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Schardijn, Amy. "AN INTRODUCTION TO BOOLEAN ALGEBRAS." CSUSB ScholarWorks, 2016. https://scholarworks.lib.csusb.edu/etd/421.
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This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures.
We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem.
Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.
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Bell, Steven. "Modular homology in the Boolean algebra." Thesis, University of East Anglia, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368141.
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Tarnoff, David. "Episode 4.05 – Introduction to Boolean Algebra." Digital Commons @ East Tennessee State University, 2020. https://dc.etsu.edu/computer-organization-design-oer/33.
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Truth tables and circuit diagrams fall short in many ways including their abilities to evaluate and manipulate combinational logic. By using algebraic methods to represent logic expressions, we can apply properties and identities to improve performance.
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Tarnoff, David. "Episode 4.06 – Properties of Boolean Algebra." Digital Commons @ East Tennessee State University, 2020. https://dc.etsu.edu/computer-organization-design-oer/34.
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Tarnoff, David. "Episode 4.07 – Identities of Boolean Algebra." Digital Commons @ East Tennessee State University, 2020. https://dc.etsu.edu/computer-organization-design-oer/35.
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We are familiar with algebraic laws such as multiply zero by anything, and we get zero. In this episode, we see how a Boolean expression containing a constant, a duplicated signal, or a signal being combined with its inverse will simplify…always.
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Van, Name Joseph Anthony. "Boolean Partition Algebras." Scholar Commons, 2013. http://scholarcommons.usf.edu/etd/4599.
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A Boolean partition algebra is a pair $(B,F)$ where $B$ is a Boolean
algebra and $F$ is a filter on the semilattice of partitions of $B$ where $\bigcup F=B\setminus\{0\}$. In this dissertation, we shall investigate the algebraic theory of Boolean partition algebras and their connection with uniform spaces. In particular, we shall show that the category of complete non-Archimedean uniform spaces
is equivalent to a subcategory of the category of Boolean partition algebras, and notions such as supercompleteness
of non-Archimedean uniform spaces can be formulated in terms of Boolean partition algebras.
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Schimanski, Nichole Louise. "Orthomorphisms of Boolean Groups." PDXScholar, 2016. http://pdxscholar.library.pdx.edu/open_access_etds/3100.
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An orthomorphism, π, of a group, (G, +), is a permutation of G with the property that the map x → -x + π(x) is also a permutation. In this paper, we consider orthomorphisms of the additive group of binary n-tuples, Zn2. We use known orthomorphism preserving functions to prove a uniformity in the cycle types of orthomorphisms that extend certain partial orthomorphisms, and prove that extensions of particular sizes of partial orthomorphisms exist. Further, in studying the action of conjugating orthomorphisms by automorphisms, we find several symmetries within the orbits and stabilizers of this action, and other orthomorphism-preserving functions. In addition, we prove a lower bound on the number of orthomorphisms of Zn2 using the equivalence of orthomorphisms to transversals in Latin squares. Lastly, we present a Monte Carlo method for generating orthomorphisms and discuss the results of the implementation.
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Geschke, Stefan. "On s-filtered [sigma-filtered] Boolean algebras." [S.l. : s.n.], 1999. http://www.diss.fu-berlin.de/2000/70/index.html.
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Hadida, Ahmed Mohamed. "A partially ordered semigroup of Boolean spaces." Diss., The University of Arizona, 1988. http://hdl.handle.net/10150/184369.
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In this thesis we are concerned with arithmetic in a certain partially ordered, commutative semigroup D. The first chapter investigates the class of countable Boolean algebras from which this semigroup arises. The elements of D correspond to the isomorphism classes of the Boolean algebras under consideration. In Chapter 2 we begin the study of the semigroup structure of D. D is axiomatically described by three groups of axioms. It is proved that these axioms are categorical. The ordering of D is used to investigate the multiplication. The set of T of torsion elements of D (elements with only finite many distinct powers), form a subsemigroup whose structure is studied. There is a natural torsion free quotient D/T whose structure is also investigated. In Chapter 3, the axioms are used to characterize elements s of T in terms of the arithmetic in the subsemigroup generated by the elements that are smaller than s. The characterization is used to determine elements of T that cover a single element. In the last part of Chapter 3, we obtain some sufficient, purely combinatorial conditions for an element to have infinite order.
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Bozeman, Alan Kyle. "Weakly Dense Subsets of Homogeneous Complete Boolean Algebras." Thesis, University of North Texas, 1990. https://digital.library.unt.edu/ark:/67531/metadc330803/.
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The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B.
Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity.
In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B).
In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras.
In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), i.e., the weak density of a homogeneous complete Boolean algebra B is at least as big as the cellularity. Also in this chapter, we introduce discernible sets. We prove that a discernible set of cardinality no greater than c(B) cannot be weakly dense.
In chapter VI, we prove the main result of this dissertation, i.e., d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}). In chapter VII, we list some unsolved problems concerning this dissertation.
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Bellini, Emanuele. "Computational techniques for nonlinear codes and Boolean functions." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/369066.
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We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds, which are an improvement of a bound by Zinoviev, Litsyn and Laihonen, are independent of other classical known theoretical bounds. Among these, we mention the Griesmer bound for linear codes, of which we provide a partial generalization for the systematic case. Our experiments show that in some cases (the majority of cases for some q) our bounds provide the best value, compared to all other closed-formula upper-bounds. We also present an algebraic method for computing the minimum weight of any nonlinear code. We show that for some particular code, using a non-standard representation of the code, our method is faster than brute force. An application of this method allows to compute the nonlinearity of a Boolean function, improving existing algebraic methods and reaching the same complexity of algorithms based on the fast Fourier transform.
