real number r0 as follows
xj = integer part(rj), j = 0, 1, 2, . . .
rj+1 = (rj − xj)nj
and
r0 ≈ x0 +
N
j=1
xj
n1n2 · · · nj
.
The approximation error is
En =
1
n1n2 · · · nN
.
Apply this approximation to r0 = 2/3 and the golden mean number with
nj = 2 for all j and N = 4.
Problem 91. Show that the full-adder can be built with nine NAND-
gates. Give the circuit. Describe the circuit using multiexpression program-
ming.
Problem 92. A boolean function f : { 0, 1 }⊗n
→ { 0, 1 }⊗m
(m ≥ n) is
periodic with period p with respect to bitwise modulo 2 addition, i.e. for
all x we have
f(x) = f(x + p).

Give an example of such a periodic boolean function.
Problem 93. Let x = x1x2 . . . xn, y = y1y2 . . . yn, z = z1z2 . . . zn be
three n-bit binary strings. Let ⊕ denote bitwise addition modulo 2, i.e.
x ⊕ y = z ⇒ for all k, zk = xk + yk mod 2
We deﬁne a scalar product of x and y by
x • y := (x1 · y1) + (x2 · y2) + · · · + (xn · yn) mod 2.
Show that this binary scalar product • is distributive over bitwise modulo
2 addition ⊕, i.e.
(x ⊕ y • z = (x • z) ⊕ (y • z).
Problem 94. Let x ∈ { 0, 1 }. Calculate
f1(x, y, z) = (x·(y+z)), f2(x, y, z) = (x·(y⊕z), f3(x, y, z) = (x+(y⊕z)).
Problem 95. Let n ∈ N0. Find the number of binary words consisting
of (n + 1) 0’s and n 1’s that can be formed such that the number of 0’s in
every subword, when read from far left to righ, is greater than the number
of 1’s in it. For n = 0 we have 0. For n = 1 we have 001.
Problem 96. Consider the unit cube with the 8 vertices
(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)
(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1).
Find a Hamilton cycle starting from (0, 0, 0).
Problem 97. Let x, y, a, b ∈ {0, 1}. Consider the function
f(x, y, a, b) =
1 if x · y = a ⊕ b
0 otherwise
where · is the AND operation and ⊕ is the XOR operation. Write a C++
program that ﬁnds the function f for all 16 possible inputs utilizing the
bitset class of the standard template library. The function plays a role
for the CHSH game.
Problem 98. Let a, b, c ∈ {0, 1} and · be the AND-operation, + be the
OR-operation, ⊕ be the XOR-operation.

(i) Find all solutions of
a · b · c = a + b + c.
(ii) Find all solutions of
a · b · c = a ⊕ b ⊕ c.
(iii) Find all solutions of
a + b + c = a ⊕ b ⊕ c.

Chapter 3
Binary Matrices
A binary matrix (also called (0, 1) matrix) is an m×n matrix in which each
entry is either zero or one. For example
0 1 0 1
1 1 0 0
is a 2×4 binary matrix. The number of m×n binary matrices is obviously
2mn
. In most cases operations on binary matrices are deﬁned in terms of
modular arithmetic mod 2. This means the elements are treated as elements
of the Galois ﬁeld GF(2) = Z2. Thus 0 ⊕ 0 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1,
1 ⊕ 1 = 0. For example in standard addition we have, for example
1 1
0 1
+
0 1
1 0
=
1 2
1 1
but in modular arithmetic mod 2 we have
1 1
0 1
⊕
0 1
1 0
=
1 0
1 1
If we have two binary matrices of the same size, then the Hadamard product
(also called Schur product) would be an entry-wise AND-operation. For
example
1 0
1 0
•
1 1
0 1
=
1 0
0 0
.
The n × n permutation matrices are binary matrices, all of whose columns
and rows have each exactly one nonzero element. An adjacent matrix in
57

