Topic
Boolean function
About: Boolean function is a research topic. Over the lifetime, 10089 publications have been published within this topic receiving 201604 citations. The topic is also known as: Boolean operation.
Papers published on a yearly basis
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Papers
TL;DR: This paper effectively solves the problem of efficiently constructing a representation of a threshold function given its Chow parameters, giving a randomized polynomial-time approximation scheme, and proves several new results of independent interest about Boolean threshold functions.
Abstract: In [Proceedings of the Second Symposium on Switching Circuit Theory and Logical Design (FOCS), 1961, pp. 34-38], Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow parameters. Providing an algorithmic version of Chow's theorem—i.e., efficiently constructing a representation of a threshold function given its Chow parameters—has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized polynomial-time approximation scheme with the following behavior: Given the Chow parameters of a Boolean threshold function $f$ over $n$ bits and any constant $\epsilon>0$, the algorithm runs in time $O(n^2\log^2n)$ and with high probability outputs a representation of a threshold function $f'$ which is $\epsilon$-close to $f$. Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include $\tilde{O}(n^2)$-time learning algorithms for threshold functions under the uniform distribution in the following models: (i) the restricted focus of attention model, answering an open question of Birkendorf et al.; (ii) an agnostic-type model. This contrasts with recent results of Guruswami and Raghavendra who show NP-hardness for the problem under general distributions; (iii) the PAC model, with constant $\epsilon$. Our $\tilde{O}(n^2)$-time algorithm substantially improves on the previous best known running time and nearly matches the $\Omega(n^2)$ bits of training data that any successful learning algorithm must use.
79 citations
16 Jul 2001
TL;DR: 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) is produced.
Abstract: The Kushilevitz-Mansour (KM)algorithm is an algorithm that finds all the "heavy" 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 an access to the membership query (MQ)oracle.
We 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 SQs to be a set of specific constant bounded product distributions. Our analogue finds all the "heavy" Fourier coefficients of degree lower than c log n (we call it BS). We use BS to learn decision trees and by adapting Freund's boosting technique we give algorithm that learns DNF in this model. Learning in this model implies learning with persistent classification noise and in some cases can be extended to learning with product attribute noise.
We develop a characterization for learnability with these extended SQs and apply it to get several negative results about the model.
79 citations
TL;DR: The first nonnormal bent function is given and even an example for a nonweakly normal bent function and a very efficient algorithm is presented that was used to verify the nonnormality of these functions.
79 citations
21 Aug 1988
TL;DR: In this article, a necessary and sufficient condition on the Walsh-spectrum of a boolean function is given, which implies that this function fulfills the Strict Avalanche Criterion, and this condition is shown to be fulfilled for a class of functions exhibiting simple spectral symmetries.
Abstract: A necessary and sufficient condition on the Walsh-spectrum of a boolean function is given, which implies that this function fulfills the Strict Avalanche Criterion. This condition is shown to be fulfilled for a class of functions exhibiting simple spectral symmetries. Finally, an extended definition of the Strict Avalanche Criterion is proposed and the corresponding spectral characterization is derived.
79 citations
Posted Content•
TL;DR: It is shown that the class of nested canalyzing functions is equal to that of unate cascade functions, which forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra.
Abstract: This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
78 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 3 |
| 2025 | 85 |
| 2024 | 130 |
| 2023 | 258 |
| 2022 | 454 |
| 2021 | 278 |