Complexity of Linear Boolean Operators

1. Auflage 2013
ISBN: 978-1-60198-726-6
Verlag: Now Publishers

Buch, Englisch, Band 25, 136 Seiten, Format (B × H): 156 mm x 234 mm

Reihe: Foundations and Trends® in Theoretical Computer Science

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 arguments-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.

Autoren/Hrsg.

Jukna, Stasys

Sergeev, Igor

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik EDV | Informatik Informatik Theoretische Informatik
• Mathematik | Informatik EDV | Informatik Informatik Berechenbarkeitstheorie, Komplexitätstheorie

Inhaltsverzeichnis

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. Acknowledgments. References