graph theory is a matrix whose rows and columns represent the vertices
and whose entries represent the edges of the graph. The adjaceny matrix
of a simple undirected graph is a binary symmetric matrix with 0 diagonal.
Problem 1. (i) Find all 2 × 2 binary matrices.
(ii) Find the determinant of all these matrices with the underlying field R.
(iii) In modular arithmetic mod 2 we define for a binary 2 × 2 matrix
det 2(A) := (a11 · a22) ⊕ (a12 · a21)
where · is the AND-operation and ⊕ the XOR operation. Find det 2(A) for
all 2 × 2 binary matrices.
Problem 2. Write a C++ program that generates all m × n binary
matrices.
Problem 3. Write a C++ program that finds the determinant of an n×n
binary matrix over the field Z2.
Problem 4. Find all binary 2 × 2 matrices with no adjacent 1s (in either
columns or rows).
Problem 5. A special case of binary matrices are the permutation ma-
trices. An n × n permutation matrix contains exactly one 1 in each row
and column. The other entries are 0. Each permutation matix has an in-
verse. The determinant of a permutation matrix is either +1 or −1. The
permutation matrices form a group under matrix multiplication with the
underlying field R. Find all 3 × 3 permutation matrices. Calculate the
matrix products of these matrices with the underlying field R and with the
underlying field Z2.
Problem 6. Consider the bitstrings 00, 01, 10, 11 and the XOR-operation
⊕. We define
(a1a0) ⊕ (b1b0) = (a1 ⊕ b1)(a0b0).
Thus we have the (group) table
⊕ 00 01 10 11
00 00 01 10 11
01 01 00 11 10
10 10 11 00 01
11 11 10 01 00
Consider the 2 × 2 matrices
I2 =
1 0
0 1
, A =
1 0
0 −1
, B =
−1 0
0 1
, C =
−1 0
0 −1

Binary Matrices 59
and matrix multiplication. Find the group table. Show that the two groups
are isomorphic.
Problem 7. Consider the bistrings 00, 01, 10, 11 and the rotation oper-
ation with
R(00) = 00, R(01) = 10, R(10) = 01, R(11) = 11.
Now the bistrings can be mapped into 2 × 2 matrices
00 →
1 0
0 1
= I2, 01 →
1 0
0 −1
, 10 →
−1 0
0 1
, 11 →
−1 0
0 −1
.
How can the rotation operation be implemented using an invertible 2 × 2
matrix.
Problem 8. Extend the previous problem to bitstrings of length 3, i.e.
000, 001, 010, 011, 100, 101, 110, 111 and the rotation
R(000) = 000, R(001) = 100, R(010) = 010, R(011) = 101
R(100) = 010, R(101) = 101, R(110) = 011, R(111) = 111
The map into the 3 × 3 diagonal matrices
000 →


1 0 0
0 1 0
0 0 1

 = I3, 001 →


1 0 0
0 0 1
0 0 −1


010 →


1 0 0
0 −1 0
0 0 1

 , 011 →


1 0 0
0 −1 0
0 0 −1

 .
100 →


−1 0 0
0 1 0
0 0 1

 = I3, 101 →


−1 0 0
0 1 0
0 0 −1


110 →


−1 0 0
0 −1 0
0 0 1

 , 111 →


−1 0 0
0 −1 0
0 0 −1

 .
Problem 9. Consider the Pauli spin matrices
σ0 = I2 =
1 0
0 1
, σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
, σ3 =
1 0
0 −1

