Self-dual Boolean Functions Research Articles

Синтез самопроверяемых цифровых устройств на основе взвешенных кодов с суммированием, проверочные символы которых описываются самодвойственными булевыми функциями

The proposed article considers sum code in a residue ring over a given modulo M of the data vector weighted bits. Test symbols are described by self-dual Boolean functions. It is proven that such codes can be constructed only under the condition M  {2, 4, 8, …} for certain arrays of weighting coefficients. The division into special classes of the weight-based sum code special case in the residue ring modulo M=4 is given. Cases with the number of data bits m=4 are considered. It has been determined that only two options for weighing with weight coefficients from arrays [1, 1, 2, 3] and [1, 2, 2, 2], without considering permutations of weights, give codes with the function selfduality property that describe check symbols. The concurrent error-detection circuit structure based on weight-based sum code in a residue ring by modulo M=4 with the m=4 data bits is given.

Самодвойственные функциональные элементы для синтеза контролепригодных цифровых систем

All self-dual analogs of elementary functional gates have been considered, the use of which allows for the synthesis of self-dual circuit implementations of arbitrary Boolean functions. In this case, two synthesis methods can be used, each one based on the property of any Boolean function to be transformed into a self-dual function using one additional variable. The first method involves replacing all non-self-dual functional gates in the device structure with self-dual analogs. The second one involves obtaining a self-dual function from the original formula. The study conducted modeling of self-dual functional gates in pulse mode of operation. It has been shown that all self-dual functional gates, except for those implementing equivalence and nonequivalence functions (modulo-2 addition), are fully self-checkable with respect to stuck-at faults when checking computations based on the belonging of the generated functions to the class of self-dual Boolean functions. However, the gates that implement the mentioned functions require additional monitoring. For them, error masking occurs due to the simultaneous distortion of signals on both combinations in a pair. This feature of these self-dual functional gates should be taken into account when developing controllable self-checking digital computing devices and systems. The article provides an example of using methods for constructing self-dual circuit implementations. The obtained results can be used in the synthesis of controllable self-dual computing devices and systems.

Properties Investigation of Self-Dual Combinational Devices with Calculation Control Based on Hamming Codes

A new approach to the synthesis of self-checking devices is considered, based on the control of calculations in testing objects using Hamming codes, the check bits of which are described by self-dual functions. In this case, the structure operates in a pulsed mode, which is actually based on the introduction of temporal redundancy when building a self-checking device. This, unfortunately, leads to some decrease in performance, however, it significantly improves the characteristics of controllability, which is especially important for devices and systems of critical use, the input data for which does not change so often. A brief review of methods for constructing built-in control circuits based on the self-duality property of calculated functions is given. The basic structures of the organization of built-in control circuits are given. The proposed ways of developing the theory of synthesis of built-in control circuits are based on checking whether or not the calculated functions belong to a class of self-dual Boolean functions. All possible values of the number of data bits for Hamming codes have been established. They will have the property of the self-duality of functions describing control bits. En-coders of such Hamming codes will be self-dual devices. Since the functions of the check bits of Hamming codes are linear, in order for them to be self-dual, it is necessary that an odd number of arguments be used in each of them. It is proved that the number of bits of code words of Hamming codes with self-dual check functions is equal to n=3+4l, l∈N0. The results of the simulations self-dual devices with built-in control circuits along two diagnostic parameters in the Multisim environment are presented. A method is proposed for modification of the structure of calculation control along two diagnostic parameters, which allows to use any linear block code (not necessarily Hamming code). It is based on retrofitting the encoder with a device for converting functions into self-dual ones. In fact, this is a code modification device. It is proved that to obtain a modified Hamming code with self-dual control functions for n≠3+4l, l∈N0; cases, it is enough to add modulo M=2 the non-self-dual control function with the function of the high data bit.

МЕТОД СИНТЕЗА САМОПРОВЕРЯЕМЫХ УСТРОЙСТВ С КОНТРОЛЕМ ВЫЧИСЛЕНИЙ ПО ДВУМ ДИАГНОСТИЧЕСКИМ ПАРАМЕТРАМ С ПРЕДВАРИТЕЛЬНЫМ СЖАТИЕМ СИГНАЛОВ ОТ ОБЪЕКТА ДИАГНОСТИРОВАНИЯ

The structure of self-checking digital devices with calculation control by two diagnostic parameters has been proposed. The belonging of being formed in in-built control circuit codeword to constant-weight code “2-out-of-4” is used as a first parameter. A second parameter is the belonging of each being calculated function to the class of self-dual Boolean functions. The specificity of being described in the article the organization structure of self-checking digital devices is the presence of the circuit of preliminary compression of signals from diagnosis object. Its usage allows to cut structural redundancy of end device down essentially. Besides, nevertheless, errors can be masked at the inputs of compression elements. The article underlines the features of the choice of being compressed outputs of diagnosis object and the algorithm is proposed allowing to minimize the risk of undetectable error occurrence at compression scheme output. The algorithm of built-in control scheme synthesis by two diagnosis parameters has been presented which takes into account conditions for the formation of full set of test combinations for testers and elements of transformation in signal correction block. The example of realization of synthesis algorithm for fully self-checking device Logisim is considered. Key features of built-in control scheme are denoted; the scheme is embodied according to the proposed structure. The way of organizing calculation control by two diagnostic parameters is of interest at the synthesis of fully self-checking digital calculation devices and systems.

