• Title/Summary/Keyword: Boolean Matrix

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Algorithm for Computing J Relations in the Monoid of Boolean Matrices (불리언 행렬의 모노이드에서의 J 관계 계산 알고리즘)

  • Han, Jae-Il
    • Journal of Information Technology Services
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    • v.7 no.4
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    • pp.221-230
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    • 2008
  • Green's relations are five equivalence relations that characterize the elements of a semigroup in terms of the principal ideals. The J relation is one of Green's relations. Although there are known algorithms that can compute Green relations, they are not useful for finding all J relations in the semigroup of all $n{\times}n$ Boolean matrices. Its computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices. The size of the semigroup of all $n{\times}n$ Boolean matrices grows exponentially as n increases. It is easy to see that it involves exponential time complexity. The computation of J relations over the $5{\times}5$ Boolean matrix is left an unsolved problem. The paper shows theorems that can reduce the computation time, discusses an algorithm for efficient J relation computation whose design reflects those theorems and gives its execution results.

EXTREME PRESERVERS OF TERM RANK INEQUALITIES OVER NONBINARY BOOLEAN SEMIRING

  • Beasley, LeRoy B.;Heo, Seong-Hee;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.113-123
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    • 2014
  • The term rank of a matrix A over a semiring $\mathcal{S}$ is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.

LINEAR OPERATORS THAT PRESERVE ZERO-TERM RANK OF BOOLEAN MATRICES

  • Kim, Seong-A.;David, Minda
    • Journal of the Korean Mathematical Society
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    • v.36 no.6
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    • pp.1181-1190
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    • 1999
  • Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterized the linear operators that preserve zero-term rank of the m×n matrices over binary Boolean algebra.

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Column ranks and their preservers of general boolean matrices

  • Song, Seok-Zun;Lee, Sang-Gu
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.531-540
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    • 1995
  • There is much literature on the study of matrices over a finite Boolean algebra. But many results in Boolean matrix theory are stated only for binary Boolean matrices. This is due in part to a semiring isomorphism between the matrices over the Boolean algebra of subsets of a k element set and the k tuples of binary Boolean matrices. This isomorphism allows many questions concerning matrices over an arbitrary finite Boolean algebra to be answered using the binary Boolean case. However there are interesting results about the general (i.e. nonbinary) Boolean matrices that have not been mentioned and they differ somwhat from the binary case.

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THE CONSTRUCTION OF FUZZY IDEMPOTENT ZERO PATTERNS BY A PROGRAM

  • Park, Se Won;Kang, Chul
    • Honam Mathematical Journal
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    • v.36 no.1
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    • pp.187-198
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    • 2014
  • The fuzzy idempotent matrices are important in various applications and have many interesting properties. Using the upper diagonal completion process, we have the zero patterns of fuzzy idempotent matrix, that is, Boolean idempotent matrices. And we give the construction of all fuzzy idempotent matrices for some dimention.

A Visual-Based Logic Minimization Method

  • Kim, Eun-Gi
    • Journal of Korea Society of Industrial Information Systems
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    • v.16 no.5
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    • pp.9-19
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    • 2011
  • In many instances a concise form of logic is often required for building today's complex systems. The method described in this paper can be used for a wide range of industrial applications that requires Boolean type of logic minimization. Unlike some of the previous logic minimization methods, the proposed method can be used to better gain insights into the logic minimization process. Based on the decimal valued matrix, the method described here can be used to find an exact minimized solution for a given Boolean function. It is a visual based method that primarily relies on grouping the cell values within the matrix. At the same time, the method is systematic to the extent that it can also be computerized. Constructing the matrix to visualize a logic minimization problem should be relatively easy for the most part, particularly if the computer-generated graphs are accompanied.

STRUCTURES OF IDEMPOTENT MATRICES OVER CHAIN SEMIRINGS

  • Kang, Kyung-Tae;Song, Seok-Zun;Yang, Young-Oh
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.721-729
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    • 2007
  • In this paper, we have characterizations of idempotent matrices over general Boolean algebras and chain semirings. As a consequence, we obtain that a fuzzy matrix $A=[a_{i,j}]$ is idempotent if and only if all $a_{i,j}$-patterns of A are idempotent matrices over the binary Boolean algebra $\mathbb{B}_1={0,1}$. Furthermore, it turns out that a binary Boolean matrix is idempotent if and only if it can be represented as a sum of line parts and rectangle parts of the matrix.

Algorithm for Efficient D-Class Computation (효율적인 D-클래스 계산을 위한 알고리즘)

  • Han, Jae-Il
    • Journal of Information Technology Services
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    • v.6 no.1
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    • pp.151-158
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    • 2007
  • D-class computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices and search for equivalent $n{\times}n$ Boolean matrices according to a specific equivalence relation. It is easy to see that even multiplying all $n{\times}n$ Boolean matrices with themselves shows exponential time complexity and D-Class computation was left an unsolved problem due to its computational complexity. The vector-based multiplication theory shows that the multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices can be done much more efficiently. However, D-Class computation requires computation of equivalent classes in addition to the efficient multiplication. The paper discusses a theory and an algorithm for efficient D-class computation, and shows execution results of the algorithm.