where we include the 2×2 identity matrix. We can associate binary numbers
with a Pauli spin matrix (including the identity matrix I2) σj (j = 0, 1, 2, 3)
via a two-dimensional binary vector r(σj) with
r(σ0) = (0 0), r(σ1) = (1 0), r(σ2) = (1 1), r(σ3) = (0 1).
The ﬁrst and second entries of this bit vector are written as r1(σj) and
r2(σj), respectively. Show that given the binary representation of a Pauli
matrix (including the identity matrix) we can recover the matrix via
σj = ir1(σj )r2(σj )
σ
r1(σj )
1 σ
r2(σj )
3 .
Problem 10. Given two binary matrices A and B. Show that the Kro-
necker product A ⊗ B is also a binary matrix.
Problem 11. Given two binary matrices A and B. Show that the direct
sum A ⊕ B is also a binary matrix.
Problem 12. For a 2 × 2 binary matrix
A =
a11 a12
a21 a22
, ajk ∈ { 0, 1 }
we deﬁne the determinant as
det A = (a11 · a22) ⊕ (a12 · a21)
where · is the AND-operation and ⊕ is the XOR-operation.
(i) Find the determinant for the following 2 × 2 matrices
1 0
0 1
,
0 1
1 0
,
1 1
0 1
,
0 1
1 1
,
1 0
1 1
,
1 1
1 0
.
(ii) Find the determinant for the following 2 × 2 matrices
0 0
0 0
,
1 0
0 0
,
0 1
0 0
,
1 0
0 0
,
0 0
0 1
1 1
0 0
,
1 0
1 0
,
0 0
1 1
,
0 1
0 1
,
1 1
1 1
.
Problem 13. The determinant of a 3 × 3 matrix
A =


a11 a12 a13
a21 a22 a23
a31 a32 a33



Binary Matrices 61
is given by
det A = a11a22a33 + a12a23a31 + a13a21a32
−a13a22a31 − a11a23a32 − a12a21a33.
For a binary matrix B we replace this expression by
det B = (b11 · b22 · b33) ⊕ (b12 · b23 · b31) ⊕ (b13 · b21 · b32)
⊕(b13 · b22 · b31) ⊕ (b11 · b23 · b32) ⊕ (b12 · b21 · b33) .
(i) Calculate the determinant for the binary matrices


1 0 0
0 1 0
0 0 1

 ,


1 1 1
0 1 1
0 0 1

 .
(ii) Calculate the determinant for the binary matrices


1 1 0
1 1 0
0 0 0

 ,


1 0 0
1 0 0
1 0 0

 ,


1 0 1
0 1 0
1 0 1

 .
Problem 14. The finite field GF(2) consists of the elements 0 and 1 (bits)
which satisfies the following addition (XOR-operation) and multiplication
(AND-operation) tables
⊕ 0 1
0 0 1
1 1 0
· 0 1
0 0 0
1 0 1
Find the determinant of the binary matrices
A =


1 0 1
0 1 0
1 0 1

 , B =


1 1 1
0 1 1
0 0 1

 .
Problem 15. An (n, k) binary linear block code is a k-dimensional sub-
space of the n-dimensional vector space
Vn := { (b0, b1, . . . , bn−1) : ∀ bj bj ∈ GF(2) }.
Here n is called the length of the code and k the dimension. An (n, k)
binary linear block code can be specified by any set of k linear independent
codewords b0, b1, . . . , bk−1. One arranges the k codewords into a k × n

binary matrix G. This matrix G is called a generator matrix for all the
codewords C. Consider the generator matrix (n = 6, k = 3)
G =


1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1

 .
Find all the codewords.
Problem 16. Consider the generator matrix ((7,4) Hamming code)
G =



1 0 0 0 0 1 1
0 1 0 0 1 0 1
0 0 1 0 1 1 0
0 0 0 1 1 1 1


 .
Find all the codewords.
Problem 17. A parity check for codewords C is an equation of the form
(a0 · b0) ⊕ (a1 · b1) ⊕ · · · ⊕ (an−1 · bn−1) = 0
which has to be satisﬁed for any b = (b0, b1, . . . , bn−1) ∈ C. The collection
of all vectors a = (a0, a1, . . . , an−1) forms a vector subspace of Vn. It is
denoted by C⊥
and is called the dual code of C. The dimension of C⊥
is n − k and C⊥
is an (n, n − k) binary linear block code. Any generator
matrix of C⊥
is a parity-check matrix for C and is denoted by H. Consider
the generator matrix (n = 6, k = 3)
G =


1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1

 .
Find the parity-check matrix H.
Problem 18. A (7, 4) Hamming code can be generated by
G =



1 0 0 0 0 1 1
0 1 0 0 1 0 1
0 0 1 0 1 1 0
0 0 0 1 1 1 1


 .
Show that the sixteen codewords are
0000000
0001111
0010110

Binary Matrices 63
0011001
0100101
0101010
0110011
0111100
1000011
1001100
1010101
1011010
1100110
1101001
1110000
1111111
Show the parity check matrix is
H =


0 1 1 1 1 0 0
1 0 1 1 0 1 0
1 1 0 1 0 0 1

 .
Problem 19. Study the Lie algebra s (2, F), where charF = 2.

