Research articleDesigning convergent cellular automata
Introduction
Cellular automata (CA) are dynamic systems in which space and time are discrete. CA consist of a number of identical cells in an array. Each cell can be in one of a number of states. The next state of each cell is determined at discrete time intervals according to the current state of the cell, the current state of the neighbouring cells and a next state rule that is identical for each cell.
Neumann (1966) and Ulam (1970) developed CA to study self-reproducing systems. Since then CA have been extensively used to study self-organisation Sipper et al., 1997, Wolfram, 1986, ecosystems Gardner, 1970, Langton, 1986 and biological systems Kansal et al., 2000, Ito and Gunji, 1994.
CA are often used to model biological systems. The CA develops with each time step, from one pattern of cell states, , to another, , according to a set of rules , the automata boundary conditions and its initial state .
CA experiments typically seek to answer one of two questions.
- (1)
How do environmental changes (simulated by changes in , and sometimes ) affect the behaviour of the biological system (modelled as a CA) as it develops (Karafyllidis, 1998)?
- (2)
How does a biological system arrive at a specific state (where n is some large time interval)(Mizas et al., 2008)?
The latter experiments are constrained by the difficulty of designing a CA to converge towards a specific state.
In seeking to model biological systems with CA various techniques have been proposed to design CA to converge towards specific patterns or behaviours, three of which are itemized below.
Ordinary differential equations (ODE). Fleischer and Barr (1993) studied the effects of cell migration, cell hereditry and inter-cellular communications using and ODE model of locally interacting cells. By using an adaptive euler-solver developed by Barzel (1992) he was able to design the model to converge to interesting patterns. Fleischer concluded that it was difficult to design cells to converge to specific patterns, and went on to suggest the use of evolution as an alternative approach.
Unsupervised evolutionary algorithms. Eggenberger (1997) used a CA to study the effects of genetic lineage and inter-cellular chemical interactions on tissue development. A genetic algorithm was used to design the CA to converge to interesting patterns. The fitness of each solution was tested against an arbitrary size of CA and the symmetry of the system about the x-axis.
Supervised evolutionary algorithms. Most recent experiments have relied on supervised evolutionary algorithms to design the CA. For instance Mizas et al. (2008) modelled mutagenesis of a DNA sequence using a CA, then used a genetic algorithm to determine the development rules such that the model developed in the same way as had been observed in the DNA. Miller and Banzhaf (2003) used a 2D, 256 cell CA to simulate the inter-cellular chemical interactions of a tissue during development. A genetic algorithm was used to design the CA to converge to a single pattern (three vertical stripes) from null or partially corrupt (up to 25%) initial conditions.
Evolutionary algorithms require significant computing resources and are limited to solving simple problems (Miller, 2000). This paper will present an alternative to evolutionary algorithms for the design of convergent CA using a deterministic rather than a stochastic approach. This work was inspired by attempts by engineers to mimick morphogenesis, the self-organizing assembly mechanism employed on tissues of biological cells during embryonic development. Those studies (Miller and Banzhaf, 2003) sought to mimick the robust convergence displayed by morphogenesis on electronic systems with the goal of creating self-assembling self-repairing circuits.
In Section 2 we will present a CA and its equivalent matrix model. In the Section3 we will describe the conditions required for a simple CA to converge upon any static pattern. In Section 4 we will present and demonstrate a means of designing this CA to converge upon a specified static pattern. In Section 5 we will describe the conditions required for a more comprehensive CA model to converge and in Section 6 we will present a means of designing this CA to converge upon a specified static pattern.
Section snippets
The Equivalent Matrix Model
The platform on which we shall analyse CA convergence is a 2D array of identical cells that use information from their nearest neighbours (above, below, to the left and right) to compute their next state.
Each cell computes its next state at the same time and does so at each discrete time step. Let us index each cell with the tuple , then describe the state of each cell with an integer, and the pattern of the entire array as a matrix, (see Fig. 1 (a)).
If is the initial pattern
Fundamental Criteria for Convergence
Given a sufficiently large n in order for the dynamic non-linear system to converge, the final pattern, , must be independent of the initial pattern, . Thus no matter what the starting pattern (where refers to the initial pattern or any pattern that might be the result of system corruption), the pattern of cell states will always return to the same stable pattern.
Thus , the coefficient of , must equal zero. For this to be so, referring to the coefficients of the states of the
Determining the Local Rule Set
Let us design a transition function for a two by two cell CA such that it converges on the pattern of a diagonal stripe from top-left to bottom-right, .
From (8) values for and d can be derived. A solution to these equations describes the local interactions between cells that will result in the desired global pattern. In the case of , if the transition function can be derived from (10).
Thus the transition function for is
Requirements for a Sum-of-products CA to Converge
Because there are more linear simultaneous equations formed from (8) than there are coefficients to solve for, a direct solution to (8) does not always exist.
A solution to this is to define a more elaborate transition function, , such that there are many more coefficients to solve the simultaneous equations for.
If k is greater than three, expanding gives a coefficient of formed by the multinomial of the transition
Determining the Local Rules for a CA to Converge to an Arbitrary Pattern
There still exist certain patterns that cannot be generated using . This is because the same two input combinations cannot map to different output values. Fig. 3 shows a pattern that cannot be formed from if the inputs are from above and to the left of each cell.
This pattern is impossible to form with alone because it would require for cell and for cell , something that is not permissible in a one-to-one mapping.
To this end additional computation must be
Demonstration
For the purposes of demonstrating this approach, a CA will be designed to converge to the final pattern shown in Fig. 6. We will then demonstrate it converging to this final pattern starting from three different initial patterns.
Using the algorithm of Fig. 5 it is possible to derive a set of rules that form and for the CA to converge to the target pattern. The solution requires a LUT with 870 entries for and a LUT with 131 entries for . Fig. 7 shows the CA converging from null
Conclusions
This paper has introduced a new technique for the design of convergent CA. We have shown that to ensure the pattern will converge regardless of initial conditions, it is necessary that the transition rules used by each cell are independent of the current cell state. In addition, the rules can only depend upon the state of one cell per axis, the cell to the left or to the right and the cell above or the cell below.
We have found it is possible to map global patterns to local transition rules
Acknowledgements
The authors would like to acknowledge the support of the EPSRC for funding the work presented in this paper.
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