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### Minimization of Boolean functions

Marie Rüdigerová
1965 Applications of Mathematics
INTRODUCTION For the minimization of combinational Boolean functions in a sum-of-product form, many methods have been developed.  ...  J.\ Minimization of Boolean Functions. Bell System Technical Journal, 35,1956, pp. 1417-44. [4] Svoboda A.\ Some Applications of Contact Grids. Stroje na zpracovani informaci, VI, 1958, pp. 9-23.  ...  l) Pointing out the essential terms of the minimal form of the function.  ...

### Symmetric Boolean Functions

A. Canteaut, M. Videau
2005 IEEE Transactions on Information Theory
We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties.  ...  Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree.  ...  Boolean functions.  ...

### Irreducible Boolean Functions [article]

Moncef Bouaziz, Miguel Couceiro, Maurice Pouzet
2008 arXiv   pre-print
Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs.  ...  This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset Ω̃.  ...  Describe the irreducible Boolean functions. Boolean functions as polynomials.  ...

### Reducible Boolean functions

J. C. C. McKinsey
1936 Bulletin of the American Mathematical Society
as a Boolean function.  ...  In this note I establish a condition that a Boolean function of n variables, say/, be reducible to a product of two Boolean functions f\ and / 2 , where ƒ involves variables not occurring in ft; and, similarly  ...  If a Boolean function J\Xi, , X n ) be given, then a necessary and sufficient condition that there exist a g and an h, so that J\%ly ' ' ' y Xn) == g\%l) ' ' ' J %PJ VP ft\Xq, ' ' ' , X n ) j is that  ...

### Plotting Boolean Functions

Walter E. Stuermann
1960 The American mathematical monthly
PLOTTING BOOLEAN FUNCTIONS WALTER E.  ...  Consider the following Boolean function of four variables and six products in disjunctive normal form, where + denotes the Boolean sum, the immediate juxtaposition of elements designates the Boolean product  ...

### Boolean Random Functions [chapter]

Dominique Jeulin
2014 Lecture notes in mathematics
The Boolean RF are a generalization of the Boolean RACS.  ...  Introduction This text reviews a family of random functions (RF) which is an extension of the binary Boolean model, and is of wide use for applications, the Boolean RF.  ...  A Boolean islands RF Z(x) with intensity is built from this cylinder primary function. Give i) the univariate and bivariate distribution functions of Z(x).  ...

### Normal Boolean functions

Pascale Charpin
2004 Journal of Complexity
introduced the normality of bent functions. His work strengthened the interest for the study of the restrictions of Boolean functions on kdimensional flats providing the concept of k-normality.  ...  Using recent results on the decomposition of any Boolean functions with respect to some subspace, we present several formulations of k-normality.  ...  A Boolean function of m variables is a function from F m 2 into F 2 ; and we denote by B m the set of all Boolean functions of m variables.  ...

### Landscape Boolean functions

Constanza Riera, Pantelimon Stănică
2019 Advances in Mathematics of Communications
In this paper we define a class of generalized Boolean functions defined on F n 2 with values in Zq (we consider q = 2 k , k ≥ 1, here), which we call landscape functions (whose class contains generalized  ...  In particular, we show that the previously published characterizations of generalized plateaued Boolean functions (which includes generalized bent and semibent) are in fact particular cases of this more  ...  Certainly, every classical Boolean function is a landscape function.  ...

### Landscape Boolean Functions [article]

Constanza Riera, Pantelimon Stanica
2018 arXiv   pre-print
In this paper we define a class of Boolean and generalized Boolean functions defined on F_2^n with values in Z_q (mostly, we consider q=2^k), which we call landscape functions (whose class containing generalized  ...  In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting.  ...  Given a generalized Boolean function f : F n 2 → Z q , the derivative D a f of f with respect to a vector a is the generalized Boolean function D a f : F n 2 → Z q defined by D a f (x) = f (x) − f (x ⊕  ...

### Quantum boolean functions [article]

Ashley Montanaro, Tobias J. Osborne
2010 arXiv   pre-print
Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions.  ...  In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I.  ...  boolean function.  ...

### Locally monotone Boolean and pseudo-Boolean functions

Miguel Couceiro, Jean-Luc Marichal, Tamás Waldhauser
2012 Discrete Applied Mathematics
We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign  ...  As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions.  ...  We are interested in the so-called Boolean functions f : B n → B and pseudo-Boolean functions f : B n → R, where n denotes the arity of f .  ...

### The Stochastic Boolean Function Evaluation Problem for Symmetric Boolean Functions [article]

Dimitrios Gkenosis, Nathaniel Grammel, Lisa Hellerstein, Devorah Kletenik
2021 arXiv   pre-print
We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions.  ...  As part of the design of the first algorithm, we prove that the goal value of any symmetric Boolean function is less than n(n+1)/2.  ...  We thank Zach Pomerantz for experiments that gave us useful insights into the goal value of symmetric functions and the anonymous referees for their comments.  ...

### Almost Boolean Functions: The Design of Boolean Functions by Spectral Inversion

John A. Clark, Jeremy L. Jacob, Subhamoy Maitra, Pantelimon Stanica
2004 Computational intelligence
The design of Boolean functions with properties of cryptographic significance is a hard task. In this paper, we adopt an unorthodox approach to the design of such functions.  ...  Our search space is the set of functions that possess the required properties. It is 'Booleanness' that is evolved.  ...  Linear Boolean Function.  ...

### Encoding Nested Boolean Functions as Quantified Boolean Formulas

Uwe Bubeck, Hans Kleine Büning
2012 Journal on Satisfiability, Boolean Modeling and Computation
We obtain an equivalence-preserving transformation in linear time from the PSPACEcomplete language of nested Boolean functions (NBF), also called Boolean programs, to prenex QBF.  ...  In this paper, we consider the problem of compactly representing nested instantiations of propositional subformulas with different arguments as quantified Boolean formulas (QBF).  ...  Definition 1 . 1 (Nested Boolean Function) A nested Boolean function (NBF) is a finite sequence D(f k ) = (f 0 , ..., f k ) of Boolean functions.  ...

### Learning Boolean Functions Incrementally [chapter]

Yu-Fang Chen, Bow-Yaw Wang
2012 Lecture Notes in Computer Science
Classical learning algorithms for Boolean functions assume that unknown targets are Boolean functions over fixed variables.  ...  Based on a classical learning algorithm for Boolean functions, we develop two learning algorithms to infer Boolean functions over enlarging sets of ordered variables.  ...  Let f be a Boolean function over x n .  ...
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