Circuits Of Functional Elements Research Articles

On the Complexity of Implementation of Characteristic Functions of the Spheres by Circuits of Functional Elements

On the Complexity of Implementation of Characteristic Functions of the Spheres by Circuits of Functional Elements

On the Complexity and Depth of Embedded in Boolean Cube Circuits That Implement Boolean Functions

A class of circuits of functional elements over the standard basis of the conjunction, disjunction, and negation elements is considered. For each circuit Σ in this class, its depth D(Σ) and dimension R(Σ) equal to the minimum dimension of the Boolean cube allowing isomorphic embedding Σ are defined. It is established that for n = 1, 2,… and an arbitrary Boolean function f of n variables there exists a circuit Σf for implementing this function such that R(Σf) ⩽ n − log2 log2n + O(1) and D(Σf) ⩽ 2n − 2 log2 log2n + O(1). It is proved that for n = 1, 2,… almost all functions of n variables allow implementation by circuits of the considered type, whose depth and dimension differ from the minimum values of these parameters (for all equivalent circuits) by no more than a constant and asymptotically no more than by a factor of 2, respectively.

About One Efficient Method of Synthesis of Boolean Formulas and Circuits of Functional Elements

About One Efficient Method of Synthesis of Boolean Formulas and Circuits of Functional Elements

Complexity of linear and majority functions in the basis of antichain functions

The complexity of realization of Boolean functions by circuits of functional elements in the basis consisting of all characteristic functions of antichains over a Boolean cube is studied. It is proved that the complexity of realization of an n-variable parity function is \(\left\lfloor {\frac{{n + 1}}{2}} \right\rfloor \) and the complexity of its negation equals the complexity of the n-variable majority function which is \(\left\lceil {\frac{{n + 1}}{2}} \right\rceil \).

Уточнение оценок показателей сложности схем и автоматизация их эффективного применения

An important feature of modern scientific and technological revolution is the rapid growth of collecting and processing information in almost all areas of science and technology. This gives rise to a large number of difficult tasks (problems) for a variety of reasons that people are not able to solve on their own because of their huge information capacity and complexity. It should be noted that in recent years there has been a steady trend of increasing the role of computers in the task, which is associated with problems of processing and presentation of data not only in quantitative terms but also in qualitative form - in the form of relationships, text description, obtained in economics, medicine, biology etc. The creation and distribution of software for data processing, which would allow users-nonprogrammers, working in various fields of science and technology, to solve problems on the basis of a computer, is also extremely relevant. Improving computer element base occurs with great predominance increase the degree of integration in comparison with other parameters. Therefore, the primary means of increasing productivity is to increase computer equipment - the number of gates used in it. In this paper we study the problem of implementing any Boolean function in the class of formulas and - circuits of functional elements in standard and Zhegalkin bases. An efficient method for the synthesis of formulas and schemes on the basis of the recurrence relations (functional equations), followed by obtaining advance analytically upper estimates of various indicators of complexity (the number of letters, number subformulas; on the number of functional elements over the depth of the formula and the depth of the circuit), including and schemes for the minimum of difficulty. If necessary to clarify the upper bounds for the complexity, the computational algorithm is being proposed.

Switching activity of Boolean circuits and synthesis of Boolean circuits with asymptotically optimal complexity and linear switching activity

For circuits of functional elements (CFEs), the concept of their switching activity is introduced, which complements the earlier analyzed static activity, or power, and models the power consumption of integrated circuits due to arising transient processes in them. For the switching activity of Boolean functions (BFs) of n variables realized by CFEs, a linear (in n) upper estimate for the Shannon function is obtained in an arbitrary finite complete basis. In addition, synthesismethods are proposed that allow one to construct such CFEs for the above functions in the standard basis {&, ∨, ¬}, whose complexity is asymptotically not greater than 2 n /n and the switching and static activities have a growth order linear with respect to n; moreover, the static activity of these CFEs satisfies new sharper estimates.

Synthesis of circuits admitting complete checking tests of constant length under inverse faults at outputs of elements

A method for synthesizing circuits of functional elements in a special basis which admit complete checking tests of length at most 4 under inverse faults at the outputs of functional elements is proposed.

Single fault detection tests for circuits of functional elements

Circuits of functional elements in arbitrary complete finite bases are considered. The realizability of any Boolean function of n variables by an irredundant circuit admitting single fault detection tests having a length linear in n for constant faults is established.

Complexity of realization of a linear Boolean function in Sheffer’s basis

The paper is focused on realization of linear Boolean functions by circuits of functional elements in the basis Open image in new window , which is the Sheffer stroke. The exact value of complexity of negation of linear function is obtained in this paper. Another result is the description of all minimal circuits realizing a linear function.

