Conditions of the Affine Extension of an Incompletely Defined Boolean Function

The paper presents conditions of extension of the weakly defined Boolean functions to their full affine form. The main goal of the analysis is a fast estimation whether a given incompletely defined function can be extended to a full affine form. If it is possible a simple algorithm of the states completion has been proposed. In such a case undefined points are clearly replaced by 0, 1 values. Spectral coefficients of a Boolean function allow to determine whether a partially defined function can be realised as affine

Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear pre- and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment. This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra directly from Boolean equations. By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can be easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequently used in digital logic design. A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC- cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed. Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don\u27t cares. Don\u27t care minterms are simply represented in the form of disjoint cubes. The links between spectral and classical methods used for designing digital circuits are described. The real meaning of spectral coefficients from Walsh and other orthogonal spectra in classical logic terms is shown. The relations presented here can be used for the calculation of different transforms. The methods are based on direct manipulations on Karnaugh maps. The conversion start with Karnaugh maps and generate the spectral coefficients. The spectral representation of multiple-valued input binary functions is proposed here for the first time. Such a representation is composed of a vector of Walsh transforms each vector is defined for one pair of the input variables of the function. The new representation has the advantage of being real-valued, thus having an easy interpretation. Since two types of codings of values of binary functions are used, two different spectra are introduced. The meaning of each spectral coefficient in classical logic terms is discussed. The mathematical relationships between the number of true, false, and don\u27t care minterms and spectral coefficients are stated. These relationships can be used to calculate the spectral coefficients directly from the graphical representations of binary functions. Similarly to the spectral methods in classical logic design, the new spectral representation of binary functions can find applications in many problems of analysis, synthesis, and testing of circuits described by such functions. A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME). Since the known fast algorithm that generates the GRME, based on the factorization of the Reed-Muller transform matrix, always starts from the truth table (minterms) of a Boolean function, then the described method has advantages due to a smaller required computer memory. Moreover, for Boolean functions, described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. By investigating a family of elementary second order matrices, new transforms of real vectors are introduced. When used for Boolean function transformations, these transforms are one-to-one mappings in a binary or ternary vector space. The concept of different polarities of the Arithmetic and Adding transforms has been introduced. New operations on matrices: horizontal, vertical, and vertical-horizontal joints (concatenations) are introduced. All previously known transforms, and those introduced in this dissertation can be characterized by two features: ordering and polarity . When a transform exists for all possible polarities then it is said to be generalized . For all of the transforms discussed, procedures are given for generalizing and defining for different orderings. The meaning of each spectral coefficient for a given transform is also presented in terms of standard logic gates. There exist six commonly used orderings of Walsh transforms: Hadamard, Rademacher, Kaczmarz, Paley, Cal-Sal, and X. By investigating the ways in which these known orderings are generated the author noticed that the same operations can be used to create some new orderings. The generation of two new Walsh transforms in Gray code orderings, from the straight binary code is shown. A recursive algorithm for the Gray code ordered Walsh transform is based on the new operator introduced in this presentation under the name of the bi-symmetrical pseudo Kronecker product . The recursive algorithm is the basis for the flow diagram of a constant geometry fast Walsh transform in Gray code ordering. The algorithm is fast (N 10g2N additions/subtractions), computer efficient, and is implemente

Walsh spectral techniques for logic synthesis FPGA

The implementation value of multi-output Boolean functions in logic synthesis FPGA can be reduced by using Walsh spectral representation. This paper proposes an algorithm for calculating the maximum coefficient of the autocorrelation function of BF without generating a truth table, using the heuristic procedure limits the maximum autocorrelation coefficients of sorting on a small subset of the function. We also suggest a spectral technique of the linear function transformation defined by disjoint cubes. This method for decomposition of BF, which allows to reducing the complexity of the linear part of the corresponding blocks about 25-55%, and the complexity of the nonlinear part of the blocks do not increase more than 10%, compared to the traditional approach

An effective cube comparison method for discrete spectral transformations of logic functions

