The Discovery of Complex Rules

After Wolfram’s proposal, other authors analyzed systems of CAs and proposed their own classifications (Gutowitz 1990; Li, Packard, and Langton 1990). Kurka (1997) proposed to classify them in 5 groups, based on the structure of their attractors. Walker (1990) based his characterization on connected Boolean networks. Many other researchers proposed classifications based on rule sets governing the CAs’ behavior (Barbe 1990; Jen 1990; McIntosh,1990; Voorhees, 1990; Wootters & Langton 1990).

Language Structures in Cellular Automata

The ingenuity of nature and the power of DNA have generated an infinite range of languages - including human language. The existence of these languages inspires us to design artificial cognitive systems whose dynamic interaction with the environment is grounded, at least to some extent, on the same basic laws. Modern scientific knowledge provides us with new opportunities to investigate and understand the logic underlying biological life. We can then use this logic to derive design principles and computational models for artificial systems. The technologies we apply in these studies provide us with new insights into the complexity of the processes underlying the evolutionary success of modern species. We have yet to fully penetrate the mysteries of these natural languages. Nonetheless, the literature suggests (Chomsky, 1957; Aronof & Rees-Miller, 2003; Bilotta & Pantano, 2006) that while the superficial features of different languages depend on different physical supports and different mechanisms, their deep structures share common rules. These constitute linguistic universals, organized at different levels of complexity, where each level has its own rules of composition. At all levels, we can consider these rules as “production rules” or even as rules of reproduction.

Emergent Structures

There are two classes of problem in the study of Cellular Automata. The forward. problem is the problem of determining the properties of the system. Solutions often consist of finding quantities that are computable on a rules table and characterizing the behavior of the rule upon repeated iterations, starting from different initial conditions. Solutions to the backwards problem begin with the properties of a system and find a rule or a set of rules which have these properties. This is especially useful in the application of Cellular Automata to the natural sciences, when researchers deal with a large collection of phenomena (Gutowitz, 1989). Another approach is to identify the basic structures of a Cellular Automaton (Adamatzky, 1995). Once these are known it becomes possible to develop specific models for particular systems and to detect general principles applicable to a wide variety of systems (Wolfram, 1984; Lam, 1998). According to Adamatzky, the identification of a system consists of two related steps, namely specification and estimation. In specification we choose a useful and efficient description of the system: perhaps an equation and a set of parameters. The second step involves the estimation of parameter values for the equation: exploiting measures of similarity.

Cellular Automata Metrics

There have been many attempts to understand complexity and to represent it in terms of computable quantities. To date, however, these attempts have had little success. Although we find complexity in a broad range of scientific domains, precise definitions escape our grasp (Bak, 1996; Morin, 2001; Prigogine & Stengers, 1984). One of the key models in complexity science is the Cellular Automaton (CA), a class of system in which small changes in the initial conditions or in local rules can provoke unpredictable behavior (Wolfram, 1984; Wolfram, 2002; Langton, 1986; 1990). The key issue, here as in other kinds of complex system, is to discover the rules governing the emergence of complex phenomena. If such rules were known we could use them to model and predict the behavior of complex physical and biological systems. Taking it for granted that complex behavior is the result of interactions among multiple components of a larger system; we can ask a number of fundamental questions.

Rhythms of Life

It is widely recognized that the birth of modern science dates to the moment when Galileo first timed physical processes taking place in space. In biology, it is only recently that scientists have felt the need for experimental and mathematical methods describing the development of living organisms in terms of dynamic processes. When we analyze the development of self-replicators, we see that they develop their characteristic patterns and self-similarity through processes that resemble a set of oscillators, operating on different time scales. The periodic behavior of these large scale processes and the relations between them both depend on information processing and on local activity. The end result is the steadily increasing complexity we observe in all biological development processes. Physiological rhythms are essential to the life of the organism. Some are maintained for its whole life and even a brief interruption signifies death. Others only operate for short periods of time - some under the control of the organism, some not.

Searching for Self-Replicating Systems

In Complexity Science (Bak, 1996; Morin, 2001; Gell-Mann, 1994; Prigogine & Stengers, 1984) and Artificial Life (Langton, 1995; Adami, 1998), almost all attempts to simulate or synthesize living systems in new media are somehow related to the influential work of John von Neumann (1966).These studies can be grouped into four basic categories (Sipper et al., 1998).

Modelling Biological Systems

What are the mechanisms underlying biological systems’ ability to transform themselves: the ability of structures to replicate for their own goals, or to meet the specific goals of the system or environment to which they belong? What kind of evolutionary process underlies the emergence of the simple structures which joined together and replicated to produce the life on earth? How can we reproduce these functions in digital machines? More generally, what are the common elements shared by complex physical, biological, social and artificial systems?

Lifelike Self-Replicators

The concept of a cellular automaton derives from John von Neumann’s studies of the logic of life. In these studies, von Neumann focused on self-replicating structures with universal computational capabilities. Given the appropriate initial conditions, a universal computer can perform any finite computation, reproducing even the most complex biological behaviors. It is well known that the ECAs described in (Wolfram, 2002) and the Game of Life (Gardner, 1970; Evans, 2003; Adachi et al., 2008) have universal computational capabilities. It has been shown, furthermore, that certain one-dimensional CAs can generate structures that are equivalent to the components of an idealized digital computer, and that, by connecting these components in different ways, it is possible to implement any kind of algorithm. In brief, these CAs are equivalent to the better known - and simpler - Turing machine and share its ability to perform universal computation (Smith 1971).

Basic Definitions

Cellular Automata (CAs) are discrete dynamic systems that exhibit chaotic behavior and self-organization and lend themselves to description in rigorous mathematical terms. The main aim of this chapter is to introduce CAs from a formal perspective. Ever since the work of von Neumann (1966), von Neumann & Burks (1970), Toffoli & Norman (1987), Wolfram (1983; 1984), and Langton (1984; 1986; 1990), specialists have recognized CAs as a model of crucial importance for complexity studies. Like other models used in the investigation of complex phenomena (Chua & Yang, 1988; Chua, 1998), a CA consists of a number of elementary components, whose interactions determine its dynamics. In this and the following chapters, we will sometimes refer to these elementary components as cells, sometimes as sites. The cells of a CA can be positioned along a straight line or on a 2 or 3-dimensional grid, creating 1-D, 2-D and 3-D CAs. Automata consisting of cells whose only possible states are 0 or 1, are Boolean Automata; automata whose cells can assume more than 2 states are multi-state CAs. In both cases, the CA contains “elementary particles” whose dynamics are governed by simple rules. These rules determine sometimes unpredictable, emergent behaviors, ranging from the simple, through the complex to the chaotic.

The Universal Constructor

At the beginning of the 1950s, John von Neumann (1966) asked himself whether it is possible to design a machine with the ability to create exact copies of itself which would themselves have the ability to produce new copies. Such a machine would have reproductive capabilities comparable to those we find in biological organisms. In this setting, von Neumann’s goal was to design a Universal.Constructor capable of reading the instructions for, and assembling, any machine the designer might seek to build. If the instructions specified a Universal Constructor, the machine would have the ability to build a copy of itself. In other words, it would be able to reproduce. If we wanted the copies of the machine to share this ability, all we would have to do would be to copy the instructions and incorporate them in the new machines.