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Sengupta, Rimli. "Lower bounds for natural functions in restricted boolean circuits." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/8269.
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Bellini, Emanuele. "Computational techniques for nonlinear codes and Boolean functions." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1376/1/00-Thesis.pdf.
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We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds, which are an improvement of a bound by Zinoviev, Litsyn and Laihonen, are independent of other classical known theoretical bounds. Among these, we mention the Griesmer bound for linear codes, of which we provide a partial generalization for the systematic case. Our experiments show that in some cases (the majority of cases for some q) our bounds provide the best value, compared to all other closed-formula upper-bounds. We also present an algebraic method for computing the minimum weight of any nonlinear code. We show that for some particular code, using a non-standard representation of the code, our method is faster than brute force. An application of this method allows to compute the nonlinearity of a Boolean function, improving existing algebraic methods and reaching the same complexity of algorithms based on the fast Fourier transform.
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Chen, Xi, and 陈曦. "On construction and control of probabilistic Boolean networks." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B48329605.
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Modeling gene regulation is an important problem in genomic research. The Boolean network (BN) and its generalization Probabilistic Boolean network (PBN) have been proposed to model genetic regulatory interactions.
BN is a deterministic model while PBN is a stochastic model. In a PBN, on one hand, its stationary distribution gives important information about the long-run behavior of the network. On the other hand, one may be interested in system synthesis which requires the construction of networks from the observed stationary distribution. This results in an inverse problem of constructing PBNs from a given stationary distribution and a given set of Boolean Networks (BNs), which is ill-posed and challenging, because there may be many networks or no network having the given properties and the size of the inverse problem is huge. The inverse problem is first formulated as a constrained least squares problem. A heuristic method is then proposed based on the conjugate gradient (CG) algorithm, an iterative method, to solve the resulting least squares problem. An estimation method for the parameters of the PBNs is also discussed. Numerical examples are then given to demonstrate the effectiveness of the proposed methods.
However, the PBNs generated by the above algorithm depends on the initial guess and is not unique. A heuristic method is then proposed for generating PBNs from a given transition probability matrix. Unique solution can be obtained in this case. Moreover, these algorithms are able to recover the dominated BNs and therefore the major structure of the network.
To further evaluate the feasible solutions, a maximum entropy approach is proposed using entropy as a measure of the fitness. Newton’s method in conjunction with the CG method is then applied to solving the inverse problem. The convergence rate of the proposed method is demonstrated. Numerical examples are also given to demonstrate the effectiveness of our proposed method.
Another important problem is to find the optimal control policy for a PBN so as to avoid the network from entering into undesirable states. By applying external control, the network is desired to enter into some state within a few time steps. For PBN CONTROL, people propose to find a control sequence such that the network will terminate in the desired state with a maximum probability. Also, the problem of minimizing the maximum cost is considered. Integer linear programming (ILP) and dynamic programming (DP) in conjunction with hard constraints are then employed to solve the above problems. Numerical experiments are given to demonstrate the effectiveness of our algorithms. A hardness result is demonstrated and suggests that PBN CONTROL is harder than BN CONTROL. In addition, deciding the steady state probability in PBN for a specified global state is demonstrated to be NP-hard.
However, due to the high computational complexity of PBNs, DP method is computationally inefficient for a large size network. Inspired by the state reduction strategies studied in [86], the DP method in conjunction with state reduction approach is then proposed to reduce the computational cost of the DP method. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method.published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy
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Jiao, Yue. "Mathematical models for control of probabilistic Boolean networks." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B41508634.
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Bowen, Richard Strong. "Minimal Circuits for Very Incompletely Specified Boolean Functions." Scholarship @ Claremont, 2010. https://scholarship.claremont.edu/hmc_theses/18.
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In this report, asymptotic upper and lower bounds are given for the minimum number of gates required to compute a function which is only partially specified and for which we allow a certain amount of error. The upper and lower bounds match. Hence, the behavior of these minimum circuit sizes is completely (asymptotically) determined.
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Oliveira, Anderson José de 1985. "Análise algébrica dos rotulamentos associados ao mapeamento do código genético." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261596.
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Orientador: Reginaldo Palazzo JúniorDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
Made available in DSpace on 2018-08-19T17:49:58Z (GMT). No. of bitstreams: 1 Oliveira_AndersonJosede_M.pdf: 1619063 bytes, checksum: 79a49301084eecde745f0e73cddfc1fa (MD5) Previous issue date: 2012
Resumo: Uma área de pesquisa em franca expansão é a modelagem matemática do código genético, por meio da qual pode-se identificar as características e propriedades do mesmo. Neste trabalho apresentamos alguns modelos matemáticos aplicados à biologia, especificamente relacionado ao código genético. Os objetivos deste trabalho são: a) caracterização da hidropaticidade dos aminoácidos através da construção de reticulados booleanos e diagramas de Hasse associados a cada rotulamento do código genético, b) proposta de um algoritmo soma com transporte para efetuar a soma entre códons, ferramenta importante em análises mutacionais, c) representação polinomial dos códons do código genético, d) comparação dos resultados dos rotulamentos A, B e C em cada uma das modelagens construídas, e) análise do comportamento dos aminoácidos em cada um dos rotulamentos do código genético. Os resultados encontrados permitem a utilização de tais ferramentas em diversas áreas do conhecimento como bioinformática, biomatemática, engenharia genética, etc, devido a interdisciplinaridade do trabalho, onde elementos de biologia, matemática e engenharia foram utilizados
Abstract: A research area in frank expansion is the mathematical modeling of the genetic code, through can identify the characteristics and properties of them. In this paper we present some mathematical models applied to biology, specifically related to the genetic code. The aims of this work are: a) a characterization of the hydropathy of the amino acids through the construction of boolean lattices and Hasse diagrams associated with each labeling of the genetic code, b) the proposal of a sum algorithm of transportation to make the sum of codons, important tool in mutational analysis, c) a polynomial representation of the codons of the genetic code, d) a comparing of the results of the A, B and C labels in each of the built modeling, e) an analysis of the behavior of the amino acids in each of the labels of the genetic code. The results allow the use of such tools in a lot of areas like bio informatics, biomathematics, genetic engineering, etc., due to the interdisciplinary of the paper, where elements of biology, mathematics and engineering were used
Mestrado
Telecomunicações e Telemática
Mestre em Engenharia Elétrica
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Vora, Rohit H. "An Algorithm for multi-output Boolean logic minimization." Thesis, Virginia Tech, 1987. http://hdl.handle.net/10919/43829.
