Introduction to Teachers

Dr. Todd Rowland
todd_rowland@wolfram.com
Wolfram Science Group
http://www.wolframscience.com/reference/education/

Here are some materials for teaching a two week course based on the material found in the book A New Kind of Science by Stephen Wolfram. We introduce some of the main basic concepts without going into some of the more complicated arguments.

Before describing the contents, let me start with some explanation. One of the biggest educational challenges today, and for the past fifty years, has been to incorporate technology into the classroom. Not only to prepare students for the changing world, but also to improve the learning process.

These materials fit both criteria. The study of NKS is a meaningful, and fundamental, use of technology, that is currently changing the perspectives of established science, as well as paving the way for future technology. At the same time, it can be done rigorously without losing its simplicity, as well as its aesthetic appeal. So it provides an ideal medium to learn critical thinking.

We start with the simplest rules. Using the simplest cases is a core NKS idea, as well as appealing to common sense. Understand the simplest cases, before moving on to the more complicated cases.  We study rules because we are interested in how things change. Because we look at simple rules, there are applications both to abstract systems like computer programs, and concrete systems like natural phenomena.

One of the main ideas in these lessons is the idea of emulation. One rule can do what another rule does, even though their definitions are different.  The most obvious cases have a direct translation from one evolution to another. But many more are less obvious.  A basic question of NKS is what simple rules can emulate other simple rules and how they do so.

In the case of rule 110, we will see evidence that it emulates a much larger rule, with a fairly complicated description. In fact, it is known that Rule 110 is a universal rule. It can emulate any other rule. The notion of universality was discovered by Alan Turing (1937), and that theoretical advance was a precursor to the actuality of the modern computer.

Computational irreducibility is shown by the example of rule 30, which we can attack using analytical methods. So far nobody has found a way to make the output of Rule 30 predictable or understandable in a fundamental way.

By contrast, we see in the preceding lessons that Rule 250 and Rule 90 make predictable patterns. In the case of Rule 250, it makes a checkerboard pattern, and several questions can be posed which are quite answerable. The pattern made by Rule 90 is more complicated by still regular enough to be predictable. Its regularity is called nesting. Some questions about Rule 90, while answerable, are quite hard even though the questions can be readily understood.

Before studying the cellular automata, it makes sense to first study those with simpler rules. The simplest ones don't depend on any neighbors, and the results are quite simple. Even then, one finds the "Not" operator. Going on to those rules which depend on their right neighbor as well, one finds more possible behavior. Still it is all predictable. At that level, one finds the operators from logic, the Boolean operators, as well as the shift operator. Along with "Not", these are all fundamental to logic and to computer science.

By going through these simple cases first, one can see, possibly through their own computer experiments, how far one has to go from the very simplest types of rules to get complicated behavior. Once we understand the concept of universality, we see that once we reach that point of irreducible behavior, we don't need to go any further. The universal Rule 110 can emulate any other more complicated rule.

It is interesting to finish then with the Principle of Computational Equivalence, which claims that almost all of the complicated , irreducible rules are computationally universal like Rule 110. We should emphasize that this is, at the present time, a principle, not a proven fact. The reader, as well as the student, is invited to think about a scientific principle and what goes into it.