Chapter 4
Reversible Logic Gates
Reversible logic gates are gates that function in both directions. CMOS
implementations of such gates are designed. A special pass transistor logic
family is applied: reversible MOS. Many diﬀerent reversible logic gates are
candidates as a universal building blocks. The controlled NOT gate, the
Fredkin gate can be implemented. They dissipate very little energy. Owing
to their use of reversible truth tables, they are even candidates for zero-
power computing. Circuit synthesis take advantage of mathematical group
theory. Algorithms have been developed for the synthesis of arbitrary re-
versible circuits.
Reversible circuits are applicable to nanotechnolgy, quantum and optical
computing as well as reducing power in CMOS implementation. In adia-
batic circuits, current is restricted to ﬂow across devices with low voltage
drop and the energy stored on their capacitors is recycled. One uses re-
versible energy recovery gate capable to realize functions { AND, OR } or
{ NAND, NOR }.
64

Reversible Logic Gates 65
Problem 1. For reversible gates the following boolean expression plays
an important role
(a11 · a22) ⊕ (a12 · a21)
where a11, a12, a21, a22 ∈ { 0, 1 }. It could be considered as the determinant
of the 2 × 2 binary matrix
a11 a12
a21 a22
.
Find the inverse of the matrix when it exists.
Problem 2. Find the truth table for the boolean function
f(a, a , b, b ) = (a · b ) ⊕ (a · b) .
Problem 3. Consider a two input gate (x, y) / two output gate (x , y )
given by
x = a · x ⊕ b · y ⊕ c
y = a · x ⊕ b · y ⊕ c
where a, b, a , b , c, c ∈ { 0, 1 }.
(i) Let a = 0, b = 1, a = 1, b = 0 and c = c = 0. Find the output (x , y )
for all possible inputs (x, y). Is the transformation invertible?
(ii) Let a = 1, b = 1, a = 1, b = 1 and c = c = 0. Find the output (x , y )
for all possible inputs (x, y). Is the transformation invertible?
Problem 4. The Feynman gate is a 2 input/2 output gate given by
x1 = x1
x2 = x1 ⊕ x2
(i) Give the truth table for the Feynman gate.
(ii) Show that copying can be implemented using the Feynman gate.
(iii) Show that the complement can be implemented using the Feynman
gate.
(iv) Is the Feynman gate invertible?
Problem 5. Consider the Toﬀoli gate
T : {0, 1}3
→ {0, 1}3
, T(a, b, c) := (a, b, (a · b) ⊕ c)
where ¯a is the NOT operation, + is the OR operation, · is the AND oper-
ation and ⊕ is the XOR operation.

1. Express NOT(a) exclusively in terms of the TOFFOLI gate.
2. Express AND(a, b) exclusively in terms of the TOFFOLI gate.
3. Express OR(a, b) exclusively in terms of the TOFFOLI gate.
4. Show that the TOFFOLI gate is invertible.
Thus the TOFFOLI gate is universal and reversible (invertible).
Problem 6. Consider the Fredkin gate
F : {0, 1}3
→ {0, 1}3
, F(a, b, c) := (a, a · b + ¯a · c, a · c + ¯a · b)
ation and ⊕ is the XOR operation.
1. Express NOT(a) exclusively in terms of the FREDKIN gate.
2. Express AND(a, b) exclusively in terms of the FREDKIN gate.
3. Express OR(a, b) exclusively in terms of the FREDKIN gate.
4. Show that the FREDKIN gate is invertible.
Thus the FREDKIN gate is universal and reversible (invertible).
Problem 7. The Toffoli gate T(x1, x2; x3) has 3 inputs (x1, x2, x3) and
three outputs (y1, y2, y3) and is given by
(x1, x2, x3) → (x1, x2, x3 ⊕ (x1 · x2))
where x1, x2, x3 ∈ { 0, 1 }, ⊕ is the XOR-operation and · the AND-operation.
Problem 8. A generalized Toffoli gate T(x1, x2, . . . , xn; xn+1) is a gate
that maps a boolean pattern (x1, x2, . . . , xn, xn+1) to
(x1, x2, . . . , xn, xn+1 ⊕ (x1 · x2 · . . . · xn))
where ⊕ is the XOR-operation and · the AND-operation. Show that the
generalized Toffoli gate includes the NOT-gate, CNOT-gate and the original
Toffoli gate.
Problem 9. The Fredkin gate F(x1; x2, x3) has 3 inputs (x1, x2, x3) and
three outputs (y1, y2, y3). It maps boolean patterns
(x1, x2, x3) → (x1, x3, x2)