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A new algorithm is presented for a guaranteed absolute minimal solution to the problem of Boolean Logic Minimization in its most generalized form of multi-output function with arbitrary cost criterion.
The proposed algorithm is shown to be tighter than the Quine-McCluskey method in its ability to eliminate redundant prime implicants, making it possible to simplify the cyclic tables. In its final form, the proposed algorithm is truly concurrent in generation of prime implicants and construction of minimal forms. A convenient and efficient technique is used for identifying existing prime implicants. Branch-and-bound method is employed to restrict the search tree to a cost cut-off value depending on the definition of cost function specified.
A most generalized statement of the algorithm is formulated for manual as well as computer implementation and its application is illustrated with an example. A program written in Pascal, for classical diode-gate cost function as well as PLA-area cost function, is developed and tested for an efficient computer implementation. Finally, various advantages of the proposed approach are pointed out by comparing it with the classical approach of Quine-McCluskey method.Master of Science
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Calderini, Marco. "On Boolean functions, symmetric cryptography and algebraic coding theory." Doctoral thesis, Università degli studi di Trento, 2015. https://hdl.handle.net/11572/369089.
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In the first part of this thesis we report results about some “linear†trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums.
We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we
investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given.
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Jiao, Yue, and 焦月. "Mathematical models for control of probabilistic Boolean networks." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B41508634.
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Chakrapani, Lakshmi Narasimhan. "Probabilistic boolean logic, arithmetic and architectures." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/26706.
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Thesis (Ph.D)--Computing, Georgia Institute of Technology, 2009.Committee Chair: Palem, Krishna V.; Committee Member: Lim, Sung Kyu; Committee Member: Loh, Gabriel H.; Committee Member: Mudge, Trevor; Committee Member: Yalamanchili, Sudhakar. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Calderini, Marco. "On Boolean functions, symmetric cryptography and algebraic coding theory." Doctoral thesis, University of Trento, 2015. http://eprints-phd.biblio.unitn.it/1498/1/template_tesi.pdf.
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In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums.
We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we
investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given.
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Gwynne, Matthew. "Hierarchies for efficient clausal entailment checking : with applications to satisfiability and knowledge compilation." Thesis, Swansea University, 2014. https://cronfa.swan.ac.uk/Record/cronfa42854.
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Falkowski, Bogdan Jaroslaw. "Spectral Methods for Boolean and Multiple-Valued Input Logic Functions." PDXScholar, 1991. https://pdxscholar.library.pdx.edu/open_access_etds/1152.
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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't cares. Don't 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't 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 implemented
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Roy, Amitabha. "Symmetry breaking and fault tolerance in boolean satisfiability /." view abstract or download file of text, 2001. http://wwwlib.umi.com/cr/uoregon/fullcit?p3024528.
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Thesis (Ph. D.)--University of Oregon, 2001.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 124-127). Also available for download via the World Wide Web; free to University of Oregon users.
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Upadrasta, Bharat. "Boolean factor analysis a review of a novel method of matrix decomposition and neural network Boolean factor analysis /." Diss., Online access via UMI:, 2009.
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Thesis (M.S.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009.Includes bibliographical references.
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Habineza, Olivier. "Graphs of integral distance and their properties." University of Western Cape, 2021. http://hdl.handle.net/11394/8430.
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Philosophiae Doctor - PhDUnderstanding the geometries of points in space has been attractive to mathematicians for ages. As a model, twelve years ago, Kurz and Meyer [32] considered point sets in the m-dimensional a ne space Fmq over a nite eld Fq with q = pr elements, p prime, where each squared Euclidean distance of two points is a square in Fq: The latter points are said to be at integral distance in Fmq , and the sets above are called integral point sets.
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Peh, Lawrence T. W. "An efficient algorithm for extracting Boolean functions from linear threshold gates, and a synthetic decompositional approach to extracting Boolean functions from feedforward neural networks with arbitrary transfer functions." University of Western Australia. Dept. of Computer Science, 2000. http://theses.library.uwa.edu.au/adt-WU2003.0013.
