不完全定义离散函数中变量数减少的代数方法

J. Astola, P. Astola, R. Stankovic, I. Tabus
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引用次数: 13

摘要

本文考虑不完全定义的离散函数,即布尔函数和多值函数,f:S→{0,1,…,q-1}式中S≤{0,1,…,q-1}n,即函数值仅在对应的完全定义函数的定义域的某个子集S上指定。我们假设函数是稀疏的,即vs vs相对于定义域的基数是“小”的。通过嵌入域{0,1,…,q-1}n,其中n为变量数,q为素数幂,在合适的环结构中,环的乘法结构可用来构造线性函数{0,1,…n, q1}{0,1,……当m > 2logq |S| + logq(n -1)时,q-1}m在S上内射。通过这种方法,我们找到了一种线性变换,它可以将变量的数量从n减少到m,并且可以用线性分解来实现一个不完全定义的离散函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An Algebraic Approach to Reducing the Number of Variables of Incompletely Defined Discrete Functions
In this paper, we consider incompletely defined discrete functions, i.e., Boolean and multiple-valued functions, f:S→{0,1,...,q-1} where S ⊆ {0,1,...,q-1}n i.e.,the function value is specified only on a certain subset S of the domain of the corresponding completely defined function. We assume the function to be sparse i.e. |S| is 'small' relative to the cardinality of the domain. We show that by embedding the domain {0,1,...,q-1}n, where n is the number of variables and q is a prime power, in a suitable ring structure, the multiplicative structure of the ring can be used to construct a linear function {0,1,...,q-1}n {0,1,...,q-1}m that is injective on S provided that m > 2logq |S| + logq(n - 1). In this way we find a linear transform that reduces the number of variables from n to m, and can be used e.g. in implementation of an incompletely defined discrete function by using linear decomposition.
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