• Title/Summary/Keyword: Boolean Matrix

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LINEAR PRESERVERS OF BOOLEAN RANK BETWEEN DIFFERENT MATRIX SPACES

  • Beasley, LeRoy B.;Kang, Kyung-Tae;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.625-636
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    • 2015
  • The Boolean rank of a nonzero $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. We investigate the structure of linear transformations T : $\mathbb{M}_{m,n}{\rightarrow}\mathbb{M}_{p,q}$ which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, $2{\leq}k{\leq}$ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.

LINEAR PRESERVERS OF SYMMETRIC ARCTIC RANK OVER THE BINARY BOOLEAN SEMIRING

  • Beasley, LeRoy B.;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1317-1329
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    • 2017
  • A Boolean rank one matrix can be factored as $\text{uv}^t$ for vectors u and v of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in v. A Boolean matrix of Boolean rank k is the sum of k Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix A of Boolean rank k is the minimum over all Boolean rank one decompositions of A of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.

CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES

  • Beasley, Leroy B.;Kang, Kyung-Tae;Song, Seok-Zun
    • The Pure and Applied Mathematics
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    • v.21 no.2
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    • pp.121-128
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    • 2014
  • The Boolean rank of a nonzero m $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with $1{\leq}k{\leq}min\{m,n\}$.

BOOLEAN RANK INEQUALITIES AND THEIR EXTREME PRESERVERS

  • Song, Seok-Zun;Kang, Mun-Hwan
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1525-1532
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    • 2011
  • The $m{\times}n$ Boolean matrix A is said to be of Boolean rank r if there exist $m{\times}r$ Boolean matrix B and $r{\times}n$ Boolean matrix C such that A = BC and r is the smallest positive integer that such a factorization exists. We consider the the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

Logic Optimization Using Boolean Resubstitution (부울 대입에 의한 논리식 최적화)

  • Kwon, Oh-Hyeong
    • Journal of the Korea Academia-Industrial cooperation Society
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    • v.10 no.11
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    • pp.3227-3233
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    • 2009
  • A method for performing Boolean resubstitution is proposed. This method is efficiently implemented using division matrix. It begins by creating an algebraic division matrix from given two logic expressions. By introducing Boolean properties and adding literals into the algebraic division matrix, we make the Boolean division matrix. Using this extended division matrix, Boolean substituted expressions are found. Experimental results show the improvements in the literal counts over well-known logic synthesis tools for some benchmark circuits.

EXTREME PRESERVERS OF RANK INEQUALITIES OF BOOLEAN MATRIX SUMS

  • Song, Seok-Zun;Jun, Young-Bae
    • Journal of applied mathematics & informatics
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    • v.26 no.3_4
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    • pp.643-652
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    • 2008
  • We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases for the Boolean rank inequalities relative to the sums and difference of two Boolean matrices or compared between their Boolean ranks and their real ranks. For these sets, we consider the linear operators that preserve them. We characterize those linear operators as T(X) = PXQ or $T(X)\;=\;PX^tQ$ with appropriate invertible Boolean matrices P and Q.

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PERMANENTS OF PRIME BOOLEAN MATRICES

  • Cho, Han-Hyuk
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.605-613
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    • 1998
  • We study the permanent set of the prime Boolean matrices in the semigroup of Boolean matrices. We define a class $M_n$ of prime matrices, and find all the possible permanents of the elements in $M_n$.

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SPANNING COLUMN RANKS OF NON-BINARY BOOLEAN MATRICES AND THEIR PRESERVERS

  • Kang, Kyung-Tae;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.507-521
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    • 2019
  • For any $m{\times}n$ nonbinary Boolean matrix A, its spanning column rank is the minimum number of the columns of A that spans its column space. We have a characterization of spanning column rank-1 nonbinary Boolean matrices. We investigate the linear operators that preserve the spanning column ranks of matrices over the nonbinary Boolean algebra. That is, for a linear operator T on $m{\times}n$ nonbinary Boolean matrices, it preserves all spanning column ranks if and only if there exist an invertible nonbinary Boolean matrix P of order m and a permutation matrix Q of order n such that T(A) = PAQ for all $m{\times}n$ nonbinary Boolean matrix A. We also obtain other characterizations of the (spanning) column rank preserver.

EXTREME SETS OF RANK INEQUALITIES OVER BOOLEAN MATRICES AND THEIR PRESERVERS

  • Song, Seok Zun;Kang, Mun-Hwan;Jun, Young Bae
    • Communications of the Korean Mathematical Society
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    • v.28 no.1
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    • pp.1-9
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    • 2013
  • We consider the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.