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[Formulae and special characters can only be approximated here. Please see the pdf version of the Abstract for an accurate reproduction.] Artificial neural networks are universal function approximators that represent functions subsymbolically by weights, thresholds and network topology. Naturally, the representation remains the same regardless of the problem domain. Suppose a network is applied to a symbolic domain. It is difficult for a human to dynamically construct the symbolic function from the neural representation. It is also difficult to retrain networks on perturbed training vectors, to resume training with different training sets, to form a new neuron by combining trained neurons, and to reason with trained neurons. Even the original training set does not provide a symbolic representation of the function implemented by the trained network because the set may be incomplete or inconsistent, and the training phase may terminate with residual errors. The symbolic information in the network would be more useful if it is available in the language of the problem domain. Algorithms that translate the subsymbolic neural representation to a symbolic representation are called extraction algorithms. I argue that extraction algorithms that operate on single-output, layered feedforward networks are sufficient to analyse the class of multiple-output networks with arbitrary connections, including recurrent networks. The translucency dimensions of the ADT taxonomy for feedforward networks classifies extraction approaches as pedagogical, eclectic, or decompositional. Pedagogical and eclectic approaches typically use a symbolic learning algorithm that takes the network’s input-output behaviour as its raw data. Both approaches construct a set of input patterns and observe the network’s output for each pattern. Eclectic and pedagogical approaches construct the input patterns respectively with and without reference to the network’s internal information. These approaches are suitable for approximating the network’s function using a probably-approximately-correct (PAC) or similar framework, but they are unsuitable for constructing the network’s complete function. Decompositional approaches use internal information from a network more directly to produce the network’s function in symbolic form. Decompositional algorithms have two components. The first component is a core extraction algorithm that operates on a single neuron that is assumed to implement a symbolic function. The second component provides the superstructure for the first. It consists of a decomposition rule for producing such neurons and a recomposition rule for symbolically aggregating the extracted functions into the symbolic function of the network. This thesis makes contributions to both components for Boolean extraction. I introduce a relatively efficient core algorithm called WSX based on a novel Boolean form called BvF. The algorithm has a worst case complexity of O(2 to power of n divided by the square root of n) for a neuron with n inputs, but in all cases, its complexity can also be expressed as O(l) with an O(n) precalculation phase, where l is the length of the extracted expression in terms of the number of symbols it contains. I extend WSX for approximate extraction (AWSX) by introducing an interval about the neuron’s threshold. Assuming that the input patterns far from the threshold are more symbolically significant to the neuron than those near the threshold, ASWX ignores the neuron’s mappings for the symbolically input patterns, remapping them as convenient for efficiency. In experiments, this dramatically decreased extraction time while retaining most of the neuron’s mappings for the training set. Synthetic decomposition is this thesis’ contribution to the second component of decompositional extraction. Classical decomposition decomposes the network into its constituent neurons. By extracting symbolic functions from these neurons, classical decomposition assumes that the neurons implement symbolic functions, or that approximating the subsymbolic computation in the neurons with symbolic computation does not significantly affect the network’s symbolic function. I show experimentally that this assumption does not always hold. Instead of decomposing a network into its constituent neurons, synthetic decomposition uses constraints in the network that have the same functional form as neurons that implement Boolean functions; these neurons are called synthetic neurons. I present a starting point for constructing synthetic decompositional algorithms, and proceed to construct two such algorithms, each with a different strategy for decomposition and recomposition. One of the algorithms, ACX, works for networks with arbitrary monotonic transfer functions, so long as an inverse exists for the functions. It also has an elegant geometric interpretation that leads to meaningful approximations. I also show that ACX can be extended to layered networks with any number of layers.
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Akishev, Galym. "Monadic bounded algebras : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics /." ResearchArchive@Victoria e-Thesis, 2009. http://hdl.handle.net/10063/915.
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McHard, Richard William. "Sperner properties of the ideals of a Boolean lattice." Diss., [Riverside, Calif.] : University of California, Riverside, 2009. http://proquest.umi.com/pqdweb?index=0&did=1957320841&SrchMode=2&sid=2&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1268340717&clientId=48051.
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Thesis (Ph. D.)--University of California, Riverside, 2009.Includes abstract. Title from first page of PDF file (viewed March 11, 2010). Available via ProQuest Digital Dissertations. Includes bibliographical references (p. 170-172). Also issued in print.
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Gologlu, Faruk. "Divisibility Properties On Boolean Functions Using The Numerical Normal Form." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/2/12605549/index.pdf.
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A Boolean function can be represented in several different forms. These different
representation have advantages and disadvantages of their own. The Algebraic Normal
Form, truth table, and Walsh spectrum representations are widely studied in
literature. In 1999, Claude Carlet and Phillippe Guillot introduced the Numerical
Normal Form. NumericalNormal Form(NNF) of a Boolean function is similar to Algebraic
Normal Form, with integer coefficients instead of coefficients from the two
element field. Using NNF representation, just like the Walsh spectrum, characterization
of several cryptographically important functions, such as resilient and bent
functions, is possible. In 2002, Carlet had shown several divisibility results concerning
resilient and correlation-immune functions using NNF. With these divisibility
results, Carlet is able to give bounds concerning nonlinearity of resilient and correlation
immune functions.
In this thesis, following Carlet and Guillot, we introduce the Numerical Normal
Form and derive the pairwise relations between the mentioned representations.
Characterization of Boolean, resilient and bent functions using NNF is also given.
We then review the divisibility results of Carlet, which will be linked to some results
on the nonlinearity of resilient and correlation immune functions.
We show the Möbius inversion properties of NNF of a Boolean function, using Gian-Carlo Rota&rsquo
s work as a guide. Finally, using a lot of the mentioned results, we prove a necessary condition on theWalsh spectrum of Boolean functions with given degree.
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Hacker, Charles Hilton, and n/a. "WinLogiLab - A Computer-Based Teaching Suite for Digital Logic Design." Griffith University. School of Engineering, 2001. http://www4.gu.edu.au:8080/adt-root/public/adt-QGU20050915.172404.