Spectral methods have been used for many applications in digital logic design, digital signal processing and telecommunications. In digital logic design they are implemented for testing of logical networks, multiplexer-based logic synthesis, signal processing, image processing and pattern analysis. New developments of more efficient algorithms for spectral transformations (Rademacher-Walsh, Generalized Reed-Muller, Adding, Arithmetic, multiple-valued Walsh and multiple-valued Generalized Reed- Muller) their implementation and applications will be described

Graph Laplacians, Nodal Domains, and Hyperplane Arrangements

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e. the connected components of the maximal induced subgraphs of G on which an eigenvector ψ does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of ψ in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures

Integrating protein structural information

Dissertação apresentada para obtenção de Grau de Doutor em Bioquímica,Bioquímica Estrutural, pela Universidade Nova de Lisboa, Faculdade de Ciências e TecnologiaThe central theme of this work is the application of constraint programming and other artificial intelligence techniques to protein structure problems, with the goal of better combining experimental data with structure prediction methods. Part one of the dissertation introduces the main subjects of protein structure and constraint programming, summarises the state of the art in the modelling of protein structures and complexes, sets the context for the techniques described later on, and outlines the main points of the thesis: the integration of experimental data in modelling. The first chapter, Protein Structure, introduces the reader to the basic notions of amino acid structure, protein chains, and protein folding and interaction. These are important concepts to understand the work described in parts two and three. Chapter two, Protein Modelling, gives a brief overview of experimental and theoretical techniques to model protein structures. The information in this chapter provides the context of the investigations described in parts two and three, but is not essential to understanding the methods developed. Chapter three, Constraint Programming, outlines the main concepts of this programming technique. Understanding variable modelling, the notions of consistency and propagation, and search methods should greatly help the reader interested in the details of the algorithms, as described in part two of this book. The fourth chapter, Integrating Structural Information, is a summary of the thesis proposed here. This chapter is an overview of the objectives of this work, and gives an idea of how the algorithms developed here could help in modelling protein structures. The main goal is to provide a flexible and continuously evolving framework for the integration of structural information from a diversity of experimental techniques and theoretical predictions. Part two describes the algorithms developed, which make up the main original contribution of this work. This part is aimed especially at developers interested in the details of the algorithms, in replicating the results, in improving the method or in integrating them in other applications. Biochemical aspects are dealt with briefly and as necessary, and the emphasis is on the algorithms and the code

Contextuality in foundations and quantum computation

Contextuality is a key concept in quantum theory. We reveal just how important it is by demonstrating that quantum theory builds on contextuality in a fundamental way: a number of key theorems in quantum foundations can be given a unifi ed presentation in the topos approach to quantum theory, which is based on contextuality as the common underlying principle. We review existing results and complement them by providing contextual reformulations for Stinespring's and Bell's theorem. Both have a number of consequences that go far beyond the evident confirmation of the unifying character of contextuality in quantum theory. Complete positivity of quantum channels is already encoded in contexts, nonlocality arises from a notion of composition of contexts, and quantum states can be singled out among more general non-signalling correlations over the composite context structure by a notion of time orientation in subsystems, thus solving a much discussed open problem in quantum information theory. We also discuss nonlocal correlations under the generalisation to orthomodular lattices and provide generalised Bell inequalities in this setting. The dominant role of contextuality in quantum foundations further supports a recent hypothesis in quantum computation, which identifi es contextuality as the resource for the supposed quantum advantage over classical computers. In particular, within the architecture of measurement-based quantum computation, the resource character of nonlocality and contextuality exhibits rather clearly. We study contextuality in this framework and generalise the strong link between contextuality and computation observed in the qubit case to qudit systems. More precisely, we provide new proofs of contextuality as well as a universal implementation of computation in this setting, while emphasising the crucial role played by phase relations between measurement eigenstates. Finally, we suggest a fine-grained measure for contextuality in the form of the number of qubits required for implementation in the non-adaptive, deterministic case.Open Acces

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum