arXiv subjects

Sudhakar Sahoo

Publications and source records attributed to Sudhakar Sahoo.

At least 19 recordsLinked to original sources

arXiv2020

Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper is to study the link between a transition function of a one-dimensional CA and IVTs. Mathematically, we have also established the algebraic structures of a set of transition functions of a one-dimensional CA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modelled using IVTs.

nlin.CG↗

arXiv2020

Two Dimensional Discrete Dynamics of Integral Value Transformations

A notion of dimension preservative map, \textit{Integral Value Transformations} (IVTs) is defined over $\mathbb{N}^k$ using the set of $p$-adic functions. Thereafter, two dimensional \textit{Integral Value Transformations} (IVTs) is systematically analyzed over $\mathbb{N} \times \mathbb{N}$ using pair of two variable Boolean functions. The dynamics of IVTs over $\mathbb{N} \times \mathbb{N}=\mathbb{N}^2$ is studied from algebraic perspective. It is seen that the dynamics of the IVTs solely depends on the dynamics (state transition diagram) of the pair of two variable Boolean functions. A set of sixteen \textit{Collatz-like} IVTs are identified in two dimensions. Also, the dynamical system of IVTs having attractor with one, two, three and four cycles are studied. Additionally, some quantitative information of \textit{Integral Value Transformations} (IVTs) in different bases and dimensions are also discussed.

math.DS↗

arXiv2016

Deterministic Computing Mechanism for Perfect Density Classification

The purpose of the present study is to search one-dimensional Cellular Automata (CA) rules which will solve the density classification task (DCT) perfectly. The mathematical analysis of number conserving functions over binary strings of length n gives an indication of its corresponding number conserving cellular automata rules (either uniform or non-uniform). The state transition diagrams (STDs) of number conserving CA rules have been analyzed where it has been found that these STDs can generate different DCT solutions. While studying the properties of STDs, an interesting classification of binary strings could be made where equal weight strings form a class and the cardinality of each class is same as the binomial coefficient nCk; n is the length and k is the weight of the binary string. Apart from STDs, other deterministic methods have been proposed to obtain the exact solution of DCT. All these exact solutions of DCT using different deterministic methods can be viewed as an improvement over the soft computing techniques used earlier to obtain approximate solutions.

nlin.CG↗

arXiv2015

Multi-Number CVT-XOR Arithmetic Operations in any Base System and its Significant Properties

Carry Value Transformation (CVT) is a model of discrete dynamical system which is one special case of Integral Value Transformations (IVTs). Earlier in [5] it has been proved that sum of two non-negative integers is equal to the sum of their CVT and XOR values in any base system. In the present study, this phenomenon is extended to perform CVT and XOR operations for many non-negative integers in any base system. To achieve that both the definition of CVT and XOR are modified over the set of multiple integers instead of two. Also some important properties of these operations have been studied. With the help of cellular automata the adder circuit designed in [14] on using CVT-XOR recurrence formula is used to design a parallel adder circuit for multiple numbers in binary number system.

cs.OH↗

arXiv2015

Carry Value Transformation (CVT) - Exclusive OR (XOR) Tree and Its Significant Properties

CVT and XOR are two binary operations together used to calculate the sum of two non-negative integers on using a recursive mechanism. In this present study the convergence behaviors of this recursive mechanism has been captured through a tree like structure named as CVT-XOR Tree. We have analyzed how to identify the parent nodes, leaf nodes and internal nodes in the CVT-XOR Tree. We also provide the parent information, depth information and the number of children of a node in different CVT-XOR Trees on defining three different matrices. Lastly, one observation is made towards very old Mathematical problem of Goldbach Conjecture.

cs.DM↗

arXiv2013

Classification of Boolean Functions where Affine Functions are Uniformly Distributed

Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where only one affine Boolean function belongs to each class. Two different methods are proposed to achieve this classification. The first method is a recursive procedure that uses the Cartesian product of sets starting from the set of 1-variable Boolean function and in the second method classification is achieved through a set of invariant bit positions with respect to an affine function belonging to that class. The invariant bit positions also provide information concerning the size and symmetry properties of the classes/sub-classes, such that the members of classes/sub-classes satisfy certain similar properties.

cs.LO↗

arXiv2012

Cellular Automata Rules and Linear Numbers

In this paper, linear Cellular Automta (CA) rules are recursively generated using a binary tree rooted at "0". Some mathematical results on linear as well as non-linear CA rules are derived. Integers associated with linear CA rules are defined as linear numbers and the properties of these linear numbers are studied.

nlin.CG↗

arXiv2011

Properties of Carry Value Transformation

The notion of Carry Value Transformation (CVT) is a model of Discrete Deterministic Dynamical System. In this paper, we have studied some interesting properties of CVT and proved that (1) the addition of any two non-negative integers is same as the sum of their CVT and XOR values. (2) While performing the repeated addition of CVT and XOR of two non-negative integers "a" and "b" (where a >= b), the number of iterations required to get either CVT=0 or XOR=0 is at most the length of "a" when both are expressed as binary strings. A similar process of addition of Modified Carry Value Transformation (MCVT) and XOR requires a maximum of two iterations for MCVT to be zero. (3) An equivalence relation is defined in the set (Z x Z) which divides the CV table into disjoint equivalence classes.

cs.DM↗

arXiv2010

2-Variable Boolean Operation -- its use in Pattern Formation

In this paper the theory of 2-Variable Boolean Operation (2-VBO) has been discussed on a pair of n-bit strings. 2-VBO serves to bring out the relation between numbers which when plot on a 2-D surface form interesting patterns; patterns that may be fixed, periodic, chaotic or complex. Some of these patterns represent natural fractals. This paper also provides mathematical analysis corresponding to each of the obtained patterns, which would aid to understanding their formation. 2-VBO is an attempt towards the production and classification of patterns which represent various mathematical models and naturally occurring phenomena.

nlin.CD↗

arXiv2009

On the Interesting World of Fractals and Their Applications to Music

In this paper we have defined one function that has been used to construct different fractals having fractal dimensions between 1.58 and 2. Also, we tried to calculate the amount of increment of fractal dimension in accordance with the base of the number systems. Further, interestingly enough, these very fractals could be a frame of lyrics for the musicians, as we know that the fractal dimension of music is around 1.65 and varies between a high of 1.68 and a low of 1.60. Further, at the end we conjecture that the switching from one music fractal to another is nothing but enhancing a constant amount fractal dimension which might be equivalent to a kind of different sets of musical notes in various orientations.

cs.OH↗

arXiv2009

On Universal Physical Reality in the Light of Quantum Consciousness

In this paper, we have first given an intuitive definition of "Consciousness" as realized by us. Next, from this intuitive definition we derived the physical definition of quantum consciousness (Quantum Consciousness Parameter or QCP). This QCP is the elementary level of consciousness in quantum particles, which are the most elementary particles in nature. Thus QCP can explain both the perceptible and non-perceptible nature and some existing postulates of physics. We conceptualize that the level of human consciousness is most complex having highest fractal dimension of 4.85 in the electroencephalographs experiment done by other research groups. On the other hand, other species are having lesser consciousness level, which can be reflected by lesser fractal dimensions. We have also explored the bio informatics of consciousness from genome viewpoints where we tried to draw an analogy of neurons with electrons and photons. Lastly, we refine the quantum mechanics in terms of QCP; we all know that in Einstein's special theory of relativity, Einstein has used the postulate "Consistency of the velocity of light irrespective of all frames of reference (inertial or non-inertial frames)". In our theoretical revelation QCP can be directly applied to get a confirmatory proof of this postulate. Thus the postulate can be framed as a law.

