Chapter 3
Derivative Operations for Classes C N
of Boolean Functions
Bernd Steinbach and Christian Posthoff
3.1 Introduction
Derivative operations for lattices of Boolean functions facilitate, e.g., the optimized
synthesis of combinational circuits by means of strong, weak, and vectorial bidecompositions [1, 7] without the need to manipulate each function separately.
This successful application of derivative operations for lattices of Boolean functions
leads to the question whether derivative operations can also be calculated for other
sets of Boolean functions without manipulating each function of the set separately.
From the solution of Boolean differential equations [5] it is known that the solution
of certain Boolean differential equations consists of special classes of Boolean
function. Hence, the exploration of derivative operations of classes of Boolean
functions is an interesting topic that may open the door for new applications in
the future. In this context a unique representation of classes of Boolean functions
is needed. The aim of this contribution is the finding of solutions for some of these
topics.
This contribution is organized as follows: We assume that the reader is familiar
with the Boolean algebra and the derivative operations of single Boolean functions;
if not, we refer to [1, 4, 8]. The explored classes C N of Boolean functions are
defined and explored in Sect. 3.2. Several theorems of Sect. 3.2 help to reduce the
effort to calculate all derivative operations for all Boolean functions of classes C N
in Sect. 3.3. We conclude this contribution in Sect. 3.4.
B. Steinbach ()
Institute of Computer Science, Freiberg University of Mining and Technology, Freiberg, Germany
e-mail: steinb@informatik.tu-freiberg.de
C. Posthoff
Department of Computing and Information Technology, The University of the West Indies, Saint
Augustine, Trinidad and Tobago
e-mail: christian@posthoff.de
© Springer Nature Switzerland AG 2020
R. Drechsler, M. Soeken (eds.), Advanced Boolean Techniques,
https://doi.org/10.1007/978-3-030-20323-8_3
51
Derivative Operations for Classes C N
of Boolean Functions
Bernd Steinbach and Christian Posthoff
3.1 Introduction
Derivative operations for lattices of Boolean functions facilitate, e.g., the optimized
synthesis of combinational circuits by means of strong, weak, and vectorial bidecompositions [1, 7] without the need to manipulate each function separately.
This successful application of derivative operations for lattices of Boolean functions
leads to the question whether derivative operations can also be calculated for other
sets of Boolean functions without manipulating each function of the set separately.
From the solution of Boolean differential equations [5] it is known that the solution
of certain Boolean differential equations consists of special classes of Boolean
function. Hence, the exploration of derivative operations of classes of Boolean
functions is an interesting topic that may open the door for new applications in
the future. In this context a unique representation of classes of Boolean functions
is needed. The aim of this contribution is the finding of solutions for some of these
topics.
This contribution is organized as follows: We assume that the reader is familiar
with the Boolean algebra and the derivative operations of single Boolean functions;
if not, we refer to [1, 4, 8]. The explored classes C N of Boolean functions are
defined and explored in Sect. 3.2. Several theorems of Sect. 3.2 help to reduce the
effort to calculate all derivative operations for all Boolean functions of classes C N
in Sect. 3.3. We conclude this contribution in Sect. 3.4.
B. Steinbach ()
Institute of Computer Science, Freiberg University of Mining and Technology, Freiberg, Germany
e-mail: steinb@informatik.tu-freiberg.de
C. Posthoff
Department of Computing and Information Technology, The University of the West Indies, Saint
Augustine, Trinidad and Tobago
e-mail: christian@posthoff.de
© Springer Nature Switzerland AG 2020
R. Drechsler, M. Soeken (eds.), Advanced Boolean Techniques,
https://doi.org/10.1007/978-3-030-20323-8_3
51