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This is, in the special case of symmetric functions, a substantial improvement over the characterization given in [LMN 89]."
Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this “complexity Waterloo” that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.
This result, along with our upper bounds, shows that for almost all Boolean function no real approximating function of small L1 norm can be found, or: almost all Boolean function has exponential L1 norm, or: for almost all Boolean function the distribution of the Fourier-coefficients is "even": they cannot be divided into two classes: one with small L1, the other with small L2 norms. Our results suggest that in the multi-party communication theory, instead of the well-studied degree of a polynomial representation of a Boolean function, its L1 norm can be an important measure of complexity."
This is the first book to cover the theory of noise sensitivity of Boolean functions with particular emphasis on critical percolation.
Here Professor Paterson brings together papers from the 1990 Durham symposium on Boolean function complexity. The participants include many well known figures in the field.
This graduate-level text gives a thorough overview of the analysis of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics.
The two internationally renowned authors elucidate the structure of "fast" parallel computation. Its complexity is emphasised through a variety of techniques ranging from finite combinatorics, probability theory and finite group theory to finite model theory and proof theory. Non-uniform computation models are studied in the form of Boolean circuits; uniform ones in a variety of forms. Steps in the investigation of non-deterministic polynomial time are surveyed as is the complexity of various proof systems. Providing a survey of research in the field, the book will benefit advanced undergraduates and graduate students as well as researchers.
Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a 'highlight application' such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Håstad's NP-hardness of approximation results, and 'sharp threshold' theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students and researchers in computer science theory and related mathematical fields.