if and only if x1 = 1, otherwise it passes the boolean pattern unchanged.
Problem 10. The generalized Fredkin gate F(x1, x2, . . . , xn; xn+1, xn+2)
is a gate is the mapping of the boolean pattern
(x1, x2, . . . , xn, xn+1, xn+2) → (x1, x2, . . . , xn, xn+2, xn+1)
if and only if the boolean product x1 · x2 · . . . · xn = 1 (· is the bitwise AND
operation), otherwise the boolean pattern passes unchanged. Let n = 2
and (x1, x2, x3, x4) = (1, 1, 0, 1). Find the output.
Problem 11. Is the gate (a, b, c ∈ { 0, 1 })
(a, b, c) → (a, a · b ⊕ c, a · c ⊕ b)
reversible?
Problem 12. Prove that the Fredkin gate is universal. A set of gates is
called universal if we can build any logic circuits using these gates assuming
bit setting gates are given.
Problem 13. The half-adder is given by
S = A ⊕ B
C = A · B.
Construct a half-adder using two Toﬀoli gates.
Problem 14. Consider the 3-input/3-output gate given by
x1 = x1
x2 = x1 ⊕ x2
x3 = x1 ⊕ x2 ⊕ x3.
(i) Give the truth table.
(ii) Is the transformation invertible.
x1 = x1
x2 = x1 ⊕ x2
x3 = x3 ⊕ (x1 · x2).

(ii) Is the gate invertible?
x1 = x1
x2 = x2
x3 = x3
x4 = x4 ⊕ x1 ⊕ x2 ⊕ x3 .
Problem 17. Show that one Fredkin gate
(a, b, c) → (a, a · b + a · c, a · c + a · b)
is sufficient to implement the XOR gate. Assume that either b or c are
available.
Problem 18. Show that the map f : {0, 1}3
→ {0, 1}3
abc xyz
000 -> 000
100 -> 100
010 -> 101
110 -> 011
001 -> 001
101 -> 010
011 -> 110
111 -> 111
is invertible. The map describes a reversible half-adder. If c = 0, then x is
the first digit of the sum a + b and y is the carry bit. If c = 1, then z is the
first digit of the sum a + b + c and y is the carry bit.
x1 = x1 ⊕ x3
x2 = x1 ⊕ x2
x3 = (x1 · x2) ⊕ (x1 · x3) ⊕ (x2 · x3).

x1 = x1 ⊕ x3
x2 = x1 ⊕ x2
x3 = (x1 + x2) ⊕ (x1 + x3) ⊕ (x2 + x3).
x1 = x1 ⊕ x3
x2 = x2 ⊕ x3 ⊕ (x1 · x2) ⊕ (x2 · x3)
x3 = x1 ⊕ x2 ⊕ x3
x4 = x4 ⊕ x3 ⊕ (x1 · x2) ⊕ (x2 · x3) .
Problem 22. The NOT, AND and OR gate form a universal set of oper-
ations (gates) for boolean algebra. The NAND operation is also universal
for boolean algebra. However these sets of operations are not reversible sets
of operations. Consider the Toﬀoli and Fredkin gates
TOFFOLI : {0, 1}3
→ {0, 1}3
, TOFFOLI(a, b, c) = (a, b, (a · b) ⊕ c)
FREDKIN : {0, 1}3
→ {0, 1}3
, FREDKIN(a, b, c) = (a, a·c+¯a·b, a·b+¯a·c)
ation and ⊕ is the XOR operations.
1. Express NOT(a) exclusively in terms of the TOFFOLI gate.
2. Express NOT(a) exclusively in terms of the FREDKIN gate.
3. Express AND(a,b) exclusively in terms of the TOFFOLI gate.
4. Express AND(a,b) exclusively in terms of the FREDKIN gate.
5. Express OR(a,b) exclusively in terms of the TOFFOLI gate.
6. Express OR(a,b) exclusively in terms of the FREDKIN gate.
7. Show that the TOFFOLI gate is reversible.
8. Show that the FREDKIN gate is reversible.
Thus the TOFFPLI and FREDKIN gates are eachuniversal and reversible
(invertible).

