# Sets, Lattices, and Classes of Logic Functions

• Christian Posthoff
• Bernd Steinbach
Chapter

## Abstract

Many applications use not only single logic functions but sets of them. It is beneficial when not each function of such a set must be computed separately. A strongly simplified computation is possible when the sets of functions satisfy certain properties. We explore sets of functions that satisfy the rules of an equivalence relation as well sets having the structure of a lattice. Very often used are partially defined functions which describe a lattice of functions that is isomorphic to a Boolean Algebra. These lattices can and must be generalized to express the results of derivative operation of a lattice of logic functions. The Boolean Differential Calculus has been extended for all derivative operations of such generalized lattice of logic functions. The derivative operations of the Boolean Differential Calculus also support the solution of logic equations with regard to variables. These operations are used to specify conditions whether a logic equation can be solved with regard to selected variables or whether such a solution function is even uniquely determined. In general, we get lattices of logic functions as solution of a logic equation with regard to variables. These lattices and their relations to each other can also be calculated using derivative operations. As special application we explain systems of reversible functions. Sets of functions can also be determined by functional equations that contain unknown functions such that lattices of functions satisfy the given equation. More general, a Boolean differential equation can be used to describe an arbitrary set of logic functions. We present a basic algorithm that calculates the set of all solution lattices of a special type of Boolean differential equations and generalize this algorithm for arbitrary sets of logic functions as solution of a Boolean differential equation.

## References

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Steinbach, B., Posthoff, C.: Boolean Differential Equations. Morgan & Claypool Publishers, San Rafael (2013). ISBN: 978-1-6270-5241-2. https://doi.org/10.2200/S00511ED1V01Y201305DCS042
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Steinbach, B., Posthoff, C.: Derivative operations for lattices of Boolean functions. In: Proceedings Reed-Muller Workshop 2013, RM, Toyama, May 2013, pp. 110–119. https://doi.org/10.13140/2.1.2398.6568

© Springer International Publishing AG 2019

## Authors and Affiliations

• Christian Posthoff
• 1
• Bernd Steinbach
• 2
1. 1.Computing and Information TechnologyUniversity of the West Indies (retired)ChemnitzGermany
2. 2.Computer ScienceTU Bergakademie Freiberg (retired)ChemnitzGermany