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This thesis presents an interactive computerised teaching suite developed for the design of combinatorial and sequential logic circuits. This suite fills a perceived gap in the currently available computer-based teaching software for digital logic design. Several existing digital logic educational software are available, however these existing programs were found to be unsuitable for our use in providing alternative mode subject delivery. This prompted the development of a Microsoft Windows TM tutorial suite, called WinLogiLab. WinLogiLab comprises of a set of tutorials that uses student provided input data, to perform the initial design steps for digital Combinatorial and Sequential logic circuits. The combinatorial tutorials are designed to show the link between Boolean Algebra and Digital Logic circuits, and follows the initial design steps: from Boolean algebra, truth tables, to Exact and the Heuristic minimisation techniques, to finally produce the combinatorial circuit. Similarly, the sequential tutorials can design simple State Machine Counters, and can model more complex Finite State Automata.
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Hacker, Charles. "WinLogiLab - A Computer-Based Teaching Suite for Digital Logic Design." Thesis, Griffith University, 2001. http://hdl.handle.net/10072/367209.
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This thesis presents an interactive computerised teaching suite developed for the design of combinatorial and sequential logic circuits. This suite fills a perceived gap in the currently available computer-based teaching software for digital logic design. Several existing digital logic educational software are available, however these existing programs were found to be unsuitable for our use in providing alternative mode subject delivery. This prompted the development of a Microsoft Windows TM tutorial suite, called WinLogiLab. WinLogiLab comprises of a set of tutorials that uses student provided input data, to perform the initial design steps for digital Combinatorial and Sequential logic circuits. The combinatorial tutorials are designed to show the link between Boolean Algebra and Digital Logic circuits, and follows the initial design steps: from Boolean algebra, truth tables, to Exact and the Heuristic minimisation techniques, to finally produce the combinatorial circuit. Similarly, the sequential tutorials can design simple State Machine Counters, and can model more complex Finite State Automata.Thesis (Masters)
Master of Philosophy (MPhil)
School of Engineering
Science, Environment, Engineering and Technology
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Ruan, Yongshao. "Efficient inference : a machine learning approach /." Thesis, Connect to this title online; UW restricted, 2004. http://hdl.handle.net/1773/7009.
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Zhou, Lixin. "Testability Design and Testability Analysis of a Cube Calculus Machine." PDXScholar, 1995. https://pdxscholar.library.pdx.edu/open_access_etds/4911.
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Cube Calculus is an algebraic model popular used to process and minimize Boolean functions. Cube Calculus operations are widely used in logic optimization, logic synthesis, computer image processing and recognition, machine learning, and other newly developing applications which require massive logic operations. Cube calculus operations can be implemented on conventional general-purpose computers by using the appropriate "model" and software which manipulates this model. The price that we pay for this software based approach is severe speed degradation which has made the implementation of several high-level formal systems impractical. A cube calculus machine which has a special data path designed to execute multiplevalued input, and multiple-valued output cube calculus operations is presented in this thesis. This cube calculus machine can execute cube calculus operations 10-25 times faster than the software approach. For the purpose of ensuring the manufacturing testability of the cube calculus machine, emphasize has been put on the testability design of the cube calculus machine. Testability design and testability analysis of the iterative logic unit of the cube calculus machine was accomplished. Testability design and testability analysis methods of the cube calculus machine are weli discussed in this thesis. Full-scan testability design method was used in the testability design and analysis. Using the single stuck-at fault model, a 98.30% test coverage of the cube calculus machine was achieved. A Povel testability design and testability analysis approach is also presented in this thesis.
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Ferreira, Rodrigo Costa. "Semântica proposicional categórica." Universidade Federal da Paraíba, 2010. http://tede.biblioteca.ufpb.br:8080/handle/tede/5678.
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Previous issue date: 2010-12-01Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
The basic concepts of what later became called category theory were introduced in 1945 by Samuel Eilenberg and Saunders Mac Lane. In 1940s, the main applications were originally in the fields of algebraic topology and algebraic abstract. During the 1950s and 1960s, this theory became an important conceptual framework in other many areas of mathematical research, especially in algrebraic homology and algebraic geometry, as shows the works of Daniel M. Kan (1958) and Alexander Grothendieck (1957). Late, questions mathematiclogics about the category theory appears, in particularly, with the publication of the Functorial Semantics of Algebraic Theories (1963) of Francis Willian Lawvere. After, other works are done in the category logic, such as the the current Makkai (1977), Borceux (1994), Goldblatt (2006), and others. As introduction of application of the category theory in logic, this work presents a study on the logic category propositional. The first section of this work, shows to the reader the important concepts to a better understanding of subject: (a) basic components of category theory: categorical constructions, definitions, axiomatic, applications, authors, etc.; (b) certain structures of abstract algebra: monoids, groups, Boolean algebras, etc.; (c) some concepts of mathematical logic: pre-order, partial orderind, equivalence relation, Lindenbaum algebra, etc. The second section, it talk about the properties, structures and relations of category propositional logic. In that section, we interpret the logical connectives of the negation, conjunction, disjunction and implication, as well the Boolean connectives of complement, intersection and union, in the categorical language. Finally, we define a categorical boolean propositional semantics through a Boolean category algebra.
