In this work we discuss selected topics on small-depth computation, presenting a few unpublished proofs along the way. The four sections contain: (1) A unified treatment of the challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over {0, 1}.(2) An unpublished proof that small bounded-depth circuits (AC0) have exponentially small correlation with the parity function. The proof is due to Klivans and Vadhan; it builds upon and simplifies previous ones. (3) Valiant's simulation of log-depth linear-size circuits of fan-in 2 by sub-exponential size circuits of depth 3 and unbounded fan-in. To our knowledge, a proof of this result has never appeared in full. (4) Applebaum, Ishai, and Kushilevitz's cryptography in bounded depth.

For any research field to have a lasting impact, there must be a firm theoretical foundation. Neural networks research is no exception. Some of the founda tional concepts, established several decades ago, led to the early promise of developing machines exhibiting intelligence. The motivation for studying such machines comes from the fact that the brain is far more efficient in visual processing and speech recognition than existing computers. Undoubtedly, neu robiological systems employ very different computational principles. The study of artificial neural networks aims at understanding these computational prin ciples and applying them in the solutions of engineering problems. Due to the recent advances in both device technology and computational science, we are currently witnessing an explosive growth in the studies of neural networks and their applications. It may take many years before we have a complete understanding about the mechanisms of neural systems. Before this ultimate goal can be achieved, an swers are needed to important fundamental questions such as (a) what can neu ral networks do that traditional computing techniques cannot, (b) how does the complexity of the network for an application relate to the complexity of that problem, and (c) how much training data are required for the resulting network to learn properly? Everyone working in the field has attempted to answer these questions, but general solutions remain elusive. However, encouraging progress in studying specific neural models has been made by researchers from various disciplines.

This book constitutes the proceedings of the Second Annual International Conference on Computing and Combinatorics, COCOON '96, held in June 1996 in Hong Kong. The 44 papers presented in the book in revised version were carefully selected from a total of 82 submissions. They describe state-of-the-art research results from various areas of theoretical computer science, combinatorics related to computing, and experimental analysis of algorithms; computational graph theory, computational geometry, and networking issues are particularly well-presented.

This book constitutes the refereed proceedings of the 8th International Conference on Theory and Applications of Models of Computation, TAMC 2011, held in Tokyo, Japan, in May 2011. The 51 revised full papers presented together with the abstracts of 2 invited talks were carefully reviewed and selected from 136 submissions. The papers address the three main themes of the conference which were computability, complexity, and algorithms and are organized in topical sections on general algorithms, approximation, graph algorithms, complexity, optimization, circuit complexity, data structures, logic and formal language theory, games and learning theory, and cryptography and communication complexity.

This book constitutes the refereed proceedings of the 14th International Symposium Fundamentals of Computation Theory, FCT 2003, held in Malmö, Sweden in August 2003. The 36 revised full papers presented together with an invited paper and the abstracts of 2 invited talks were carefully reviewed and selected from 73 submissions. The papers are organized in topical sections on approximibility, algorithms, networks and complexity, computational biology, computational geometry, computational models and complexity, structural complexity, formal languages, and logic.

Proving lower bounds on the amount of resources needed to compute specific functions is one of the most active branches of theoretical computer science. Significant progress has been made recently in proving lower bounds in two restricted models of Boolean circuits. One is the model of small depth circuits, and in this book Johan Torkel Hastad has developed very powerful techniques for proving exponential lower bounds on the size of small depth circuits' computing functions. The techniques described in Computational Limitations for Small Depth Circuitscan be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority. The main tool used in the proof of the lower bounds is a lemma, stating that any AND of small fanout OR gates can be converted into an OR of small fanout AND gates with high probability when random values are substituted for the variables. Hastad also applies this tool to relativized complexity, and discusses in great detail the computation of parity and majority in small depth circuits. Contents:Introduction. Small Depth Circuits. Outline of Lower Bound Proofs. Main Lemma. Lower Bounds for Small Depth Circuits. Functions Requiring Depth k to Have Small Circuits. Applications to Relativized Complexity. How Well Can We Compute Parity in Small Depth? Is Majority Harder than Parity? Conclusions. John Hastad is a postdoctoral fellow in the Department of Mathematics at MIT Computational Limitations of Small Depth Circuitsis a winner of the 1986 ACM Doctoral Dissertation Award.

The communication complexity of two-party protocols is an only 15 years old complexity measure, but it is already considered to be one of the fundamen tal complexity measures of recent complexity theory. Similarly to Kolmogorov complexity in the theory of sequential computations, communication complex ity is used as a method for the study of the complexity of concrete computing problems in parallel information processing. Especially, it is applied to prove lower bounds that say what computer resources (time, hardware, memory size) are necessary to compute the given task. Besides the estimation of the compu tational difficulty of computing problems the proved lower bounds are useful for proving the optimality of algorithms that are already designed. In some cases the knowledge about the communication complexity of a given problem may be even helpful in searching for efficient algorithms to this problem. The study of communication complexity becomes a well-defined indepen dent area of complexity theory. In addition to a strong relation to several funda mental complexity measures (and so to several fundamental problems of com plexity theory) communication complexity has contributed to the study and to the understanding of the nature of determinism, nondeterminism, and random ness in algorithmics. There already exists a non-trivial mathematical machinery to handle the communication complexity of concrete computing problems, which gives a hope that the approach based on communication complexity will be in strumental in the study of several central open problems of recent complexity theory.

This book constitutes the refereed proceedings of the 16th International Conference on Foundations of Software Technology and Theoretical Computer Science, FST&TCS '96, held in Hyderabad, India, in December 1996. The volume presents 28 revised full papers selected from a total of 98 submissions; also included are four invited contributions. The papers are organized in topical sections on computational geometry, process algebras, program semantics, algorithms, rewriting and equational-temporal logics, complexity theory, and type theory.

This book constitutes the refereed proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science, MFCS 2006. The book presents 62 revised full papers together with the full papers or abstracts of 7 invited talks. All current aspects in theoretical computer science and its mathematical foundations are addressed, from algorithms and data structures, to complexity, automata, semantics, logic, formal specifications, models of computation, concurrency theory, computational geometry and more.

This volume constitutes the refereed proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011, held in Warsaw, Poland, in August 2011. The 48 revised full papers presented together with 6 invited talks were carefully reviewed and selected from 129 submissions. Topics covered include algorithmic game theory, algorithmic learning theory, algorithms and data structures, automata, grammars and formal languages, bioinformatics, complexity, computational geometry, computer-assisted reasoning, concurrency theory, cryptography and security, databases and knowledge-based systems, formal specifications and program development, foundations of computing, logic in computer science, mobile computing, models of computation, networks, parallel and distributed computing, quantum computing, semantics and verification of programs, and theoretical issues in artificial intelligence.