4th International Conference
SCIENCE AND HIGHER EDUCATION IN FUNCTION OF
SUSTAINABLE DEVELOPMENT
SED 2011
7th and 8th of October 2011, Uzice, Serbia
BOOLEAN EQUATIONS AND BOOLEAN INEQUATIONS
Dragić Banković
Faculty of Science, Kragujevac, Serbia, dragic@kg.ac.rs
Summary: In this paper the main results from the fields of Boolean equations
and Boolean inequations
will be presented. Some new results from the generalized systems of equations will be described as well.
Keywords: Boolean function, Boolean equation, Boolean inequation, general solution
1. INTRODUCTION
Boolean equations have been studied from the end of the nineteenth century till today. Many authors researched
Boolean equations. Particularly, great contribution to the field of Boolean equation was given by Löwenheim.
There are various forms of the solutions of Boolean equation. The basic facts of Boolean equations and their
solutions can be found in Rudeanu’s book [7]. The results published after 1974 are presented in [8]. Boolean
inequations were considered first by Schröder [8]. The newer results can be found in [2] and [3].
2. BOOLEAN FUNCTIONS
Let
be a Boolean algebra and n be a natural number.
Definition 1. Let
then
=
···
In the sequel
Then
.
and
( ,…,
)
will be omitted.
Theorem 1. The function
form
where
. If
is Boolean if and only if it can be written in the canonical disjunctive
means the union over the all
For example, a function
.
can be written as
.
3. BOOLEAN EQUATIONS
A Boolean equation in unknowns over a Boolean algebra is the equation of the form
are Boolean function. Similarly, Boolean subsumption in n unknowns is of the form
where
Theorem 2 [7]. Every Boolean equation, subsumption or system of Boolean equations/subsumption is
equivalent to a single Boolean equation of the form
To solve a Boolean equation
determine the set
Theorem 3 [7]. Let
if and only if
means to determine all
. Note that
such that
holds i.e. to
be a Boolean functions. The equation f(X) = 0 is consistent (has a solution)
where
means the product over the all
Let
Definition 2. Let
scalar form
if and only if, for every
be Boolean functions and
The formula
or in
defines the general solution of the consistent Boolean equation
,
One can prove that the previous formula is equivalent to
Theorem 4 [6]. Assume that the equation
is consistent. Then
(a)
(b)
for all
Note that the formula
Theorem 5 [4]. Let
for all
represents the general solution of
be a Boolean function. If
, i.e. the formula
then
represents the general solution of
Method of successive eliminations is very popular method for solving Boolean equations. The main idea for the
method of successive eliminations can be presented using Boolean equation with two unknowns. Namely, the
equation with two unknowns
can be written as
(1)
The equation (1), by
Substituting
, is consistent if and only if
This
equation
by
Theorem
3,
is
and
consistent
if
its
general
and
only
solution
i.e.
if
is
in (1) we get the equation by , which can be easy solved.
Let
and
Theorem 6 [1]. Let
be a Boolean function. If
)
where, for every
]
is consistent then, for every
,
is a permutation of
4. GENERALIZED SYSTEMS OF BOOLEAN EQUATIONS
Many problems of optimization in databases reduce to the solution of systems consisting of Boolean equations
and Boolean inequations. Generally speaking, when we refer to a system of equations of any kind, we
understand that those equations are linked by conjunction. The idea of a generalized system of equations is to
link those equations by any logical operation, not merely conjunction.
Definition 2. The generalized systems of Boolean equations (GSBE) over a Boolean algebra B are defined as
follows:
1. every Boolean equation
is a GSBE;
2. the negation, conjunction and disjunction of any GSBE’s is a GSBE;
3. every GSBE is obtained by applying rules 1. and 2. finitely many times.
Unfortunately, the problem of solving GSBE in an arbitrary Boolean algebra is far from being solved in a
satisfactory manner. There are partial results only.
Let
be a Boolean function. The relation
is called Boolean inequation.
Theorem 7 [2]. Let
e a Boolean function. If
for all
and
then
.
Example 1. Let
that satisfies
. Consider the following Boolean inequation
In accordance with Theorem 7, we get
. It is obvious
Taking
the formula
gives
or
. One can check that the set of all the
solutions of the inequation
is
. Note that the formula
, obtained by Theorem 7,
doesn't determine all the solutions of the inequation
. One can prove that there is not Boolean
function
which determine all the solutions of
To describe all the solutions of an inequation
the Boolean function with two variables.
, where
is a Boolean function, we shall use
Remark. Note that
In accordance with Remark 1, to solve a Boolean inequation
.
Let
Theorem 8 [7]. Let
if and only if
, where
means to determine all
such that
. One can prove that
be a Boolean function . The inequation
is consistent (has a solution)
≠ 0.
Theorem 9 [2]. Let
. Then, for every
Theorem 10 [2] Let
surjective.
. The function
,
defined by
is
Remark 2. Theorem 9 shows that the values of the function g "covers" the set B. Some elements of are the
solutions of the inequation
, while the rest of them are the solutions of the equation
. The formula
determines all the solutions of Boolean inequation
and all the solutions of Boolean equation
. Namely, if we take
we get
i.e. we get the formula which determines all the solutions of equation
, by
Theorem 4. For
, we get the formula which determines all the solutions of inequation
, by
Theorem 9.
Theorem 11 [2] Let
be a Boolean function. If
is consistent then, for every
where
represents the general solution of equation
In order to solve inequation
Theorem 12 [3]. Let
has a solution. Let
for every
,
Theorem 13. Let
.
, the various form of general solution of
can be used.
be Boolean functions. Suppose that the system
expresses the general solution of equation
. Then,
. Then
The proof of this theorem is based on the following equivalences
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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