Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Acknowledgments
- Contributors
- Acronyms and Abbreviations
- Boolean Models and Methods in Mathematics, Computer Science, and Engineering
- Part I Algebraic Structures
- 1 Compositions and Clones of Boolean Functions
- 2 Decomposition of Boolean Functions
- Part II Logic
- Part III Learning Theory and Cryptography
- Part IV Graph Representations and Efficient Computation Models
- Part IV Applications in Engineering
2 - Decomposition of Boolean Functions
from Part I - Algebraic Structures
Published online by Cambridge University Press: 05 June 2013
- Frontmatter
- Contents
- Preface
- Introduction
- Acknowledgments
- Contributors
- Acronyms and Abbreviations
- Boolean Models and Methods in Mathematics, Computer Science, and Engineering
- Part I Algebraic Structures
- 1 Compositions and Clones of Boolean Functions
- 2 Decomposition of Boolean Functions
- Part II Logic
- Part III Learning Theory and Cryptography
- Part IV Graph Representations and Efficient Computation Models
- Part IV Applications in Engineering
Summary
Introduction
The basic step in functional decomposition of a Boolean function f : {0, 1}n ↦ {0, 1} with input variables N = {x1, x2, … xn} is essentially the partitioning of the set N into two disjoint sets A = {x1, x2, …, xp} (the “modular set”) and B = {xp+1, …, xn} (the “free set”), such that f = F(g(xA), xB). The function g is called a component (subfunction) of f, and F is called a composition (quotient) function of f. The idea here is that F computes f based on the intermediate result computed by g and the variables in the free set. More complex (Ashenhurst) decompositions of a function f can be obtained by recursive application of the basic decomposition step to a component function or to a quotient function of f.
Functional decomposition for general Boolean functions has been introduced in switching theory in the late 1950s and early 1960s by Ashenhurst, Curtis, and Karp [1, 2, 20, 23, 24]. More or less independent from these developments, decomposition of positive functions has been initiated by Shapley [36], Billera [5, 4], and Birnbaum and Esary [12] in several contexts such as voting theory (simple games), clutters, and reliability theory. However, the results in these areas are mainly formulated in terms of set systems and set operations.
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- Publisher: Cambridge University PressPrint publication year: 2010
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