Os conceitos básicos do que mais tarde seria chamado de teoria das categorias são introduzidos no artigo General Theory of Natural Equivalences (1945) de Samuel Eilenberg e Saunders Mac Lane. Já em meados da década de 1940, esta teoria é aplicada com sucesso ao campo da topologia. Ao longo das décadas de 1950 e 1960, a teoria das categorias ostenta importantes mudanças ao enfoque tradicional de diversas áreas da matemática, entre as quais, em especial, a álgebra geométrica e a álgebra homológica, como atestam os pioneiros trabalhos de Daniel M. Kan (1958) e Alexander Grothendieck (1957). Mais tarde, questões lógico-matemáticas emergem em meio a essa teoria, em particular, com a publica ção da Functorial Semantics of Algebraic Theories (1963) de Francis Willian Lawvere. Desde então, diversos outros trabalhos vêm sendo realizados em lógica categórica, como os mais recentes Makkai (1977), Borceux (1994), Goldblatt (2006), entre outros. Como inicialização à aplicação da teoria das categorias à lógica, a presente dissertação aduz um estudo introdutório à lógica proposicional categórica. Em linhas gerais, a primeira parte deste trabalho procura familiarizar o leitor com os conceitos básicos à pesquisa do tema: (a) elementos constitutivos da teoria das categorias : axiomática, construções, aplicações, autores, etc.; (b) algumas estruturas da álgebra abstrata: monóides, grupos, álgebra de Boole, etc.; (c) determinados conceitos da lógica matemática: pré-ordem; ordem parcial; equivalência, álgebra de Lindenbaum, etc. A segunda parte, trata da aproximação da teoria das categorias à lógica proposicional, isto é, investiga as propriedades, estruturas e relações próprias à lógica proposicional categórica. Nesta passagem, há uma reinterpreta ção dos conectivos lógicos da negação, conjunção, disjunção e implicação, bem como dos conectivos booleanos de complemento, interseção e união, em termos categóricos. Na seqüência, estas novas concepções permitem enunciar uma álgebra booleana categórica, por meio da qual, ao final, é construída uma semântica proposicional booleana categórica.
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Yeum, Ji-A. "Probability of Solvability of Random Systems of 2-Linear Equations over GF(2)." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1230149164.
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Van, Name Joseph. "Boolean Partition Algebras." Thesis, University of South Florida, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3560193.
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A Boolean partition algebra is a pair (B, F ) where B is a Boolean algebra and F is a filter on the semilattice of partitions of B where [special characters omitted] F = B \ {0}. In this dissertation, we shall investigate the algebraic theory of Boolean partition algebras and their connection with uniform spaces. In particular, we shall show that the category of complete non-Archimedean uniform spaces is equivalent to a subcategory of the category of Boolean partition algebras, and notions such as supercompleteness of non-Archimedean uniform spaces can be formulated in terms of Boolean partition algebras.
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Fiszer, Robert Adrian. "Synthesis of Irreversible Incompletely Specified Multi-Output Functions to Reversible EOSOPS Circuits with PSE Gates." PDXScholar, 2014. https://pdxscholar.library.pdx.edu/open_access_etds/2109.
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As quantum computers edge closer to viability, it becomes necessary to create logic synthesis and minimization algorithms that take into account the particular aspects of quantum computers that differentiate them from classical computers. Since quantum computers can be functionally described as reversible computers with superposition and entanglement, both advances in reversible synthesis and increased utilization of superposition and entanglement in quantum algorithms will increase the power of quantum computing.
One necessary component of any practical quantum computer is the computation of irreversible functions. However, very little work has been done on algorithms that synthesize and minimize irreversible functions into a reversible form. In this thesis, we present and implement a pair of algorithms that extend the best published solution to these problems by taking advantage of Product-Sum EXOR (PSE) gates, the reversible generalization of inhibition gates, which we have introduced in previous work [1,2].
We show that these gates, combined with our novel synthesis algorithms, result in much lower quantum costs over a wide variety of functions as compared to our competitors, especially on incompletely specified functions. Furthermore, this solution has applications for milti-valued and multi-output functions.
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Boris, Šobot. "Games on Boolean algebras." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2009. http://dx.doi.org/10.2298/NS20090907SOBOT.
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The method of forcing is widely used in set theory to obtain various consistency proofs. Complete Boolean algebras play the main role in applications of forcing. Therefore it is useful to define games on Boolean algebras that characterize their properties important for the method. The most investigated game is Jech’s distributivity game, such that the first player has the winning strategy iff the algebra is not (ω, 2)-distributive. We define another game characterizing the collapsing of the continuum to ω, prove several sufficient conditions for the second player to have a winning strategy, and obtain a Boolean algebra on which the game is undetermined. Forsing je metod široko korišćen u teoriji skupova za dokaze konsistentnosti. Kompletne Bulove algebre igraju glavnu ulogu u primenama forsinga. Stoga je korisno definisati igre na Bulovim algebrama koje karakterišu njihove osobine od značaja za taj metod. Najbolje proučena je Jehova igra, koja ima osobinu da prvi igrač ima pobedničku strategiju akko algebra nije (ω, 2)-distributivna. U tezi definišemo još jednu igru, koja karakteriše kolaps kontinuuma na ω, dokazujemo nekoliko dovoljnih uslova da bi drugi igraš imao pobedničku strategiju, i konstruišemo Bulovu algebru na kojoj je igra neodređena.
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Gusmai, Daniel Martins. "Calculadora das classes residuais." reponame:Repositório Institucional da UFABC, 2018.
Find full textAbstract:
Orientador: Prof. Dr. Eduardo GuéronDissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018.
Calculadoras são aparelhos comuns no cotidiano do homem moderno, contudo, os conceitos matemáticos envolvidos em sua concepção ainda são conhecidos por poucos. Durante séculos, a obstinação da humanidade em construir máquinas capazes de computar de forma autônoma resultou tanto no surgimento dos atuais computadores, como também em um magnífico legado de conhecimentos matemáticos agregados a tal conquista. Conteúdos tais como congruências e álgebra booleana suscitaram a revolução dos sistemas informatizados e tem sido amplamente explorados por meio de inúmeras aplicações, nossa trajetória perpassou pela aritmética modular, o teorema de Euler-Fermat e as classes residuais, além de bases numéricas, tópicos de eletrônica digital e funções booleanas, com foco no desenvolvimento de circuitos lógicos e o engendrar de componentes eletrônicos, que configuram a base para idealização e construção de calculadoras que efetuem as operações aritméticas em bases arbitrárias, objetivo preponderante deste trabalho. O esmiuçar das etapas de construção das calculadoras, viabiliza o aprofundamento dos conceitos matemáticos que a fomentaram. A abordagem dos temas supracitados culmina para aprimorar e evidenciar a aplicabilidade da matemática à essência da era moderna.