Chapter 5
Floating Point Numbers
In C, C++, Java and Perl we have two types of floatimg point number,
namely float and double. For the data type float with 32 bits the value
is stored as we have
sign bit, 8 bit exponent, 23 bit mantissa
i.e.
byte 1 byte 2 byte 3 byte 4
SXXX XXXX XMMM MMMM MMMMM MMMM MMMM MMMM
For the data type double we have 64 bits in C++. The value of double is
stored as
sign bit, 11 bit exponent, 52 bit mantissa
This means
byte 1 byte 2 byte 3 byte 4 byte 8
SXXX XXXX XXXX MMMM MMMM MMMM MMMM MMMM ... MMMM MMMM
We can also do bitwise manipulations of floating point numbers, for exam-
ple on the data type
Problem 1. Write a C++ progran that changes a bit in the floating
point number double using , is the bitwise OR | and the shift operation
<<.
70

Floating Point Numbers 71
// floating.cpp
#include <iostream>
int main(void)
{
double x = 3.14159;
int* p = (int*) &x;
*(p+1) = *(p+1)^(1<<31); // short cut *(p+1) ^= (1<<31)
cout << "x = " << x << endl;
return 0;
}
Note that the data type double takes 64 bit and the sign bit is at bit
position 63. The data type int takes 32 bits, << is the shift left operation
and ^ is the XOR operation.
Problem 3. C++ and C provide six operators for bit manipulations.
These may only be applied to integral operands, that is char, short, int
and long,whether signed or unsigned.
& bitwise AND
| bitwise OR
^ bitwise XOR
~ one’s complement (unary)
<< left shift
>> right shift
Thus these operators cannot be applied to double and float. Write a
C++ program that can do the bitwise operations AND, OR, XOR, NOT
on the data type double.
Problem 4. Write a C++ program using the bitset class which con-
verts the memory representation of an arbitrary data type to a bitset of
appropiate size and vice versa. In other words store the sizeof(T) bytes
for an instance of T in a bitset. Since float and int are both 32-bit (on 32
bit architectures) use the program to ﬁnd an integer with the same binary
representation as a given value of type float.
Problem 5. What is ouput of the following program
// btod.cpp
#include <bitset>
#include <cassert>
#include <iostream>

template <class T> class size;
template <> class size<char>
{
public:
static const size_t bits = numeric_limits<char>::digits
+ ((numeric_limits<char>::is_signed) ? 1:0);
static const size_t chars = 1;
};
template <class T> class size
{
public:
static const size_t bits = sizeof(T) * size<char>::bits;
static const size_t chars = sizeof(T);
};
template <> class size<int>
{
public:
static const size_t bits = numeric_limits<int>::digits
+ ((numeric_limits<int>::is_signed) ? 1:0);
static const size_t chars = sizeof(int);
};
template <class T, const size_t n>
T map_bitset_to(const bitset<n> &b)
{
char *cp;
T d;
size_t i, j, k = 0;
assert(n >= size<T>::bits);
for(i=0, cp=(char*)&d; i < size<T>::chars; i++, cp++)
{
*cp = 0;
for(j=0; j < size<char>::bits; j++, k++)
if(b.test(k)) *cp |= (1<<j);
}
return d;
}
int main(void)
{

