Introduction to finite fields and their applications: Preface

Communication Complexity: Basics

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.

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Boolean Function Complexity: Advances and Frontiers

An investigation is made of relays whose reliability can be described in simple terms by means of probabilities. It is shown that by using a sufficiently large number of these relays in the proper manner, circuits can be built which are arbitrarily reliable, regardless of how unreliable the original relays are. Various properties of these circuits are elucidated.

Reliable Circuits Using Less Reliable Relays

Introduction to finite fields and their applications: Factorization of Polynomials

This book is a concise, self-contained, up-to-date introduction to extremal combinatorics for nonspecialists. There is a strong emphasis on theorems with particularly elegant and informative proofs, they may be called gems of the theory. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science: methods of extremal set theory, the linear algebra method, the probabilistic method, and fragments of Ramsey theory. No special knowledge in combinatorics or computer science is assumed the text is self-contained and the proofs can be enjoyed by undergraduate students in mathematics and computer science. Over 300 exercises of varying difficulty, and hints to their solution, complete the text. This second edition has been extended with substantial new material, and has been revised and updated throughout. It offers three new chapters on expander graphs and eigenvalues, the polynomial method and error-correcting codes. Most of the remaining chapters also include new material, such as the KruskalKatona theorem on shadows, the LovszStein theorem on coverings, large cliques in dense graphs without induced 4-cycles, a new lower bounds argument for monotone formulas, Dvir's solution of the finite field Kakeya conjecture, Moser's algorithmic version of the Lovsz Local Lemma, Schning's algorithm for 3-SAT, the SzemerdiTrotter theorem on the number of point-line incidences, surprising applications of expander graphs in extremal number theory, and some other new results.

Extremal Combinatorics: With Applications in Computer Science

Boolean Function Complexity

By the complexity of a graph we mean the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of boolean functions.We prove some lower bounds on the complexity of explicitly given graphs. This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic framework. We also formulate several combinatorial problems whose solution would have intriguing consequences in computational complexity.

On Graph Complexity

A method for obtaining lower bounds on the contact circuit complexity of explicitly defined Boolean functions is given It appears as one of possible concretizations of a more general “convolutional” approach to the lower bounds problem worked out by the author in 1984 [12] The method is based on an appropriate notion of “inner information” or “entropy” of finite objects (circuits, Boolean functions, etc) Lower bounds on the complexity are obtained by means of entropy-preserving embeddings of circuits into the more restricted ones This allows to prove in a uniform and easy way that contact circuits, which are local in a sense that the function computed by a subcircuit weakly depends on the whole circuit, require 2 Ω(√n) or even 2 Ω( n log n ) contacts to compute some explicitly defined n -argument Boolean functions from NP

Stasys Jukna

Papers

Extremal Combinatorics: With Applications in Computer Science

Boolean Function Complexity: Advances and Frontiers

Boolean Function Complexity

On Graph Complexity

Entropy of contact circuits and lower bounds on their complexity