Calculators are common apparatuses in the everyday of modern man, however, the mathematical concepts involved in its conception are still known by few. For centuries, mankind¿s obstinacy in building machines capable of computing autonomously resulted in both the emergence of current computers and a magnificent legacy of mathematical knowledge added to such achievement. Contents such as congruences and Boolean algebra have aroused the revolution of computerized systems and it has been extensively explored through numerous applications, our trajectory ran through modular arithmetic, Euler-Fermat¿s theorem and residual classes, as well as numerical bases, topics of digital electronics and Boolean functions, focusing on the development of logic circuits and the generation of electronic components, which form the basis for the design and construction of calculators that perform arithmetic operations on arbitrary bases, a preponderant objective of this work. The to detail of the construction steps of the calculators, enables the deepening of the mathematical concepts that fomented it. The approach to the aforementioned themes culminates in improving and evidencing the applicability of mathematics to the essence of the modern era.
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Seedat, Ebrahim. "A study of maximum and minimum operators with applications to piecewise linear payoff functions." Thesis, Rhodes University, 2013. http://hdl.handle.net/10962/d1001457.
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The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
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Aleksandar, Pavlović. "Sequential Topologies on Boolean Algebras." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2009. http://dx.doi.org/10.2298/NS20090113PAVLOVIC.
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A priori limit operator>. maps sequence of a set X into a subset of X.There exists maximal topology on X such that for each sequence x there holds>.(x) C limx. The space obtained in such way is always sequential.If a priori limit operator each sequence x which satisfy lim sup x = lim inf xmaps into {lim sup x}, then we obtain the sequential topology Ts. If a priori 'limitoperator maps each sequence x into {lim sup x}, we obtain topology denoted byaT. Properties of these topologies, in general, on class of Boolean algebras withcondition (Ii) and on class of weakly-distributive b-cc algebras are investigated.Also, the relations between these classes and other classes of Boolean algebras areconsidered.A priori limit operator A svakom nizu elemenata skupa X dodeljuje nekipodskup skupa X. Tada na skupu X postoji maksimalna topologija takva da zasvaki niz x vazi A(X) c lim x. Tako dobijen prostor je uvek sekvencijalan.Ako a priori limit operator svakom nizu x koji zadovoljava uslov lim sup x =liminfx dodeljuje skup {limsupx} onda se, na gore opisan nacin, dobija tzv.sekvencijalna topologija Ts. Ako a priori limit operator svakom nizu x dodeljuje{lim sup x}, dobija se topologija oznacena sa OT. Ispitivane su osobine ovihtopologija, generalno, na klasi Bulovih algebri koje zadovoljavaju uslov (Ii) inaklasi slabo-distributivnih i b-cc algebri, kao i odnosi ovih klasa prema drugimklasama Bulovih algebri.
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Yan, Catherine Huafei. "The theory of commuting Boolean algebras." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/43462.
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Bruns, Corey T. "Variations of independence in boolean algebras." Connect to online resource, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3303844.
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Kwuida, Leonard. "Dicomplemented Lattices." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2004. http://nbn-resolving.de/urn:nbn:de:swb:14-1101148726640-29266.
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Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassenThe aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a "negation" on formal concepts, "concept algebras" are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on "Contextual Logic", a Formal Concept Analysis approach, based on concepts as units of thought
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Zaïdi, Mehdi. "Dualité pour le fragment existentiel de la logique du premier ordre sur les mots." Thesis, Université Côte d'Azur, 2022. https://tel.archives-ouvertes.fr/tel-03882130.
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Cette thèse s'intéresse à l'application de méthodes de dualité topologique à des problèmes de l'informatique théorique. Un des objectifs finaux de cette démarche est l'obtention de résultats en théorie de la complexité, via l'étude d'objets topologiques caractérisant les différentes classes de complexité. La logique est ce qui est à l'interface entre ces deux domaines en apparence très éloignés, plus particulièrement un sous-domaine de la théorie des modèles finis : la logique sur les mots. Il est possible de donner une description de certaines classes de complexité comme des familles de langages,potentiellement non réguliers, sur un alphabet fini.Très peu de résultats de dualité sont connus pour les fragments de la logique du premier ordre sur les mots décrits par des langages qui sortent du cadre régulier.Notre contribution est l'étude détaillée d'un tel fragment. Pour un entier k ≥ 1 fixé, nous considérons l'algèbre de Boole BΣ1[Nuk ]. Celle-ci correspond au fragment de logique sur les mots consistant en les combinaisons Booléennes de propositions définies en utilisant un bloc d'au plus k quantificateurs existentiels, les prédicats sur les lettres et les prédicats numériques uniformes d'arité l ∈ {1, ..., k}. Nous fournissons une étude détaillée de l'espace dual à cette algèbre de Boole, pour tout k ≥ 1, et nous donnons plusieurs caractérisations de ses points. Dans le cas particulier où k = 1, nous sommes capables de construire une famille d'équations ultrafiltre qui caractérise l'algèbre de Boole BΣ1[Nu 1 ]This thesis fits in the area of research that investigates the application of topological duality methods to problems that appear in theoretical computer science.One of the eventual goals of this approach is to derive results in computational complexity theory by studying appropriate topological objects which characterise them.The link which relates these two seemingly separated fields is logic, more precisely a subdomain of finite model theory known as logic on words. It allows for a description of complexity classes as certain families of languages, possibly non-regular, on a finite alphabet.Very little is known about the duality theory relative to fragments of first-order logic on words which lie outside of the scope of regular languages. The contribution of ourwork is a detailed study of such a fragment. Fixing an integer k ≥ 1, we consider the Boolean algebra BΣ1[Nuk ]. It corresponds to the fragment of logic on words consisting in Boolean combinations of sentences defined by using a block of at most k existential quantifiers, letter predicates and uniform numerical predicates of arity l ∈ {1, ..., k}.We give a detailed study of the dual space of this Boolean algebra, for any k ≥ 1, andprovide several characterisations of its points. In the particular case where k = 1, weare able to construct a family of ultrafilter equations which characterise the Boolean algebra BΣ1[Nu 1 ]
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Vormbrock, Björn [Verfasser]. "The Structure of Double Boolean Algebras / Björn Vormbrock." Aachen : Shaker, 2006. http://d-nb.info/1186585730/34.