Floating Point Numbers 73
bitset<size<double>::bits>
b(string("0111111111111111111111111111111111111111111111110000000000000000"));
cout << map_bitset_to<double>(b) << endl;
return 0;
}
Problem 6. What is the output of the following program
// double.cpp
#include <iostream>
#include <limits>
struct double_bits {
unsigned long mantissa : 52; /* need to modify for 32-bit */
unsigned int exponent : 11;
unsigned int sign : 1;
};
union udouble {
double d;
double_bits bits;
};
int main(void)
{
union udouble u;
cout.precision(numeric_limits<double>::digits);
u.d = 3.14;
cout << u.d << endl;
cout << u.bits.sign << "t" << u.bits.exponent
<< "t" << u.bits.mantissa << endl;
u.bits.sign = 1;
u.bits.exponent++;
u.bits.mantissa--;

return 0;
}

Chapter 6
Cellular Automata
Problem 1. We consider a one-dimensional ring of N sites labelled se-
quentialy by the index i staring from zero, i.e. i = 0, 1, . . . , N − 1. We
impose periodic boundary conditions, i.e. N ≡ 0. Each site i can take the
values ai = 0 or ai = 1. Let ai evolve as a function ai(t) of discrete time
steps t = 0, 1, 2, . . . according to the map
ai(t + 1) = (ai−1(t) + ai+1(t)) mod 2.
Since the map involves the sum ai−1 + ai+1 mod 2, it is a nonlinear map.
Give a C++ implementation for this map.
Problem 2. Consider the initial value problem of the two-dimensional
cellular automata
xi,j(t+1) = (xi+1,j(t)+xi,j+1(t))⊕xi−1,j(t)⊕xi,j−1(t)⊕xi,j(t), t = 0, 1, 2, . . .
where i, j are the coordinates of the pixel and −M ≤ i ≤ +M, −M ≤ j ≤
+M and M is a positive integer. Here + denotes the OR-operation and
⊕ the XOR-operation. Periodic boundary conditions are imposed. Write
a C++ program that implements this cellular automate. Apply the bitset
class of the Standard Template Library.
Problem 3. Consider the initial value problem for the cellular automata
xj(t + 1) = xj−1(t) ⊕ xj(t) ⊕ xj+1(t), t = 0, 1, 2, . . .
where j = {−15, −14, . . . , 0, . . . , 16} and cyclic boundary condition. The
initial values at t = 0 are x0(0) = 1 otherwise 0. Write a C++ program
75

that implements this cellular automata using the bitset class. Display the
results.
Problem 4. Consider the one-dimensional cellular automata with rule
62. Find the map.
Problem 5. (i) Consider the one-dimensional elementary cellular au-
tomata
111 110 101 100 011 010 001 000
0 0 0 1 0 1 1 0
which is rule 94. Show that starting with a random initial condition the
system settles down in a stationary state.
(ii) Consider the one-dimensional elementary cellular automata
111 110 101 100 011 010 001 000
0 1 1 1 1 0 1 0
which is rule 50. Show that starting with a random initial condition the
system settles down in a state of period 2.

Chapter 7
String Manipulations
Problem 1. Let s1, s2 be two strings. Write a C++ function squeeze(s1,s2)
that deletes each character in s1 that matches any character in the string
s2.
Problem 2. Let s1 and s2 be two strings. Write a function any(s1,s2),
which returns the first location in the string s1 where any character from
the string s2 occurs, or -1 if s1 contains no characters from string s2.
Problem 3. A substring of a string s consists only of consecutive char-
acters from s. A subsequence of a string s is simply an (ordered) sequence
of characters (not necessarily consecutive) from s. For example, if s =
”ATTGCTA”, then ”AGCA” and ”ATTA” are subsequences, but there
are not substrings. A common subsequence of two strings is a subsequence
of both a them. For example, consider ”ATCTGAT” and ”TGCATA”.
Then ”TCTA” is a common subsequence of the two strings. The longest
common subsequence problem is given two strings s and t find the longest
common subsequence of s and t.
(i) Write a C++ program that finds the longest common subsequence of
two given strings.
(ii) Write a C++ program that finds the longest common substrings of two
given strings.
Problem 4. Use the string class of C++ and declare an array of strings
string* sa = new string[N];
77