Application of Election Functions to Estimate the Number of Monotone Self-Dual Boolean functions

One of the problems of modern discrete mathematics is R. Dedekind problem on the number of monotone boolean functions. For other precomplete classes, general formulas for the number of functions of the classes had been found, but it has not been found so far for the class of monotone boolean functions. Within the framework of this problem, there are problems of a lower level. One of them is the absence of a general formula for the number of boolean functions of intersection $MS$ of two classes --- the class of monotone functions and the class of self-dual functions. In the paper, new lower bounds are proposed for estimating the cardinality of the intersection for both an even and an odd number of variables. It is shown that the election function of an odd number of variables is monotone and self-dual. The election function of an even number of variables is determined. Free election functions, which are functions with fictitious variables similar in properties to election functions, are introduced. Then the union of a set of election functions and a set of free election functions is considered, and the cardinality of this union is calculated. The resulting value of the cardinality is proposed as a lower bound for $|MS|$. For the class $MS$ of monotone self-dual functions of an even number of variables, the lower bound is improved over the bounds proposed earlier, and for functions of an odd number of variables, the lower bound for $|MS|$ is presented for the first time.

A study on monotone self-dual Boolean functions

This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form (IDNF) do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of variables and disjuncts are examined. An algorithm is proposed to test whether a monotone Boolean function in IDNF with n variables and n disjuncts is self-dual. The runtime of the algorithm is O(n3).

The Number of Monotone and Self-Dual Boolean Functions

Abstract In the present paper we study properties of pre-complete class of Boolean functions - monotone Boolean functions. We discuss interval graph, the abbreviated d.n.f., a minimal d.n.f. and a shortest d.n.f. of this function. Then we present a d.n.f. with the highest number of conjunctionsand we determinate the exact number of them. We count the number of monotone Boolean functions with some special properties. In the end we estimate the number of Boolean functionthat are monotone and self-dual at the same time.

Synthesis of logic units for calculating self-dual symmetric Boolean functions

A problem of synthesizing logic circuits of units is considered. Here, an arbitrary self-dual symmetric Boolean function depending on three, five, or seven variables is realized in a single output. The constructive complexity of the synthesized logic circuits is determined by the number of inputs of logic elements, while the operating speed is influenced by the depth of the circuit. The logic circuits exceed the existing analogues in terms of complexity and operating speed.

MacWilliams duality and a Gleason-type theorem on self-dual bent functions

We prove that the MacWilliams duality holds for bent functions. It enables us to derive the concept of formally self-dual Boolean functions with respect to their near weight enumerators. By using this concept, we prove the Gleason-type theorem on self-dual bent functions. As an application, we provide the total number of (self-dual) bent functions in two and four variables obtaining from formally self-dual Boolean functions.

Invariant continuous aggregation functions

Continuous aggregation functions, being invariant under any monotone bijective transformation, are discussed and characterised. As a basic tool for this characterisation, self-dual -valued capacities or their counterparts, self-dual monotone Boolean functions, are exploited. A generating role of is discussed. The relevant role of simple medians in describing monotone threshold Boolean functions is made apparent.

On the fractional chromatic number of monotone self-dual Boolean functions

We compute the exact fractional chromatic number for several classes of monotone self-dual Boolean functions. We characterize monotone self-dual Boolean functions in terms of the optimal value of an LP relaxation of a suitable strengthening of the standard IP formulation for the chromatic number. We also show that determining the self-duality of a monotone Boolean function is equivalent to determining the feasibility of a certain point in a polytope defined implicitly.

Exploiting Regularities for Boolean Function Synthesis

The "regularity" of a Boolean function can be exploited for decreasing its minimization time. It has already been shown that the notion of autosymmetry is a valid measure of regularity, however such a notion has been studied thus far either in the theoretical framework of self-dual Boolean functions, or for the synthesis of a particular family of three-level logic networks. In this paper we show that the degree of autosymmetry of an arbitrary function can be computed implicitly in a very efficient way, and autosymmetry can then be exploited in any logic minimization context. Our algorithms make crucial use of Binary Decision Diagrams. A set of experimental results on the synthesis of standard benchmark functions substantiates the practical relevance of our theoretical results.

Decompositions of positive self-dual boolean functions

A coterie, which is used to realize mutual exclusion in distributed systems, is a family C of subsets such that any pair of subsets in C has at least one element in common, and such that no subset in C contains any other subset in C. Associate with a family of subsets C a positive Boolean function f c such that f c ( x) = 1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f c is dual-minor, and is a non-dominated (ND) coterie if and only if f c is self-dual. We study in this paper the decomposition of a positive self-dual function into smaller positive self-dual functions, as it explains how to represent and how to construct the corresponding ND coterie. A key step is how to decompose a positive dual-minor function f into a conjunction of positive self-dual functions f 1, f 2,…, f k. In addition to the general condition for this decomposition, we clarify the condition for the decomposition into two functions f 1, and f 2, and introduce the concept of canonical decomposition. Then we present an algorithm that determines a minimal canonical decomposition, and a very simple algorithm that usually gives a decomposition close to minimal. The decomposition of a general self-dual function is also discussed.