scholarly journals Effective algorithm for construction of terms of minimum computational complexity

2016 ◽  
pp. 21-33
Author(s):  
Dmitriy Daderkin ◽  
Keyword(s):  
Computational Complexity ◽  
Effective Algorithm

scholarly journals Liveness characteristic analysis of a class of Petri nets

2018 ◽  
Vol 10 (6) ◽  
pp. 168781401878148
Author(s):  
Miao Liu ◽  
Zhou He
Keyword(s):  
Computational Complexity ◽  
Petri Nets ◽  
Flexible Manufacturing ◽  
Flexible Manufacturing Systems ◽  
Manufacturing Systems ◽  
Effective Algorithm ◽  
Finite Capacity ◽  
Characteristic Analysis ◽  
Control Actions ◽  
Sequential Processes

Petri nets are an effective tool for analyzing and modeling the dynamic behavior of flexible manufacturing systems. Finite capacity systems of simple sequential processes with resources are an important subclass of Petri nets, for which this article gives a liveness characteristic analysis. First, an effective algorithm for deciding the liveness of finite capacity systems of simple sequential processes with resources is developed by analyzing the relation between the structural properties of resource subnets and the strict minimal siphons. Then, a liveness condition of finite capacity systems of simple sequential processes with resources is accordingly established. Based on the proposed liveness condition, an algorithm for configuring an initial marking for a finite capacity systems of simple sequential processes with resources is given, and therefore, a live finite capacity systems of simple sequential processes with resources net with a configured initial marking can be obtained, which avoids the siphon enumerations and the addition of any control actions. It is shown that the computational complexity of both the developed liveness deciding and the initial marking configuration algorithms is polynomial. Examples are finally provided to illustrate the mentioned results.

scholarly journals An Effective Algorithm for the Synthesis of Irreducible Polynomials over a Galois Fields of Arbitrary Characteristics

2021 ◽  
Vol 20 ◽  
pp. 508-519
Author(s):  
Anatoly Beletsky
Keyword(s):  
Computational Complexity ◽  
High Performance ◽  
Large Degree ◽  
Search Method ◽  
Effective Algorithm ◽  
Irreducible Polynomials ◽  
Computing Systems ◽  
Galois Fields ◽  
Average Performance ◽  
Very High

The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary characteristics. The number of testing steps coincides with the degree of the synthesized polynomials. Upon completion of testing, the polynomial is classifieds as either irreducible or composite. If the degree of the synthesized polynomials is small (no more than two dozen), the formation of a set of tested polynomials is carried out using the exhaustive search method. For large values of the degree, the test polynomials are generating by statistical modeling. The developed algorithm allows one to synthesize binary irreducible polynomials up to 2Kbit on personal computers of average performance

scholarly journals Computational Analysis of Quantitative Characteristics of some Residual Properties of Solvable Baumslag-Solitar Groups

2021 ◽  
Vol 28 (2) ◽  
pp. 136-145
Author(s):  
Elena Alexandrovna Tumanova
Keyword(s):  
Computational Complexity ◽  
Computational Analysis ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Effective Algorithm ◽  
Residual Properties ◽  
Quantitative Characteristics ◽  
Algebraic Cryptography ◽  
Almost All ◽  
Element Set

Let $G_{k}$ be defined as $G_{k} = \langle a, b;\ a^{-1}ba = b^{k} \rangle$, where $k \ne 0$. It is known that, if $p$ is some prime number, then $G_{k}$ is residually a finite $p$-group if and only if $p \mid k - 1$. It is also known that, if $p$ and $q$ are primes not dividing $k - 1$, $p < q$, and $\pi = \{p,\,q\}$, then $G_{k}$ is residually a finite $\pi$-group if and only if $(k, q) = 1$, $p \mid q - 1$, and the order of $k$ in the multiplicative group of the field $\mathbb{Z}_{q}$ is a $p$\-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let $f_{k}(x)$ be the number of sets $\{p,\,q\}$ such that $p < q$, $p \nmid k - 1$, $q \nmid k - 1$, $(k, q) = 1$, $p \mid q - 1$, the order of $k$ modulo $q$ is a $p$\-number, and $p$, $q$ are chosen among the first $x$ primes. We state that, if $2 \leq |k| \leq 10000$ and $1 \leq x \leq 50000$, then, for almost all considered $k$, the function $f_{k}(x)$ can be approximated quite accurately by the function $\alpha_{k}x^{0.85}$, where the coefficient $\alpha_{k}$ is different for each $k$ and $\{\alpha_{k} \mid 2 \leq |k| \leq 10000\} \subseteq (0.28;\,0.31]$. We also investigate the dependence of the value $f_{k}(50000)$ on $k$ and propose an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion. The results obtained may have applications in the theory of computational complexity and algebraic cryptography.

scholarly journals An Algorithm for Fast Multiplication of Kaluza Numbers

2021 ◽  
Vol 11 (17) ◽  
pp. 8203
Author(s):  
Aleksandr Cariow ◽  
Galina Cariowa ◽  
Janusz P. Paplinski
Keyword(s):  
Computational Complexity ◽  
Effective Algorithm ◽  
Real Numbers ◽  
Fast Multiplication ◽  
Interesting Solution

This paper presents a new algorithm for multiplying two Kaluza numbers. Performing this operation directly requires 1024 real multiplications and 992 real additions. We presented in a previous paper an effective algorithm that can compute the same result with only 512 real multiplications and 576 real additions. More effective solutions have not yet been proposed. Nevertheless, it turned out that an even more interesting solution could be found that would further reduce the computational complexity of this operation. In this article, we propose a new algorithm that allows one to calculate the product of two Kaluza numbers using only 192 multiplications and 384 additions of real numbers.

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