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Emergence, Week 5
Welcome to the on-line forum associated with the Biology 361 = Computer Science 361 at Bryn Mawr College. Its a way to keep conversations going between course meetings, and to do so in a way that makes our conversations available to other who may in turn have interesting thoughts to contribute to them. Leave whatever thoughts in progress you think might be useful to others, see what other people are thinking, and add thoughts that that in turn generates in you.
As always, you can leave whatever thoughts occurred to you this week. But if you need something to get you started ...
What's your reaction to Wolfram's explorations of 1-D cellular automata? How does it compare with/relate to your own?
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Ch2-3 Wolfram
I enjoyed reading through these chapters because it presented the information to me in a way that I can more easily relate to. The screen captures included in the reading made me excited to see the vastness of what these simple programs can execute. Personally, I'd like to extend pxcor and pycor to see more of what Wolfram discusses.
He mentions in the reading that the reason why we have so many bugs in programs is that there are bound to be bugs in these simple programs that have vast results. Visualizing simple programs and what they yield is still a difficult concept for me, especially in programming...as we want a deterministic outcome.
Wolfram's chapters
From chapters 2 and 3 it seems that Wolfram groups behavior of the cellular automata according to three types: one group consisted of cellular automata that demonstrated pure repitition and a very simple pattern, a second group consisted of those with many details but had a regular nested structure overall, and a third group where there does not appear to be much regularity and the behavior of the cellular automata seems random.
Wolfram notes common themes between the 256 rules: one common behavior he states is where a pattern consisting of a single cell or group of cells persists such as in rules 2, 4, 103 and 123. Another common feature of some patterns is that they remain a fixed size while other patterns go on forever. Of the patterns that continue to grow some are repetitive while others are not. Repition and nesting patters Wolframs states are common patterns among the cellular automata. What is interesting to note is that Wolfram claims that only 10 out of the 256 rules yield apparent randomness. Even more interesting is rule 110 which is unique in that it has parts that demonstrate both regularity and irregularity.
Another degree of complexity is introduced when you look at rules which involve three colors rather than two. The total number of possible rules in this case would be much greater. What is significant about looking at rules which involve more than two colors is that they produce behavior similar to cellular automata that only involve two colors. This lead Wolfram to state that, “having more complicated underlying rules has not led to much greater complexity in overall behavior.” This is interesting to me because it would seem that logically involving more colors and having more complicated rules would result in more complicated behavior. It also seems like since many more rules are possible with more colors involved it would be hard to find several common themes. I wonder what the “essential ingredents” of the elementary elementary rules are that producecomplicated behavior.
Cellular automata and Wolfram's exploration
My initial thought on Wolfram’s exploration of 1-D cellular automata is that I like the way that he has organized and separated the different rules and that it just makes everything look simpler. His framework is useful and better than what I had come up with before I saw his hierarchical categorization scheme for the rules.
From Wolfram’s reading and from looking at different rules, I thought about how models can seem to get infinitely more complex, but that this is sort of an illusion. Given what we have said about complexity and how it is a result of something simple, it makes more sense to think and accept that complexity has a limit. Specifically, I am thinking about a threshold for complex behavior and what Wolfram says about how adding more complicated rules does not add much greater complexity to overall behavior. In line with this idea is the notion that there is no apparent correlation between the complexity of the rules and the behavior they produce.
Last week I talked about the relevance of cellular automata in explaining real world phenomenon and I like how Wolfram explains that “the basic themes of repetition, nesting, randomness, and localized structures…are actually very general, and in fact represent the dominant themes in the behavior of a vast range of different systems” (106). This is sort of what I was trying to explain…the fact that because cellular automata reveal things such as complexity, simplicity, randomness, disorder, and order (elements that are present across different phenomena) makes them a useful tool that can be used to make sense of other behaviors and systems.
