Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
strong emergence idea illustrated with multivalued CAs
A recent article by Paul Davies prompted some discussion about the distinction between "weak emergence" and "strong emergence". Weak emergence seems to accurately describe the basic phenomenon the NKS book discusses repeatedly, that simply underlying rules can give rise to surprisingly complicated structures, which seem to follow higher level rules of their own, that aren't readily predictable from the underlying rule. Think of particles in a class 4 CA, for example. Their variety, motions, and collision results are all determinable from the underlying rule and previous pattern, true. But in practice the only way to tell how this configuration is going to interact with that one, is to directly simulate it. And it is whole configurations that interact, in ways not obvious beforehand from the simple, local rule.
I wanted a clear sense of what strong emergence would mean compared to that more familiar, weak variety. Strong emergence is distinguished from weak, by the local underlying rule not determinining all the details of the evolution. It is not an exclusively bottomup system, in other words. Some global attribute of the overall behavior "reaches down" and selects this local evolution rather than that one. Convinced that any sufficiently clear idea can be translated into running code, I decided to illustrate this notion with a variety of modified cellular automaton, in which the local rule sometimes "underspecifies" the behavior. Or, in pattern matching terms, where there are "blanks" in the output lines as far as the local rule is concerned, with the way those blanks are filled in determined by something else.
The attached notebook illustrates the idea with three different set ups. The simplest just uses the functional form of the CellularAutomaton function in Mathematica to create a multivalued CA in which one rule case is "ambiguous", its actual behavior determined randomly. Next I use the ICA set up to switch which way the rule is applied from line to line, according to an external sequence. I can then search through possible orders of application, looking for the overall CA evolution that maximizes or minimizes some global property. Last, I combine the two, with each application of the ambiguous case of the rule specified by an external sequence. This is like "calling" the series of heads and tails in the random version. I can then search through some simple sequences of those "flips", looking for the one that maximizes this or minimizes that.
Here is the conclusion section of the attached notebook, what multivalued CAs can tell us about varieties of emergence.
Conclusion  what multivalued CAs can tell us about varieties of emergence
There is nothing wrong in principle with models whose underlying, atomic rules underspecify the exact sequence of states the system will visit. When a fully deterministic rule successfully models a system's behavior, it may be preferable as a more exhaustive, convincing, simple, or intuitive explanation of the behaviors seen in the overall system. From NKS, we know that quite involved behaviors can readily arise from simple rules, even entirely deterministic local rules that fully specify the system's behavior from the bottom up. In ordinary rule 110, we see structures arise that we would not readily expect from the underlying rule, and these structures move about and interact in complicated ways. We can't readily forsee the outcome of their interactions, by any simpler means that just watching them and seeing what they actually do. This is a case of "weak emergence", in Davies' terms. A fully deterministic rule does surprising things, depending sensitively on the exact details of its initial conditions.
But an atomic rule need not fully specify the subsequent behavior. Stochastic rules are possible, that usually behave deterministically, but that have some ambiguous cases that can go either way, randomly. A whole array of possible evolutions are compatible with a microrule of this sort. The determining portion of the rule constrains the system to behave in some subset of ways, from the ground up. But does not determine the exact sequence of states the system visits, from a given initial condition. There is no difficulty constructing NKS models of this sort, or examining their properties. Each particular evolution needs some way of deciding what outcome is actually to be followed in the ambiguous cases. But the overall behavior of the rule can be viewed as a branching array of possible evolutions.
Which branch is actually followed may not be specified by the atomic, bottom up rule, itself. It might be random  the simplest case of underspecificity in any rule. Or it might be the result of some external force or internal but hidden variable, interacting with the system, as in the classic understanding of a "machine with input". Whether this input is local or global, internal or external, can be varied as a modeling parameter. There is nothing inherently mysterious about this, it can be given a clear
operational meaning. Clear and distinct ideas can become running code. A sequence of apparently random determinations from the standpoint of an underlying constraintrule, might be determined globally, by some larger minimum or maximum principle. This too can be operationalized and modeled, as e.g. selecting a maximum out of a possibility space set by the underlying local rule, according to some larger measure. When this is what actually determines the evolution of a system, one can say strong emergence in Davies' sense of the term is occurring. An "end" or attractor makes the final selection of one actual evolution from the top down, out of a set of possibilities constrained, but not fully specified or determined, from the ground up.
Having shown these ideas can be distinguished and given clear sense, I go on to note the range of behaviors actually seen in simple rules given this extra degree of freedom, and the computational effort involved in determining their evolution. The first thing to notice is that the same gross classes of behavior are seen. It is true one can see a greater variety of behaviors within a single system, corresponding to an effectively more complicated rule, in terms of its number of states and the like. We know
that some simple deterministic rules are universal in the infinite size limit. And we see in practice that some of them do things too complicated to readily predict beforehand. The underspecified rules illustrated here are similarly difficult to track without a large computational effort, and conceptually their space of possible evolutions blows up more rapidly. In either case, that space is large, compared to the number of possible states of each subelement, the simplicity of the underlying rule, etc.
The next thing to notice is the globally decided rules, in which the local deterministic part underspecifies the actual path of the system, and some global measure has to be calculated from a larger structure to determine which possibility is actually realized, require much greater computational effort to evolve, not less computational effort. In general one needs to examine every possible sequence of underdetermined, allowed outcomes, wash all of them through some measure or higher level function, and compare all the results. It is certainly possible that something like this goes on in this or that system  for example, say in some economic model in which it is known the relevant microactors within the model are striving to maximize some global function. But it is essentially always computationally easier to apply the local rules directly, than to monkey with their exact outcome in some detailed, finely tuned, nuanced way, so as to maximize some global measure.
Thus, strong emergence can be distinguished from weak emergence, and given a clear operational sense. They may be hard to distinguish in practice, but we can formally model each and test the typical behavior each gives. However, when there are limited computational resources within some system, that tends to favor weak emergence explanations over strong emergence explanations, for complex behaviors exhibited by that system. Other things being equal, strong emergence can be expected to require greater computational effort, not less.
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