Sudhakar Sahoo

Implementation of Basic Arithmetic Operations Using Cellular Automaton

This paper presents hardware architecture to perform the basic arithmetic operation addition using cellular automata (CA). This age old problem of addition were previously solved by ripple circuit or carry look ahead circuit or by using a combination of them. Each of these circuits is purely combinational in nature and their complexity is centered on the number of logic gates and the associated gate delays. On the contrary, in our CA based design the complexity is mainly centered on the number of clock cycles required to finish the computation instead of the gate delays.

Investigation of the global dynamics of cellular automata using Boolean derivatives

Global dynamics of a non-linear Cellular Automaton (CA), is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In this paper, efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian matrices. A few new theorems on Hamming Distance between Boolean functions as well as on Jacobian matrices of cellular automata are proposed and proved. Moreover, a classification of Boolean functions based on the nature of deviation from linearity has been suggested with a view to grouping them together to classes/subclasses such that the members of a class/subclass satisfy certain similar properties. Next, an error vector, which cannot be captured by the Jacobian matrix, is identified and systematically classified. This leads us to the concept of modified Jacobian matrix whereby a quasi-affine representation of a non-linear cellular automaton is introduced.

Properties of Carry Value Transformation

Carry Value Transformation (CVT) is a model of discrete deterministic dynamical system. In the present study, it has been proved that (1) the sum of any two nonnegative integers is the same as the sum of their CVT and XOR values. (2) the number of iterations leading to either or does not exceed the maximum of the lengths of the two addenda 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 shown to exist on which divides the CV table into disjoint equivalence classes.

Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed

The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties.

Carry Value Transformation (CVT): It's 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. Finally it is shown that CVT can also be used for the production of periodic as well as chaotic patterns.

Multi-number CVT-XOR Arithmetic Operations in Any Base System and Its Significant Properties

Carry Value Transformation (CVT) is one of the modified structures of Integral Value Transformations (IVTs) and coming under the category of discrete dynamical system. Earlier in [5] it has been proved that the addition of two non-negative integers is equal to the addition of their CVT values and XOR values and is true in any base of the number 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 this 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.

Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

Partitioning 1-variable Boolean functions for various classification of n-variable Boolean functions

This paper addresses all possible equivalence classes of 1-variable Boolean functions and from these classes using recursion and Cartesian product of sets, 15 different ways of classifications of n-variable Boolean functions are obtained. The properties with regard to the size and the number of classes for these 15 different ways are also elaborated.

Design of a Parallel Adder Circuit for a Heavy Computing Environment and the Performance Analysis of Multiplication Algorithm

Firstly, this study proposed a new parallel adder circuit model in Carry Value Transformation (CVT)-Exclusive OR (XOR) paradigm. Secondly, an efficient multiplication algorithm is discussed along with its performance analysis on various inputs selection. Our design of proposed model for the addition of many integer pairs using parallel Cellular Automata Machines (CAMs) can perform the addition in a much better way with setting a preprocessing testing logic in it. CVT and XOR operations together can do the efficient addition of two non-negative integers for any bulk inputs using CAM. Multiplication is the repetitive addition process, which could be designed using recursive use of CAM. Our analysis up to 10 bits selection of all integer pairs suggest that the recursive use of CAM for multiplication becomes much faster in real life scenario for any types of inputs. Further exponential operation is highly needed for various fields of computer science which is also described in this paradigm.