Humphrey has us consider a more complex two dimensional example. We are to imagine that
Over a wide range of initial states of the CA, appropriate updating rules can produce randomly distributed arrays of colored cells, stable patterns that persist across time, and dynamic patterns that evolve over time.
We are to suppose that the given transformation instructions will eventually produce the following form [5]:
To arrive at this form, we presumably had to carry-out the transformation operations many many times. What is important is that we cannot deduce that this form will come about merely by knowing the initial conditions. We always need to know the formation's preceding step, and its preceding step, and so on until we arrive back at the starting situation. Astronomers, however, can predict the next solar eclipses without needing to know the sun's positions in between eclipses. Such predictions are computationally "compressible," where cellular automata are "incompressible."
I shall examine weak emergence in detail in a moment, but the essence of the idea is that a state of a system is weakly emergent just in case that state can be produced only through a step-by-step simulation of the system. In other words, the process that leads up to the state is computationally incompressible. In yet other words, unlike the prediction of future solar eclipses for which the computational difficulty of prediction is almost independent of how far into the future the eclipse will take place, predictions of future states of computationally incompressible systems must run through each of the intermediate time steps between the initial state and the predicted state. Letting the computational model work out its own development is thus the only effective way to discover how the system’s states evolve. The philosophical motivation for accepting this criterion as capturing a certain kind of emergence draws on the philosophical tradition that emphasizes the essential unpredictability of emergent phenomena. (emphasis mine)
Properties of Pattern Emergence
For there to be emergence, something must emerge from something else. The bow-tie pattern above emerged from the pattern of its rule-based re-iterations. If instead we made a stamp with that image, and stamped it on paper, we would not say that the formation emerged on the paper.
It is another token of the same pattern, but that token is not emergent because it is generated instantaneously.
This example tells us three things:
1) Pattern emergence is largely a "historical" phenomenon: "whether an instance of a pattern is emergent or not depends essentially upon the process that generated it." Thus we cannot look at the synchronic elements of some formation and determine whether or not it is emergent.
2) It cannot be synchronic, because that means we can look at a formation and determine the emergence. It also means that two of the same formations would bring about the same emergence. But we see that both the stamp and the automata have the same formation, but the stamp did not result from a pattern emergence.
3) If we looked at the two identical bow-tie patterns, one from automata, the other from an ink-stamp, and if we say there is the same formation, we are dealing with a type of formation. But we saw that in the one instance there was pattern emergence, but in the other instance there was not. Hence, "pattern emergence is about tokens or instances of patterns, not about types."
Weak Emergence
So we see with the automata that there are lower level circumstances and higher level properties. We call the lower level circumstances micro-facts and the higher level we call the macro-level. On the micro-level of cellular automata, there are only squares. So the property of being "bow-tie shaped" can only occur on the macro-level. We call such a property that can only appear on the higher level a nominally emergent property. We obtained it only by running the simulation that re-iterated the transformation rules. And we may characterize the individual square on the micro-level only in terms of its location and such intrinsic traits as it being black or white. The macro-level entity then is the aggregate of all the micro-constitutent states taken together along with their spatial relations. Because the whole of the macro-level structure can be reduced to the conglomeration of the micro-level states and locations, we call such a system a "locally reducible system."
Humphreys then defines weak emergence as (citing Bedau):
Assume that P is a nominally emergent property possessed by some locally reducible system S. Then P is weakly emergent if and only if P is derivable from all of S’s micro facts but only by simulation.’
We need to run the simulation, because the pattern must be computationally incompressible. Hence the bow-tie structure exemplifies weak emergence.
For the most part, this definition captures the sorts of emergences found in dynamical systems theory and in complexity theory. However, to fully apply, we need to supplement it in two ways.
1) We need to distinguish end states that are non-random from those which have "predicates picking out genuine macro-level properties."
1a) If the pattern begins random and ends random, that does not qualify as a novel emergence. For, even though each random pattern on its own is unique, all random patterns are of the type "random." So nothing is new when random transitions to random.
1b) If the pattern begins structured and ends ordered, then here also there would not be an emergence. For, we consider emergences to result from self-organization. But a turn to chaos is a self-disorganization. One thing that makes emergences so interesting is that they defy the Second Law of Thermodynamics (entropy). So we will not consider a random outcome from an ordered beginning to be emergent. It must have structure to be emergent. But there is a continuum between order and chaos, structure and random, so there is no clear way to make this distinction.
2) Maintaining the automata in a perpetually random state is not difficult to achieve. So we should not devalue the concept of weak emergence by including such cases.
Hence Humphreys offers this revised definition for weak emergence:
P is a non-random property of the system S that is distinct from any property possessed by the initial state of S.
Micro-stable and Micro-dynamic Patterns
Humphreys says that synchronic features are neither sufficient nor necessary conditions for pattern emergence.
Broadly speaking, there are two types of pattern emergences.
1) Micro-stable patterns.
Micro-stable patterns emerge when the computational process no longer develops because it has reached a stable non-random pattern which the transformation principles no longer alter. Our bow-tie pattern is such a case where there was a terminal form that maintained its exact shape.
2) Micro-dynamic patterns.
Micro-dynamic patterns emerge when a non-random pattern emerges, and that pattern itself cycles invariantly, on account of the transformation substitutions falling into a consistent pattern. To view a micro-dynamic pattern, scroll to the bottom of this page, and run the simulator.
We may distinguish three sub-types of micro-dynamic patterns.
2a) Recirculating autonomy
The micro-dynamic pattern exhibits recirculating autonomy if its same consistent parts cycle around in a constant pattern. We see this for example when we heat a fluid between two hot plates. Inter-locking columns of circulating fluid cycle in their cellular regions. These are calledBénard convection cells. [6]
Animation can be found here.
Another example of recirculating autonomy is Couette flow. This comes-about when we place a fluid between two concentric rotating cylinders that are moving at different speeds. "Vortex rolls form when the velocity gradient exceeds a critical value."
No comments:
Post a Comment