As far as the size of the world in which the cellular automata operate in, I agree that the size can be limiting and that this is something that prevents the observer from drawing meaning/sense of the rule or seeing the bigger picture. As far as what we should be paying most attention to—the beginning, middle, or ultimate output, I think that everything is important. The rules are just as important as the outcome, which can be simple or complex. For some rules, there is an ultimate output, but for others an end is not entirely clear as the patterns seem to continue to change infinitely. So I think that we should not say that the output is more important than the elements that come in between. We can essentially draw a conclusion from all of the outputs. However, some conclusions may be repetitive or simply not as interesting as the ones from later outputs.Cellular Automata in Organisms and the Issue of Scale
On page 84 of A New Kind of Science, Wolfram depicts a variation on cellular automata involving branching patterns on theoretical trees, an idea that is further developed beginning on page 400.
I think it is interesting to think about how organisms could grow and develop according to simple rules like those of the cellular automata. It’s harder to think about humans growing and developing like that, but with plants it’s fairly simple. I’ve been growing plants for a few weeks now for one of my other classes and I spent some time today thinking about those plants in this light. It makes a lot of sense that plants can grow and develop by following a set of if-then rules. Plants tend to be somewhat linear in their layout, so it’s easier to think of them this way. Animals are a little more complex, but if you break them down and look at the smallest parts, they too operate essentially on a set of if-then rules as well. In my mind, it all comes down to the issue of scale. I think it’s interesting how simple small things can look complex on a large scale (like animals) and seemingly complex things can seem simple on a large scale (like rule 30). As we mentioned briefly in class, the size of the world in which the cellular automata operate is rather limiting. If the world could be made large enough to really see some of the long-term results of the rules, I think we’d surprise ourselves even more. It also brings me back to the question I asked last week: should we concentrate on the ultimate output, or is the initial output and the first few steps just as important? Are they important in different ways?
Wolfram's Reading
In this book, “A New Kind of Science,” Stephen Wolfram lists all the patterns we get with the 1-D cellular automata. As we know by now, there are 256 total possibilities; some are simple, while others are pretty complex. I agree with his statement: “What is perhaps most bizarre about the pictures is just how little trace they ultimately show of the simplicity of the underlying cellular automaton rule that was used to produce them.” (p.39)
Wolfram also talks about three-color totalistic cellular automaton where there are more rules and they are more complex – even though the rules are more complex, the resulting behavior is not. “... beyond a certain point, adding complexity to the underlying rules for a system does not ultimately lead to more complex behavior.” (p. 62)
How is that “certain point” determined? In the case of cellular automata, it is the 256 possibilities in which the resulting color can be either black or white. There are 8 possible on or off options/switches, giving us the total 256 possibilities. I was wondering how and why he came up with this particular number. Can the first 128 rules be seen as the point after which behavior does not get more complex after all? If, on the other hand, there are 9 switches instead giving us 512 possibilities, do we get new kinds of complex behavior, or are the behaviors from rule 257 to 512 simply a repeat of the first 256 rules?
Thoughts on Wolfram
Wolfram's listing of all 256 elementary cellular automata reveals some intersting patterns. For example, the first column shows that starting from rule 0, every eighth rule generates one of only two patterns, whcih occur in pairs. That is, rules 0, 8, 32, 40, 64, 72, etc. all create a single pattern, and rules 16, 14, 48, 56, 80, 88, etc. all create a different single pattern. This 'pattern of patterns' could be described, considering only those rules which are multiples of eight (8n), as 'AABBAABBAABBAABB...' The third column (8n+2) shows a similar but slightly more complex meta-pattern whcih could be described as 'AABBAACCAABBAACC...' Of course, the chart only considers each rule's behaviour when starting from a single 'on' cell, and the pattern may not hold true for other starting conditions. Wolfram also describes a fairly simple hierarchical categorization scheme for the rules, shown below (click for larger view):
Wolfram also suggests that a different, more complex type on cellular automaton (three-state, totalistic), although it has a much larger number of possible rules, still exhibits the same basic patterns. In his words, "Some od the patterns are definitely more complicated than those seen in elementary rules. But at the level of overall behavior, there are no fundamental differences" (A New Kind of Science, p. 65). Most of the chapter consists of descriptios and depictions of various different types of systems, and how patterns created by each type fit into the above categories. The number of different systems and possible patterns seems somewhat unmanageable, but Wolfram suggests that all may be simply categorized as shown above. He concludes, of course, by humbly acknowledging his own brilliance, but he does seem to have created a basic categorical framework that we can use.