On minimal circuits for linear functions over some bases

In this research we consider realisations of linear Boolean functions by circuits of functional elements over the basis B which consists of two-input functional elements realising nonlinear functions and an inverter. The weight of a two-input element over the basis B is set equal to one, and the weight of the inverter is zero.

A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials

This work suggests a method for deriving lower bounds for the complexity of polynomials with positive real coefficients implemented by circuits of functional elements over the monotone arithmetic basis . Using this method, several new results are obtained. In particular, we construct examples of polynomials of degree in each of the variables with coefficients 0 and 1 having additive monotone complexity and multiplicative monotone complexity as . In this form, the lower bounds derived here are sharp.Bibliography: 72 titles.

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics Lk−(f2n) ∼ n min{k+1, p} is obtained for the sequence of Boolean functions \(f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}\) for any fixed k, p ≥ 1 and growing n, here Lk− (f2n) is the inversion complexity of realization of the function f2n by k-self-correcting circuits of functional elements in the basis B = {&, −}, p is the weight of a reliable invertor.

A method for synthesis of easily-testable circuits in some basis admitting single fault detection tests of constant length

A method of synthesis of irredundant circuits of functional elements in the basis {x&y, x⊕y, 1, \(\bar x\)(y ∨ z) ∨ x(y ∼ z)} realizing arbitrary Boolean functions and admitting unit verification tests of the length not exceeding 4 is proposed in the paper for inverse or constant (stuck-at) faults at outputs of gates.

The minimal circuits for linear Boolean functions

The realization of linear Boolean functions by circuits of functional elements in the basis {x&y,x ∨ y, \(\bar x\)} is considered. The main result is a structure description of all minimal circuits realizing linear Boolean functions.

10.1007/s11488-008-1008-9

A new method for studying mathematical models of discrete logical units based on functional equations is proposed. Representing Boolean functions in the class of formulas (circuits of functional elements with or without branching), we obtain the values of complexity indices in the number of characters, subformulas, and in the depth of a superposition formula (the number of functional elements and the circuit depth). The results are applied in the logical-computational approach to the intellectualization of synthesizing logical units on the basis of digital integrated circuits.

Complexity of self-correcting circuits for some sequence of Boolean functions

k-Self-correcting circuits of functional elements in the basis {x1&x2, \( \bar x \)} are considered. It is assumed that constant faults on outputs of functional elements are of the same type. Inverters are supposed to be reliable in service. The weight of each inverter is equal to 1. Conjunctors can be reliable in service, or not reliable. Each reliable conjunctor implements a conjunction of two variables and has a weight p > k + 2. Each unreliable conjunctor implements a conjunction in its correct state and implements a Boolean constant δ (δ ∈ {0, 1}) otherwise. Each unreliable conjunctor has the weight 1. It is stated that the complexity of realization of monotone threshold symmetric functions \( f_2^n \left( {x_1 ,...,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_1 x_j ,n = 3,4 \), ... by such circuits asymptotically equals (k + 3)n.

The complexity of depth-two information networks

The lower bound Ω(n log2n) for the complexity of an arbitrary depth-two information network with n inputs and n outputs is proved providing the inputs are independent, the outputs are independent, and the total information of any input and any output is n times less than the entropy of any input or output. A similar estimate for Boolean depth-two circuits of functional elements is obtained as a corollary.

Mathematical models for intellectualization of synthesis of discrete logical control units

A new method for studying mathematical models of discrete logical units based on functional equations is proposed. Representing Boolean functions in the class of formulas (circuits of functional elements with or without branching), we obtain the values of complexity indices in the number of characters, subformulas, and in the depth of a superposition formula (the number of functional elements and the circuit depth). The results are applied in the logical-computational approach to the intellectualization of synthesizing logical units on the basis of digital integrated circuits.

The depth of Boolean functions realized by circuits over an arbitrary basis

Realization of Boolean functions by circuits of functional elements is considered over arbitrary complete bases (including infinite ones). The depth of a circuit means the maximal number of functional elements forming an oriented chain going from inputs of the circuit to its output. It is shown that for any basis B the growth order of the Shannon function of depth DB(n) for n → ∞ is equal to 1, log2n, or n, and the latter case appears if and only if the basis B is finite.

Complexity of Boolean functions in intelligent systems for synthesis of digital integrated circuits

The representation of an arbitrary Boolean function over different bases in classes of formulas and circuits of functional elements (with and without branching) is considered. It is accompanied by derivation of the corresponding (for different bases) estimates of the following complexity indices such as the number of letters and formula length, the number of subformulas and superposition formula depth, and the number of functional elements in the circuit and its depth. The obtained knowledge should be applied to intelligent systems of integrated circuit synthesis.