Fill this array with strings and concatenate them.
Problem 5. Consider two strings of the same length. Write a C++
program using the string class that finds the Hamming distance.
Problem 6. Given two strings not necessarily of the same length. Write
a C++ program that implements the Levenshtein distance (edit distance).
Problem 7. Let A be a finite alphabet and let A∗
be the set of all finitely
long words that can be a written in this alphabet. One denotes by AN
and
AZ
be the sets of all semi-infinite and infinite sequences of letters from A.
Let f be a map from A → A∗
that associates with any letter in A a finite
word.
(i) Let A = {a, b} and the map (Fibonacci sequence)
a → f(a) = ab, b → f(b) = a.
Give a C++ implementation.
(ii) Let A = {a, b} and the map (Thue-Morse sequence)
a → f(a) = ab, b → f(b) = ba.
(iii) Let A = {a, b} and the map (period-doubling sequence)
a → f(a) = ab, b → f(b) = aa.
(iv) Let A = {a, b, c, d} and the map (Rudin-Shapiro sequence)
a → f(a) = ac, b → f(b) = dc, c → f(c) = ab, d → f(d) = db.
Problem 8. Consider the recursion
Sk+1 = Sm
k · Sn
k−1, k ≥ 2, S1 = B, S2 = A
where · stands for concatenation and n, m are positive integers. One defines
n = 1, m = 1 Golden mean or Fibonacci chain
n = 1, m = 2 Silver mean
n = 1, m = 3 Bronze mean
n = 2, m = 1 Copper mean
n = 3, m = 1 Nickel mean

String Manipulations 79
Give a C++ implementation utilizing the string class. Count the numbers
of A’s and B’s at each step.

Bibliography 81
Bibliography
Books
Steeb, W.-H.
Introduction to Assembly Language and C++
International School for Scientﬁc Computing, 2008
Hardy Y. and Steeb W.-H.
Classical and Quantum Computing with C++ and Java Simulations
Birkhauser-Verlag, Boston (2002)
Steeb W.-H.
Matrix Calculus and Kronecker Product with Applications and C++ Pro-
grams
World Scientiﬁc Publishing, Singapore (1997)
Mendelson E.
Boolean Algebra and Switching Circuits
Schaum’s Outline Series (1970)
Warren H. S.
Hacker’s Delight
Addison-Wesly, Boston (2003)
Papers

Index
AND-gate, 2
Associative, 5
Binary dot product, 7
Binary linear block code, 61
Binary matrices, 61
Boolean derivatives, 52
Boolean function, 1
Cantor sequence, 41
Determinant, 65
Direct sum, 60
Disjunctive normal form, 8
Doz function, 33
Dual code, 62
Encoder, 6
Encryption, 34
Feynman gate, 65
Fibonacci sequence, 78
Fredkin gate, 66, 67
Full adder, 6, 7, 47, 54
Galois field, 44
Generalized Fredkin gate, 67
Generalized Toffoli gate, 66
Generator matrix, 62
Gray code, 35
Hadamard vectors, 49
Half-adder, 67
Hamilton cycle, 55
Hamming code, 62
Hamming distance, 8, 32, 78
J-K flip-flop, 41
Kronecker product, 60
Levenshtein distance, 78
Majority logic gate, 36
Multiexpression programming, 47, 54
NAND-gate, 2
NOR-gate, 2
NOT-gate, 2
OR-gate, 2
Parity check, 62
Parity-check matrix, 62
Period-doubling sequence, 78
Permutation matrices, 58
Precedence, 2
Reed-Muller expansion, 40
Rudin-Shapiro sequence, 78
Scalar product, 32
Shannon entropy, 46
Shannon’s expansion, 40
Shift map, 44
Subsequence, 77
Substring, 77
Sum of products, 6
Thue-Morse sequence, 42, 52, 78
Toffoli gate, 65, 66
Truth table, 1
Universal gates, 2
XOR-gate, 2
82

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