Stasys Jukna and Igor Sergeev (2013), "Complexity of Linear Boolean Operators", Foundations and TrendsÂ® in Theoretical Computer Science: Vol. 9: No. 1, pp 1-123. http://dx.doi.org/10.1561/0400000063

© 2013 S. Jukna and I. Sergeev

Computational aspects of combinatorics and graph theory, Computational complexity, Computational Models and Complexity

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**In this article:**

How to compute a linear Boolean operator by a small circuit using only unbounded fanin addition gates? Because this question is about one of the simplest and most basic circuit models, it has been considered by many authors since the early 1950s. This has led to a variety of upper and lower bound argumentsâ€”ranging from algebraic (determinant and matrix rigidity), to combinatorial (Ramsey properties, coverings, and decompositions) to graph-theoretic (the superconcentrator method). We provide a thorough survey of the research in this direction, and prove some new results to fill out the picture. The focus is on the cases in which the addition operation is either the boolean OR or XOR, but the model in which arbitrary boolean functions are allowed as gates is considered as well.

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1. Introduction

2. General Upper Bounds

3. General Lower Bounds

4. Complexity of Some Basic Matrices

5. Complexity Gaps

6. Bounds for General Circuits

7. Conclusion and Open Problems

Acknowledgements

References

How to compute a linear Boolean operator by a small circuit using only unbounded fanin addition gates? Because this question is about one of the simplest and most basic circuit models, it has been considered by many authors since the early 1950s. This has led to a variety of upper and lower bound argumentsâ€”ranging from algebraic (determinant and matrix rigidity), to combinatorial (Ramsey properties, coverings, and decompositions) to graph-theoretic (the superconcentrator method).

*Complexity of Linear Boolean Operators* is the first thorough survey of the research in this area. The focus is on cases
where the addition operation is either the Boolean OR or XOR, but the model in which arbitrary Boolean functions are allowed as gates is
considered as well. The survey is intended for students and researchers in discrete mathematics and theoretical computer science. No special
background in computational complexity is assumed and the text is also accessible to senior undergraduates. The insightfulness of the
arguments presented here invites the reader to delve deeper and hopefully conquer this "complexity Waterloo": to prove a superlinear
lower bound for XOR circuits.