physics.gen-ph↗

arXiv2009

Theory of Rule 6 and its Application to Round Robin Tournament

In this paper we have used one 2 variable Boolean function called Rule 6 to define another beautiful transformation named as Extended Rule-6. Using this function we have explored the algebraic beauties and its application to an efficient Round Robin Tournament (RRT) routine for 2k (k is any natural number) number of teams. At the end, we have thrown some light towards any number of teams of the form nk where n, k are natural numbers.

cs.DM↗

arXiv2009

Classification of Cellular Automata Rules Based on Their Properties

This paper presents a classification of Cellular Automata rules based on its properties at the nth iteration. Elaborate computer program has been designed to get the nth iteration for arbitrary 1-D or 2-D CA rules. Studies indicate that the figures at some particular iteration might be helpful for some specific application. The hardware circuit implementation can be done using opto-electronic components [1-7].

cs.DM↗

arXiv2009

Theory of Carry Value Transformation (CVT) and its Application in Fractal formation

In this paper the theory of Carry Value Transformation (CVT) is designed and developed on a pair of n-bit strings and is used to produce many interesting patterns. One of them is found to be a self-similar fractal whose dimension is same as the dimension of the Sierpinski triangle. Different construction procedures like L-system, Cellular Automata rule, Tilling for this fractal are obtained which signifies that like other tools CVT can also be used for the formation of self-similar fractals. It is shown that CVT can be used for the production of periodic as well as chaotic patterns. Also, the analytical and algebraic properties of CVT are discussed. The definition of CVT in two-dimension is slightly modified and its mathematical properties are highlighted. Finally, the extension of CVT and modified CVT (MCVT) are done in higher dimensions.

cs.DM↗

arXiv2009

Solutions on 1D and 2D Density Classification Problem Using Programmable Cellular Automata

This paper presents solutions to Density Classification Task (DCT) using a variant of Cellular Automata (CA) called Programmable Cellular Automata (PCA). The translation property as well as the density preserving property of fundamental CA rules in 1D and 2D, and the advantage of PCA are embedded together to obtain the DCT solution. The advantage of PCA over standard CA is reported. A general 2D DCT of arbitrary shapes and sizes, its applicability and its solution using PCA is newly introduced.

nlin.CG↗

arXiv2008

Issues on drawing the State Transition Diagram for arbitrary Cellular Automata

This paper proposes several algorithms and their Cellular Automata Machine (CAM) for drawing the State Transition Diagram (STD) of an arbitrary Cellular Automata (CA) Rule (any neighborhood, uniform/ hybrid and null/ periodic boundary) and length of the CA n. It also discusses the novelty, hardware cost and the complexities of these algorithms.

nlin.CG↗

arXiv2008

Characterization Of any Non-linear Boolean function Using A Set of Linear Operators

Global dynamics of a non-linear Cellular Automata is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In the past efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian Matrices. In this paper two different efforts have been made: first we try to systematize non-linear CA evolution in the light of deviant states and non-deviant states. For all the non-deviant states the nearest linear rule matrix is applicable where as for the deviant states we have a set of other matrices. Second using algebraic manipulation, an efficient algorithm is proposed by which every Non-linear Boolean function can be characterized by a sequence of binary matrices.

cs.CC↗

arXiv2008

Encompression Using Two-dimensional Cellular Automata Rules

In this paper, we analyze the algebraic structure of some null boundary as well as some periodic boundary 2-D Cellular Automata (CA) rules by introducing a new matrix multiplication operation using only AND, OR instead of most commonly used AND, EX-OR. This class includes any CA whose rule, when written as an algebra, is a finite Abelean cyclic group in case of periodic boundary and a finite commutative cyclic monoid in case of null boundary CA respectively. The concept of 1-D Multiple Attractor Cellular Automata (MACA) is extended to 2-D. Using the family of 2-D MACA and the finite Abelian cyclic group, an efficient encompression algorithm is proposed for binary images.

cs.DM↗