• Title/Summary/Keyword: Boolean Matrix

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A Study on the Efficient Multiplication with All m$\times$k Boolean Matrices (모든 m$\times$k 불리언 행렬과의 효율적 곱셈에 관한 연구)

  • Han, Jae-Il
    • The Journal of the Korea Contents Association
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    • v.6 no.2
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    • pp.27-33
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    • 2006
  • Boolean matrices are applied to a variety of areas and used successfully in many applications, and there are many researches on boolean matrices. Most researches deal with the multiplication of boolean matrices, but all of them focus on the multiplication of two boolean matrices and very few researches deal with the multiplication between many n$\times$m boolean matrices and all m$\times$k boolean matrices. The paper discusses the existing optimal algorithms for the multiplication of two boolean matrices are not suitable for the multiplication between a n$\times$m boolean matrix and all m$\times$k boolean matrices, establishes a theory that enables the efficient multiplication of a n$\times$m boolean matrix and all m$\times$k boolean matrices, and shows the execution results of a multiplication algorithm designed with this theory.

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RANK PRESERVER OF BOOLEAN MATRICES

  • SONG, SEOK-ZUN;KANG, KYUNG-TAE;JUN, YOUNG-BAE
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.501-507
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    • 2005
  • A Boolean matrix with rank 1 is factored as a left factor and a right factor. The perimeter of a rank-1 Boolean matrix is defined as the number of nonzero entries in the left factor and the right factor of the given matrix. We obtain new characterizations of rank preservers, in terms of perimeter, of Boolean matrices.

THE BOOLEAN IDEMPOTENT MATRICES

  • Lee, Hong-Youl;Park, Se-Won
    • Journal of applied mathematics & informatics
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    • v.15 no.1_2
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    • pp.475-484
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    • 2004
  • In general, a matrix A is idempotent if $A^2$ = A. The idempotent matrices play an important role in the matrix theory and some properties of the Boolean matrices are examined. Using the upper diagonal completion process, we give the characterization of the Boolean idempotent matrices in modified Frobenius normal form.

LINEAR OPERATORS THAT PRESERVE PERIMETERS OF BOOLEAN MATRICES

  • Song, Seok-Zun;Kang, Kyung-Tae;Shin, Hang-Kyun
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.355-363
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    • 2008
  • For a Boolean rank 1 matrix $A=ab^t$, we define the perimeter of A as the number of nonzero entries in both a and b. The perimeter of an $m{\times}n$ Boolean matrix A is the minimum of the perimeters of the rank-1 decompositions of A. In this article we characterize the linear operators that preserve the perimeters of Boolean matrices.

Boolean Factorization (부울 분해식 산출 방법)

  • Kwon, Oh-Hyeong
    • Journal of the Korean Society of Industry Convergence
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    • v.3 no.1
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    • pp.17-27
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    • 2000
  • A factorization is an extremely important part of multi-level logic synthesis. The number of literals in a factored form is a good estimate of the complexity of a logic function. and can be translated directly into the number of transistors required for implementation. Factored forms are described as either algebraic or Boolean, according to the trade-off between run-time and optimization. A Boolean factored form contains fewer number of literals than an algebraic factored form. In this paper, we present a new method for a Boolean factorization. The key idea is to build an extended co-kernel cube matrix using co-kernel/kernel pairs and kernel/kernel pairs together. The extended co-kernel cube matrix makes it possible to yield a Boolean factored form. We also propose a heuristic method for covering of the extended co-kernel cube matrix. Experimental results on various benchmark circuits show the improvements in literal counts over the algebraic factorization based on Brayton's co-kernel cube matrix.

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THE COMPETITION INDEX OF A NEARLY REDUCIBLE BOOLEAN MATRIX

  • Cho, Han Hyuk;Kim, Hwa Kyung
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.2001-2011
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    • 2013
  • Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an $n{\times}n$ Boolean matrix A is the smallest positive integer q such that $A^{q+i}(A^T)^{q+i}=A^{q+r+i}(A^T)^{q+r+i}$ for some positive integer r and every nonnegative integer i, where $A^T$ denotes the transpose of A. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix.

Efficient Multiplication of Boolean Matrices and Algorithm for D-Class Computation (D-클래스 계산을 위한 불리언 행렬의 효율적 곱셈 및 알고리즘)

  • Han, Jae-Il;Shin, Bum-Joo
    • Journal of Korea Society of Industrial Information Systems
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    • v.12 no.2
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    • pp.68-78
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    • 2007
  • D-class is defined as a set of equivalent $n{\times}n$ boolean matrices according to a given equivalence relation. The D-class computation requires the multiplication of three boolean matrices for each of all possible triples of $n{\times}n$ boolean matrices. However, almost all the researches on boolean matrices focused on the efficient multiplication of only two boolean matrices and a few researches have recently been shown to deal with the multiplication of all boolean matrices. The paper suggests a mathematical theory that enables the efficient multiplication for all possible boolean matrix triples and the efficient computation of all D-classes, and discusses algorithms designed with the theory and their execution results.

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SEPARABILITY OF DISTINCT BOOLEAN RANK-1 MATRICES

  • SONG SEOK-ZUN
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.197-204
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    • 2005
  • For two distinct rank-1 matrices A and B, a rank-1 matrix C is called a separating matrix of A and B if the rank of A + C is 2 but the rank of B + C is 1 or vice versa. In this case, rank-1 matrices A and B are said to be separable. We show that every pair of distinct Boolean rank-l matrices are separable.

A Study on Multiplying an n × n Boolean Matrix by All n × n Boolean Matrices Successively (하나의 n 차 정사각 불리언 행렬과 모든 n 차 정사각 불리언 행렬 사이의 연속곱셈에 관한 연구)

  • Han, Jae-Il
    • Proceedings of the Korea Contents Association Conference
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    • 2006.05a
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    • pp.459-461
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    • 2006
  • The successive multiplication of all $n{\times}n$ boolean matrices is necessary for applications such as D-class computation. But, no research has been performed on it despite many researches dealing with boolean matrices. The paper suggests a theory with which successively multiplying a $n{\times}n$ boolean matrix by all $n{\times}n$ boolean matrices can be done efficiently, applies it to the successive multiplication of all $n{\times}n$ boolean matrices and shows its execution results.

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