Pabitra Pal Choudhury

VLSI architecture of a cellular automata machine

Abstract In the past, Cellular Automata based models and machines [1] have been proposed for simulation of physical systems without any analytical insight into the behaviour of the underlying simulation machine. This paper makes a significant departure from this traditional approach. An elegant mathematical model using simple matrix algebra is reported in this paper for characterizing the behaviour of two-dimensional nearest neighbourhood linear cellular automata with null and periodic boundary conditions. Based on this mathematical model, a VLSI architecture of a Cellular Automata Machine (CAM) has been proposed. Interesting applications of CAM in the fields of image analysis and fractal image generation are also reported.

Text compression using two-dimensional cellular automata

Abstract This paper presents an elegant mathematical model using simple matrix algebra for characterising the behaviour of two-dimensional nearest neighbourhood linear cellular automata with periodic boundary conditions. Based on this mathematical model, the VLSI architecture of a Cellular Automata Machine (CAM) has been proposed for text compression. Experimental results of comparisons with adaptive Huffman coding scheme also presented.

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.

Matrix algebraic formulae concerning some exceptional rules of two-dimensional cellular automata

In the past, cellular automata based models and machines [The Theory of Self-Reproducting Automata, University of Illinois Press, Urbana, 1996; Rev. Mod. Phys. 55 (1983) 601, Am. Math. Month. 97 (1990) 24; Matrix and Linear Algebra, Prentice-Hall, India, 1991; TRE Trans. Circuits CT-6 (1959) 45; Cellular Automata Machines, MIT Press, Cambridge, 1987] were proposed for simulation of physical systems, but without any analytical insight into the behaviour of the underlying simulation process. The set of papers [Int. J. Comput. Math. Appl. 33 (1997) 79; Int. J. Comput. Math. Appl. 37 (1999) 115; Matrix algebraic formulae concerning a particular rule of two dimensional cellular automata, Inf. Sci., submitted] made a significant departure from this traditional approach. In the mentioned papers, a simple and precise mathematical model using matrix algebra built on GF(2) was reported for characterising the behaviour of two-dimensional nearest neighbourhood linear cellular automata with null and periodic boundary conditions. As a sequel, in the present paper an attempt has been made to characterise a number of exceptional transformations or rules, each of which behaving uniquely, not matching with any other rules. Thus this set of exceptional rules demand special attention.

Characterisation of a particular hybrid transformation of two-dimensional cellular automata

Abstract Nowadays, one-dimensional (1D) cellular automata (CA) based models and machines were demonstrated for various applications [1–7]. 2D CA based machines have not yet been used fruitfully mainly due to the lack of analytical insight into the behaviour of the underlying evolutionary process. The set of papers [8–11] dealt with the behaviour of the uniform 2D CAs. In the present paper, for the first time an attempt has been made to characterize a particular hybrid linear 2D CA transformation in GF (2) using matrix algebra. Also we deal with the synthesis problem of this particular hybrid transformation.

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.

A statistical analysis of an algorithm's complexity

Abstract In doing the statistical analysis of bubble sort program, we compute its execution times with various parameters. The statistical analysis endorses the specific quadratic pattern of the execution time on the number of items to be sorted. Next, a cursor along the future direction is indicated.

Underlying mathematics in diversification of human olfactory receptors in different loci

As per conservative estimate, approximately 51–105 Olfactory Receptors (ORs) loci are present in human genome occurring in clusters. These clusters are apparently unevenly spread as mosaics over 21 pairs of human chromosomes. Olfactory Receptor (OR) gene families which are thought to have expanded for the need to provide recognition capability for a huge number of pure and complex odorants, form the largest known multigene family in the human genome. Recent studies have shown that 388 full length and 414 OR pseudo-genes are present in these OR genomic clusters. In this paper, the authors report a classification method for all human ORs based on their sequential quantitative information like presence of poly strings of nucleotides bases, long range correlation and so on. An L-System generated sequence has been taken as an input into a star-model of specific subfamily members and resultant sequence has been mapped to a specific OR based on the classification scheme using fractal parameters like Hurst exponent and fractal dimensions.

DNA sequence evolution through Integral Value Transformations.

In deciphering the DNA structures, evolutions and functions, Cellular Automata (CA) plays a significant role. DNA can be thought as a one-dimensional multi-state CA, more precisely four states of CA namely A, T, C, and G which can be taken as numerals 0, 1, 2 and 3. Earlier, Sirakoulis et al. (2003) reported the DNA structure, evolution and function through quaternary logic one dimensional CA and the authors have found the simulation results of the DNA evolutions with the help of only four linear CA rules. The DNA sequences which are produced through the CA evolutions, however, are seen by us not to exist in the established databases of various genomes although the initial seed (initial global state of CA) was taken from the database. This problem motivated us to study the DNA evolutions from more fundamental point of view. Parallel to CA paradigm we have devised an enriched set of discrete transformations which have been named as Integral Value Transformations (IVT). Interestingly, on applying the IVT systematically, we have been able to show that each of the DNA sequence at various discrete time instances in IVT evolutions can be directly mapped to a specific DNA sequence existing in the database. This has been possible through our efforts of getting quantitative mathematical parameters of the DNA sequences involving fractals. Thus we have at our disposal some transformational mechanism between one DNA to another.

Quantitative Description of Genomic Evolution of Olfactory Receptors

We investigate how the evolutionary network is associated among Human, Chimpanzee and Mouse with regards to their genomic information. We provide a quantitative description of genomic evolution through indexes based on fractals and mathematical morphology. These indexes are carefully chosen and reveal quantitatively the similarity or differences among the structure of the concerned sequences. They also reveal how the sequences have evolved over the course of time. We have considered olfactory receptors (ORs) as our case study. These ORs do function in different species with the subtle differences in between the structures of DNA sequences. Such differences are quantified in this paper.