Boolean function

Abstract: This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research.

Abstract: Modern systems engineering (e. g. switching circuits design) and operations research (e. g. reliability systems theory) use Boolean functions with increasing regularity. For practitioners and students in these fields books written for mathe maticians are in several respects not the best source of easy to use information, and standard books, such as, on switching circuits theory and reliability theory, are mostly somewhat narrow as far as Boolean analysis is concerned. Further more, in books on switching circuits theory the relevant stochastic theory is not covered. Aspects of the probabilistic theory of Boolean functions are treated in some works on reliability theory, but the results deserve a much broader interpre tation. Just as the applied theory (e. g. of the Laplace transform) is useful in control theory, renewal theory, queueing theory, etc., the applied theory of Boolean functions (of indicator variables) can be useful in reliability theory, switching circuits theory, digital diagnostics and communications theory. This book is aimed at providing a sufficiently deep understanding of useful results both in practical work and in applied research. Boolean variables are restricted here to indicator or O/l variables, i. e. variables whose values, namely 0 and 1, are not free for a wide range of interpretations, e. g. in digital electronics 0 for L ==low voltage and 1 for H == high voltage."

Abstract: We study a corpus of particular Boolean functions: the idempotents. They enable us to construct functions which achieve the best possible tradeoffs between the cryptographic fundamental properties: balancedness, correlation-immunity, a high degree and a high nonlinearity (that is a high distance from the affine functions). They all represent extremely secure cryptographic primitives to be implemented in stream ciphers.

Abstract: Optimal O(log n/log log n) lower bounds on the time for CRCW PRAMS with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven. A strict time hierarchy of explicit Boolean functions of n bits on such machines that holds up to O(log n/log log n) time is also exhibited. That is, for every time bound T within this range a function is exhibited that can be easily computed using polynomial resources in time T but requires more than polynomial resources to be computed in time T - 1. Finally, it is shown that almost all Boolean functions of n bits require log n - log log n + O(1) time when the number of processors is at most polynomial in n. The bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines.