VLSI architecture of a cellular automata machine

Abdul Raouf Khan

1997, … & Mathematics with …

Abstract

ln the past, Cellular Automata based models and machines [I] 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.

Figures (13)

where ‘f’ is known as the rule of a CA.

LEMMA 3. The equivalent one-dimensional map matrix for any Rule R can be represented as [0] , {2}, [71], {Z2] [2 + Ti], [2 + Ta], (S], and [7+ 5] PRooF. For getting the first column of the above matrix, we utilize

Now, considering this matrix as a state of the 2-D CA, we apply Rule 2 (right dependency) tc obtain the first (0) column of Tr, (matrix T for Rule 2). LEMMA 5. All primary rules other than Rule 1 are nongroup CA.

LEMMA 6. If A is any matrix of the form As a consequence of these results, we can prove the following theorem which was first arrived at by Sutner [13] in connection with o-game. The same result was derived by Barua and Ramkr- ishnan [14] who viewed this 170N rule as superposition of 1-D vertical and horizontal dependency transformations. However, using the above simple matrix algebraic formulations, we can easily deduce the following results.

Figure 2. Architecture for weighted random pattern generation. is 2* — 1, where k is the number of bits of the CA, then we can obtain the probability of 1 at any bit to be equal to 0.5 by loading the CA with any nonzero seed. On the other hand, if C is of nonmaximal length, then the probability of 1 will be different from 0.5. Thus, corresponding to each cycle C in the state transition diagram of any group m x n 2-D CA, we can define the following tuple r = (cycle length, probability of 1 in bit position (1,1),..., probability of 1 in bit position (m,n), seed). Thus, if we load the CA with the seed and look at any bit position (i, 7), we can distinctly tell the probability of getting 1 in that position—this is possible because of the matrix algebraic characterization of 2-D CA. We can generate a program for the 2-D CA by specifying the rules, the initial seed, and the number of cycles for which the 2-D CA has to be evolved. On running the 2-D CA with such a program, wide varieties of probability of 1 at different bit positions can be generated. The basic architecture is shown in Figure 2.

After running Rule 170N, the resultant configuration is shown in Figure 5.

Next, successively applying Rule 32N, we can obtain the required thinned image (i.e., 1’s in : particular column and 0’s elsewhere) (Figure 9). f

In this paper, we have developed an analytical tool based on matrix algebra to characterize all the nearest neighbourhood 2-D CA transformations and highlighted its application in the fields of VLSI testing and cryptology. Next, we have proposed an architecture of a programmable CAM which suits VLSI implementation. Applications of this CAM-based simulation engine for image analysis and fractal pattern generation have also been reported. Further, efforts are being made to simulate various physical/physiological procedures with the help of this simulation engine— specifically, we are investigating to derive a one-to-one correspondence between a CAM program and the actual physical procedure that transforms a set of healthy human cells (tissue) into a set of cancerous cells. This may open up new avenues in cancer research. recon”, ©, EXAMPLE 6. Let us consider a 32 x 32 2-D CA CAM with a uniform rule of 170P. Since this is a single attractor transformation, we can easily generate any standard fractal patterns from any garbage information by applying XNOR’s in selected positions and then running the 2-D CA for 4 or more number of cycles. We are presenting two such fractal patterns (Figure 10). |