scholarly journals And/or tree probabilities of Boolean functions

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Danièle Gardy ◽  
Alan Woods
Keyword(s):  
Wide Class ◽  
Boolean Functions ◽  
Probability Distributions ◽  
Fixed Number ◽  
Constant Function ◽  
International Audience ◽  
Class Of Functions

International audience We consider two probability distributions on Boolean functions defined in terms of their representations by $\texttt{and/or}$ trees (or formulas). The relationships between them, and connections with the complexity of the function, are studied. New and improved bounds on these probabilities are given for a wide class of functions, with special attention being paid to the constant function $\textit{True}$ and read-once functions in a fixed number of variables.

scholarly journals Random Boolean expressions

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Danièle Gardy
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Probability Distributions ◽  
Fixed Number ◽  
Actual Computation ◽  
Main Challenge ◽  
International Audience ◽  
Boolean Expressions

International audience We examine how we can define several probability distributions on the set of Boolean functions on a fixed number of variables, starting from a representation of Boolean expressions by trees. Analytic tools give us a systematic way to prove the existence of probability distributions, the main challenge being the actual computation of the distributions. We finally consider the relations between the probability of a Boolean function and its complexity.

scholarly journals On the sensitivity of cyclically-invariant Boolean functions

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Sourav Chakraborty
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Transitive Group ◽  
60Th Birthday ◽  
Special Issue ◽  
Natural Class ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Class Of Functions ◽  
Block Sensitivity

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper we construct a cyclically invariant Boolean function whose sensitivity is Theta(n(1/3)). This result answers two previously published questions. Turan (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Omega(root n). Kenyon and Kutin (2004) asked whether for a "nice" function the product of 0-sensitivity and 1-sensitivity is Omega(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Omega(n(1/3)). Hence for this class of functions sensitivity and block sensitivity are polynomially related.

scholarly journals No Shannon effect on probability distributions on Boolean functions induced by random expressions

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Bernhard Gittenberger
Keyword(s):  
Probability Distribution ◽  
Boolean Functions ◽  
Probability Distributions ◽  
Uniform Probability ◽  
Uniform Probability Distribution ◽  
International Audience ◽  
Almost All

International audience The Shannon effect states that "almost all'' Boolean functions have a complexity close to the maximal possible for the uniform probability distribution. In this paper we use some probability distributions on functions, induced by random expressions, and prove that this model does not exhibit the Shannon effect.

CLASS OF BOOLEAN FUNCTIONS CONSTRUCTED USING SIGNIFICANT BITS OF LINEAR RECURRENCES OVER THE RING ℤ2n

2019 ◽  
Vol 6 (2) ◽  
pp. 90-94
Author(s):  
Hernandez Piloto Daniel Humberto
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Linear Recurrences ◽  
Maximum Period ◽  
Non Linear ◽  
The Given ◽  
Class Of Functions

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

scholarly journals ON THE COMPLEXITY OF THE EVALUATION OF TRANSIENT EXTENSIONS OF BOOLEAN FUNCTIONS

2012 ◽  
Vol 23 (01) ◽  
pp. 21-35
Author(s):  
JANUSZ BRZOZOWSKI ◽  
BAIYU LI ◽  
YULI YE
Keyword(s):  
Boolean Functions ◽  
Hazard Detection ◽  
Np Complete ◽  
The Given ◽  
Class Of Functions

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We study a class of functions for which, instead of evaluating the extensions on a given set of transients, it is possible to get the same values by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.

Generalized Lotka-Volterra (GLV) Models of Stock Markets

2000 ◽  
Vol 03 (01n04) ◽  
pp. 301-322 ◽  
Author(s):  
Sorin Solomon
Keyword(s):  
Power Law ◽  
Wide Class ◽  
Stock Markets ◽  
Probability Distributions ◽  
Economic Systems ◽  
Lévy Flights ◽  
Levy Flights ◽  
General Method

The Generalized Lotka-Volterra (GLV) model: [Formula: see text] provides a general method to simulate, analyze and understand a wide class of phenomena that are characterized by power-law probability distributions: [Formula: see text] and truncated Levy flights fluctuations [Formula: see text]. We show how the model applies to economic systems.

EXISTENCE OF STRONG TRACES FOR GENERALIZED SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

2005 ◽  
Vol 02 (04) ◽  
pp. 885-908 ◽  
Author(s):  
E. YU. PANOV
Keyword(s):  
Wide Class ◽  
Conservation Law ◽  
Generalized Solutions ◽  
Flux Vector ◽  
Scalar Conservation Laws ◽  
Generalized Entropy ◽  
Scalar Conservation Law ◽  
Nondegeneracy Conditions ◽  
Scalar Conservation ◽  
Class Of Functions

In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.

scholarly journals Optimization results for a generalized coupon collector problem

2016 ◽  
Vol 53 (2) ◽  
pp. 622-629 ◽  
Author(s):  
Emmanuelle Anceaume ◽  
Yann Busnel ◽  
Ernst Schulte-Geers ◽  
Bruno Sericola
Keyword(s):  
Probability Distribution ◽  
Uniform Distribution ◽  
Closed Subset ◽  
Probability Distributions ◽  
Fixed Number ◽  
Coupon Collector ◽  
The One

Abstract In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

scholarly journals Reflection of Functions and Linear Prognosis

2021 ◽  
Author(s):  
Andrei Valerianovich Pavlov
Keyword(s):  
Linear Transformation ◽  
Wide Class ◽  
Symmetric Case ◽  
Regular Functions ◽  
Class Of Functions

Periodicity of wide class of functions as a result of reflection of even and arbitrary regular functions is proved. By consideration of new scalar work in space of linear shell of initial n vectors the equivalence of values of two different scalar productions is proved. The example of linear transformation is considered on plane for the symmetric case, resulting in possibility to make to use the orthogonal sides of rhombus at projection on the plane of its parties.

scholarly journals Quantum conditional query complexity

2017 ◽  
Vol 17 (7&8) ◽  
pp. 541-567
Author(s):  
Imdad S.B. Sardharwalla ◽  
Sergii Strelchuk ◽  
Richard Jozsa
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Query Complexity ◽  
Equivalence Testing ◽  
Conditional Probabilities ◽  
Underlying Distribution ◽  
New Type ◽  
Testing Equivalence ◽  
Oracle Access

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.

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