Implementation complexity of Boolean functions with a small number of ones

2021 ◽  
Vol 31 (4) ◽  
pp. 271-279
Author(s):  
Nikolay P. Redkin
Keyword(s):  
Boolean Functions ◽  
Circuit Implementation ◽  
Shannon Function

Abstract We consider the class F n, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class F n, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis { x & y , x ‾ } $ \{x\&y,\overline x\} $ (or in the basis { x ∨ y , x ‾ } ) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.

A generalization of Shannon function

2018 ◽  
Vol 28 (5) ◽  
pp. 309-318 ◽  
Author(s):  
Nikolay P. Redkin
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Shannon Function ◽  
Conceptual Content

Abstract When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the considered basis. In this paper we propose a generalization of such a Shannon function in the form of an upper bound that is taken over all functionally complete bases. This generalization gives an idea of the complexity of implementing Boolean functions in the “worst” bases for them. The conceptual content of the proposed generalization is demonstrated by the example of a conjunction.

On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates

2019 ◽  
Vol 29 (4) ◽  
pp. 241-254 ◽  
Author(s):  
Igor' S. Sergeev
Keyword(s):  
Boolean Functions ◽  
Shannon Function ◽  
Bounded Depth Circuits

Abstract We obtain estimates for the complexity of the implementation of n-place Boolean functions by circuits and formulas built of unbounded fan-in conjunction and disjunction gates and either negation gates or negations of variables as inputs. Restrictions on the depth of circuits and formulas are imposed. In a number of cases, the estimates obtained in the paper are shown to be asymptotically sharp. In particular, for the complexity of circuits with variables and their negations on inputs, the Shannon function is asymptotically estimated as $2\cdot {{2}^{n/2}};$this estimate is attained on depth-3 circuits.

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.

On the Signal-to-Noise Ratio for Boolean Functions

2020 ◽  
Vol E103.A (12) ◽  
pp. 1659-1665
Author(s):  
Yu ZHOU ◽  
Wei ZHAO ◽  
Zhixiong CHEN ◽  
Weiqiong WANG ◽  
Xiaoni DU
Keyword(s):  
Boolean Functions ◽  
Signal To Noise Ratio ◽  
Signal To Noise ◽  
Noise Ratio
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