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Alves, Carla Maria Carneiro. "Lógica e álgebras booleanas." Master's thesis, Universidade Lusíada, 2002. http://hdl.handle.net/10198/4537.
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O principal objectivo deste trabalho consistiu em investigar e aprofundar os
conhecimentos na área da Lógica clássica “booleana” e suas álgebras. Nesse sentido
desenvolveu-se um trabalho organizado em capítulos, designados por:
- capítulo I - Lógica proposicional;
- capítulo II - Álgebras booleanas;
- capítulo III - Conclusões e considerações finais.
No primeiro capítulo desenvolveu-se a Lógica proposicional, essencialmente do
ponto de vista semântico. Fez-se ainda uma breve revisão dos conteúdos considerados
mais pertinentes para a investigação. Os principais tópicos desenvolvidos, neste
capítulo, foram: a sintaxe, a semântica, as formas normais, os conjuntos completos de
conectivos, o lema de interpolação e o teorema da compacidade e algumas suas
aplicações.
No segundo capítulo desenvolveram-se as Álgebras booleanas. Deu-se particular
ênfase aos conceitos de álgebra e de topologia, relacionados com as álgebras booleanas.
Também neste capítulo foi feita uma revisão prévia acerca de alguns conteúdos
considerados relevantes para o estudo. Para além de algumas definições e propriedades
das álgebras booleanas, foram também abordados os seguintes subtemas: átomos,
homomorfismos de álgebras booleanas, ideais e filtros e ainda o espaço de Stone. Os
principais tópicos desenvolvidos foram: uma parte da álgebra e da topologia (alguns
conteúdos necessários para o estudo), algumas definições de álgebras booleanas, átomos
nas álgebras booleanas, homomorfismos, isomorfismos, subálgebras, ideais, filtros,
teorema de Stone.
O terceiro capítulo constitui uma síntese do trabalho e uma reflexão sobre o
modo como decorreu a investigação bem como sugestões e possíveis implicações para
investigações futuras. The main objective of this work was to investigate and deepen our knowledge
about Logic classical “boolean” and its algebras. For that, we considered two chapters
of Logic, Propositional Logical and Boolean Algebras.
For that purpose, we developed a work that was organized in the following
chapters:
- chapter I - Propositional Logic;
- chapter II – Boolean Algebras;
- chapter III – conclusions and final considerations.
The first chapter is about Propositional Logic, essentially from the semantical
point of view. The main developed topics were, in this chapter, syntax, semantics,
normal forms, connective full groups, interpolation lemma and the compactness
theorem and some applications.
The second chapter is about Boolean Algebras. We emphasized algebra and
topology concepts, which are related to Boolean algebras. In this chapter we also made
a previous survey about some concepts that were relevant for this study. Besides some
definitions and properties of Boolean algebras, the following subthemes were also
considered: atoms, homomorphisms of Boolean algebras, ideals and filters and also
Stone’s space. The main topics developed were: a part of algebra and of topology (some
contents that were necessary for the study), some Boolean algebra definitions, atoms in
Boolean algebras, homomorphisms, isomorphisms, subalgebras, ideals, filters, and
Stone’s theorem.
The third chapter consists on a synthesis of the work and a reflexion about the
way the investigation was taken and also on some suggestions and possible implications
for future investigations.
50
Sarabi, Andisheh. "Logic Synthesis with High Testability for Cellular Arrays." PDXScholar, 1994. https://pdxscholar.library.pdx.edu/open_access_etds/4752.
Full textAbstract:
The new Field Programmable Gate Array (FPGA) technologies and their structures have opened up new approaches to logic design and synthesis. The main feature of an FPGA is an array of logic blocks surrounded by a programmable interconnection structure. Cellular FPGAs are a special class of FPGAs which are distinguished by their fine granularity and their emphasis on local cell interconnects. While these characteristics call for specialized synthesis tools, the availability of logic gates other than Boolean AND, OR and NOT in these architectures opens up new possibilities for synthesis. Among the possible realizations of Boolean functions, XOR logic is shown to be more compact than AND/OR and also highly testable. In this dissertation, the concept of structural regularity and the advantages of XOR logic are used to investigate various synthesis approaches to cellular FPGAs, which up to now have been mostly nonexistent. Universal XOR Canonical Forms, Two-level AND/XOR, restricted factorization, as well as various Directed Acyclic Graph structures are among the proposed approaches. In addition, a new comprehensive methodology for the investigation of all possible XOR canonical forms is introduced. Additionally, a new compact class of XOR-based Decision Diagrams for the representation of Boolean functions, called Kronecker Functional Decision Diagrams (KFDD), is presented. It is shown that for the standard, hard, benchmark examples, KFDDs are on average 35% more compact than Binary Decision Diagrams, with some reductions of up to 75% being observed.