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In this paper we show that the computation of the determinant requires an exponential number of multiplications if the commutativity of indeterminates is not allowed. The determinant can be computed in polynomial time with the commutation of indeterminates. Hence the use of commutativity can reduce a computation of exponential complexity to a computation of polynomial complexity.
In an earlier paper [1], the author showed that certain problems involving sparse polynomials and integers are NP-hard. In this paper we show that many related problems are also NP-hard. In addition, we exhibit some new NP-complete problems. Most of the new results concern problems in which the nondeterminism is "hidden". That is, the problems are not explicitly stated in terms of one of...
We employ elementary results from the theory of several complex variables to obtain a quadratic lower bound on the complexity of computing the mean distance between points in the plane. This problem has 2N inputs and a single output and we show that exactly N(N-1)/2 square roots must be computed by any program over +, -, ×, ÷,) √, log and comparisons, even allowing an arbitrary field of constants...
In this paper, an investigation of the parallel arithmetic complexity of matrix inversion, solving systems of linear equations, computing determinants and computing the characteristic polynomial of a matrix is reported. The parallel arithmetic complexity of solving equations has been an open question for several years. The gap between the complexity of the best algorithms (2n + 0(1), where n is the...
This paper investigates the properties and utilizations of one- and two-dimensional NAND gate cellular arrays. Both irredundant and redundant one-dimensional cascades are investigated. The cascade's output function is obtained in closed form, and a test and synthesis procedure is developed. Both irredundant and redundant two-dimensional arrays are examined, and an arbitrary two-dimensional array is...
A physico-mathematical basis is used to establish bounds TD(n) on the time needed to compute n-argument functions by spatially distributed primitive devices or composite systems D. The axioms used concern the speed, packing density and noise threshold of the energy with which any computing device detects or alters the physical representation of information. The principal result is that TD(n) grows...
This paper develops techniques for establishing a lower bound on the number of arithmetic operations necessary for sets of simple expressions. The techniques are applied to matrix multiplication. A modification of Strassen's algorithm is developed for multiplying n × p matrices by p × q matrices. The techniques are used to prove that this algorithm minimizes the number of multiplications for a few...
This paper describes a new approach to the design of combinational logic using large-scale-integrated (LSI) circuit technology. A simple "prototype" logic function of n binary variables is imbedded within an array of at most (n+1) rows and columns. The cells of this array contain 2-input exclusive-OR gates, and its rows are fed by the input variables and logical "1". Its column...
An n-dimensional iterative array of finitestate machines (abbreviated nD) is a special type of real-time tape acceptor. The principal results are as follows: 1. nD's have equivalent forms With simplified stencils and length k encodings of the input alphabet. 2. The set of palindromes and the set of tapes of the form ττ are accepted by 1D's. 3. The sets of tapes accepted by nD's are a Boolean algebra...
The inherent problems of data transmission in a strictly feedforward line have been discussed in the literature. In such a line, where the stored data are indexed forward by control pulses moving in a direction away from the data source, if time variations exist in the delays of successive stages then there is always a nonzero probability that two successive control pulses will eventually appear at...
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