Boolean function

Abstract: A multi-threshold element is one in which several thresholds are used to separate the true inputs from the false inputs. Many circuit elements and configurations can be described by this model. An approach, based on conventional single-threshold threshold elements, is developed for the analysis and synthesis of multithreshold threshold elements. It is shown that the basic properties of such elements are similar to conventional threshold elements, and that k-threshold threshold-element realizability of an arbitrary n-variable Boolean function can be related to conventional threshold-element realizability of a related (n+k-1)-variable Boolean function. Foundations for two basically different methods for the synthesis of a single-element realization of an arbitrary Boolean function are developed, as are procedures for transforming such a realization into both two-level and multilevel loop-free networks of k-threshold threshold elements k?1. Every element in the networks has the identical weight vector for the independent variables, which is some-times desirable. The transformation technique is a useful approach to the synthesis of functions by networks of conventional threshold elements. It is proved that if the given function requires a k-threshold threshold element, then at least [k/2+I] conventional threshold elements in a two-level network or [1+log 2 k] such elements in a multilevel network are required. Transformations are given for corresponding minimum-gate networks. Electronic-circuit realizations of multi-threshold elements and some logical-design applications of the multi-threshold approach to network design are discussed. The latter indicate that this approach can be easy to use and can result in economical realizations.

Abstract: Three of the most important criteria for cryptographically strong Boolean functions are the balancedness, the nonlinearity and the propagation criterion. This paper studies systematic methods for constructing Boolean functions satisfying some or all of the three criteria. We show that concatenating, splitting, modifying and multiplying sequences can yield balanced Boolean functions with a very high nonlinearity. In particular, we show that balanced Boolean functions obtained by modifying and multiplying sequences achieve a nonlinearity higher than that attainable by any previously known construction method. We also present methods for constructing highly nonlinear balanced Boolean functions satisfying the propagation criterion with respect to all but one or three vectors. A technique is developed to transform the vectors where the propagation criterion is not satisfied in such a way that the functions constructed satisfy the propagation criterion of high degree while preserving the balancedness and nonlinearity of the functions. The algebraic degrees of functions constructed are also discussed, together with examples illustrating the various constructions.

Abstract: The Kushilevitz-Mansour (KM) algorithm is an algorithm that finds all the "large" Fourier coefficients of a Boolean function. It is the main tool for learning decision trees and DNF expressions in the PAC model with respect to the uniform distribution. The algorithm requires access to the membership query (MQ) oracle. The access is often unavailable in learning applications and thus the KM algorithm cannot be used. We significantly weaken this requirement by producing an analogue of the KM algorithm that uses extended statistical queries (SQ) (SQs in which the expectation is taken with respect to a distribution given by a learning algorithm). We restrict a set of distributions that a learning algorithm may use for its statistical queries to be a set of product distributions with each bit being 1 with probability ρ, 1/2 or 1-ρ for a constant 1/2 > ρ > 0 (we denote the resulting model by SQ-Dρ). Our analogue finds all the "large" Fourier coefficients of degree lower than clog(n) (we call it the Bounded Sieve (BS)). We use BS to learn decision trees and by adapting Freund's boosting technique we give an algorithm that learns DNF in SQ-Dρ. An important property of the model is that its algorithms can be simulated by MQs with persistent noise. With some modifications BS can also be simulated by MQs with product attribute noise (i.e., for a query x oracle changes every bit of x with some constant probability and calculates the value of the target function at the resulting point) and classification noise. This implies learnability of decision trees and weak learnability of DNF with this non-trivial noise. In the second part of this paper we develop a characterization for learnability with these extended statistical queries. We show that our characterization when applied to SQ-Dρ is tight in terms of learning parity functions. We extend the result given by Blum et al. by proving that there is a class learnable in the PAC model with random classification noise and not learnable in SQ-Dρ.

Abstract: This paper addresses the problem of computing the area complexity of a multi-output combinational logic circuit, given only its functional description, i.e., Boolean equations, where area complexity is measured in terms of the number of gates required for an optimal multilevel implementation of the combinational logic. The proposed area model is based on transforming the given, multi-output Boolean function description into an equivalent single-output function. The model, is empirical, and results demonstrating its feasibility and utility are presented. Also, a methodology for converting the gate count estimates, obtained from the area model, into capacitance estimates is presented. High-level power estimates based on the total capacitance estimates and average activity estimates are also presented.