Self-dual Functions Research Articles

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.

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.

Quantum Spectral Problems and Isomonodromic Deformations

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2times 2 linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé mathrm {III}_3 equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (epsilon _1+epsilon _2=0). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (epsilon _2=0). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the epsilon _2rightarrow 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) and Multi-Species IRW

We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP$(2j)$) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have $n>1$ species of particles. In addition, we allow up to $2j$ particles to occupy each site in the multi-species SEP$(2j)$. The duality functions for the multi-species SEP$(2j)$ and the multi-species IRW come from unitary intertwiners between different $*$-representations of the special linear Lie algebra $\mathfrak{sl}_{n+1}$ and the Heisenberg Lie algebra $\mathfrak{h}_n$, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP$(2j)$ and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.

О метрических свойствах множества самодуальных бент-функций

О метрических свойствах множества самодуальных бент-функций

Metrical properties of self-dual bent functions

In this paper we study metrical properties of Boolean bent functions which coincide with their dual bent functions. We propose an iterative construction of self-dual bent functions in $$n+2$$ variables through concatenation of two self-dual and two anti-self-dual bent functions in n variables. We prove that minimal Hamming distance between self-dual bent functions in n variables is equal to $$2^{n/2}$$. It is proved that within the set of sign functions of self-dual bent functions in $$n\geqslant 4$$ variables there exists a basis of the eigenspace of the Sylvester Hadamard matrix attached to the eigenvalue $$2^{n/2}$$. Based on this result we prove that the sets of self-dual and anti-self-dual bent functions in $$n\geqslant 4$$ variables are mutually maximally distant. It is proved that the sets of self-dual and anti-self-dual bent functions in n variables are metrically regular sets.

Обнаружение неисправностей в комбинационных схемах на основе самодвойственного дополнения до равновесных кодов

A new method of combinational circuits concurrent checking is considered based on Boolean complement of the working functions of the diagnosis object to the constant-weight code words with parallel checkout of the bits functions belonging to the self-dual functions class of the Boolean logic. The described approach to the organization of concurrent error-detection (CED) systems allows to increase the detection ability in comparison with the traditional checkout by constant-weight code. In this case, the undetectable error can be a such error, which will lead to the preservation of the code word weight generated at the outputs of the Boolean complement block, if the distortions of all the functions bits on the opposite sets is coincide. This condition is more stringent than the checkout of the vector belonging a pre-selected code, it means that it gives a greater probability of detecting distortion arising from errors. The described method allows checkout of combinational circuit based on constant-weight codes with the same number of single and zero bits. In this case, the priority is in the widely used «2-out-of-4» code, which has a simple checker structure. A detailed description of the new CED system structure is given, and algorithms for the extension of definition of the values of the checkout functions that provide testability of technical diagnostic tools are given.

Self-Duality of Markov Processes and Intertwining Functions

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitary equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.

О некоторых свойствах самодуальных бент-функций

О некоторых свойствах самодуальных бент-функций

Factorized Duality, Stationary Product Measures and Generating Functions

We find all self-duality functions of the form D(ξ,η)=∏xd(ξx,ηx)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} D(\\xi , \\eta )= \\prod _{x} d(\\xi _x, \\eta _x) \\end{aligned}$$\\end{document}for a class of interacting particle systems. We call these duality functions of simple factorized form. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in Giardinà et al. (J Stat Phys 135:25–55, 2009) and, more recently, in Franceschini and Giardinà (Preprint, arXiv:1701.09115, 2016) . Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all simple factorized self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.

The Hamming Distance Spectrum Between Self-Dual Maiorana–McFarland Bent Functions

A bent function is self-dual if it is equal to its dual function. We study the metric properties of the self-dual bent functions constructed on using available constructions. We find the full Hamming distance spectrum between self-dual Maiorana–McFarland bent functions. Basing on this, we find the minimal Hamming distance between the functions under study.

Self-duality and shock dynamics in the n-species priority ASEP

We construct all invariant measures of the n-species priority asymmetric simple exclusion process with reflecting boundaries and prove reversibility. Using the symmetry of the generator of the process under the quantum algebra Uq[gl(n+1)] we derive self-duality functions. From these we obtain in explicit form the time evolution on Z of a family of measures with K shocks in terms of the transition probability of a shock exclusion process with K coloured particles with particle-dependent hopping rates and nearest-neighbour colour exchange. This process is a gas of particles that form a bound state, corresponding to shock coalescence on macroscopic scale.

Moving redundancy of tolerant elements

Redundancy is one of the primary ways of improving dependability. In particular, structural redundancy is used. In such cases fail-safe operation of elements, devices and systems can be ensured. Fail-safety can enable mitigation of both faults and failures. The paper examines the matter of increasing dependability by means of the so-called sliding redundancy that ensures the health of systems of n elements with m redundant elements that can replace any of the main elements. It is proposed to improve sliding redundancy through recovery of elements out of a number of failed elements that have retained some functionality (basis). For example, the basis of the logical (Boolean) function in terms of Post’s theorem is available if such function is not a zero-preserving function, not a one-preserving function, not a self-dual function, not a line function, not a monotone function. Previously, the author proposed the so-called functionally complete tolerant logical functions (FCTF) that do not only possess functional completeness but retain it under the specified failure model. Then even a failed element remains functionally complete, yet with reduced capabilities, e.g. becomes a 2OR-NOT, though the FCTF can be implemented with an element 2AND-2OR-NOT. In this case the recovery of the original function requires several 2OR-NOT elements. However, the diagnostics of such elements and their reconfiguration in case of failure are problematic. This approach can be interpreted with logic recovery of programmable logic devices (PLD) that is based on the so-called Look Up Tables (LUT) that are memory devices based on 16:1 multiplexers. The circuit is a transmitting transistor tree. If they fail, the healthy half of LUT can be used. By means of reconfiguration using standard PLD facilities that contain local and global connections matrix, such “semi-LUTs” can be transformed into LUTs whose functions are equivalent to initial ones. That equals to an increase of the number of redundant elements. Sliding redundancy with recovery of elements out of several failed ones that retained the basis can be used in critical system applications when repair or replacement of elements is impossible. The article proposes a formula that takes such recovery into consideration, analyzes the special features of such redundancy and evaluates the advantages for dependability.

О множестве расстояний Хэмминга между самодуальными бентфункциями

О множестве расстояний Хэмминга между самодуальными бентфункциями

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

AbstractWe study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by $$q\in (0,1)$$ q ∈ ( 0 , 1 ) and where at most $$2j\in \mathbb {N}$$ 2 j ∈ N particles per site are allowed. The process is constructed from a $$(2j+1)$$ ( 2 j + 1 ) -dimensional representation of a quantum Hamiltonian with $$U_q(\mathfrak {sl}_2)$$ U q ( sl 2 ) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several self-duality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure.

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.

Boolean functions with MacWilliams duality

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree n?k for any nonnegative integer k and n ? N with some positive integer N under a certain condition.

A stronger LP bound for formula size lower bounds via clique constraints

We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan and the theory of stable set polytopes. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko. Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy.

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.