A.S.P.

Attractors, Superposition, and Probability

S.Roof, feb 2003

"Is this a system?" ...hypothethical quote of Bertlansky by rasputin11 ... or was it the Furry Freak Bros? ... at any rate, the state of the art in systems theory has made great progress.

In recent emails I discussed the connection between attractors, superposition and probability. Most of the effort was to understand and explicate some of the ideas suggested by Eldon's superposition web page. I have recently discovered a beautiful and even more succint way of stating these relations. First though, I would like to say that 'ASP' as a shortened iconoture of oroborus is most apt. The systems I mostly discuss these days are discrete dynamical networks which are closed in the sense that the interacting parts receive their inputs from some neighborhood of other interacting parts of the system. All dynamic systems can probably seen to be partially closed in this sense. Closure is the snake consuming it's tail i.e. oroborus i.e. recursion. We could add R for recursion and get R.A.S.P. but that might be considered going over the top eh eh. Anyway, let me tell you about the new method of visualization. This new visualization gives a direct hook for connecting probability to attractors and superposition. To get there, I'll discuss superposition and attractors first.

Superposition

Previously, I mentioned that attractors appear to limit possibilities and thus the superpositions converge on interesting acualities. I mentioned superposition as a functional summation of influences. Now though, I have a clearer way of stating these ideas. When the parts of a system interact in a deterministic way, we may say that the future state of a part is a function of it's previous state and any outside influences from parts it is connected to. Superposition may thus be stated as the summary combination of functional inputs whether we are talking about neural nets, cellular automata, or whatever. If the system is functionally deterministic, there can only be one resultant at any given step in time. This resultant state depends on previous states by way of superposition of inputs.

From now on I will use the word 'superposition' in a very general sense to mean this functional determinism from the many inputs to one output. This covers a lot of cases. The superposition could be simple linear summation of waves, non-linear weighted sigmoidal squashing of multiple inputs, multiplicative, or any other functionality whatsoever. The idea is that multiple inputs are functionally combined into one deterministic output.

From a temporal standpoint, superposition may be described as the upstream side of dynamic flow. From a black box perspective, the box does a superposition of input signals to generate an output. From a cellular automata perspective, superposition is the neighborhood lookup value that is input into a particular CA rule set. These visualizations are appropriate when applied to any part of a system or the system as a whole. This way of defining superposition removes some of the magic or mystery from its useage. The magic of emergent dynamic behaviour thus seems to come more from the functionality and connectivity of the system, but ultimately the magic is a result of global constraints, synchrony and synergism and not easily defined.

Attractors

Discrete dynamic systems have behaviours which can be loosely classified into 3 categories: order, complexity, and chaos. Complexity is the border realm between order and chaos. Behaviour of these systems is usually depicted by state-space diagrams which plot points for system states as a function of time. Phase change is a term that applies to boundaries between clusters or organized areas of the state space. Attractors and attractor basins are the terms for these organized areas where order is emergent over time. Discrete systems can be used to model natural systems computationally as simulations. Much research has been done on such systems. Complex behaviours are extremely difficult to analyze and study however. Most progress has been made by actually running such models forward in time and using descriptive techniques. That is how the ideas of attractors and attractor basins came into useage. There were direct pictorial demonstrations to look at.

Attractor basins emerge functionally in time in a deterministic fashion, but in most cases there is no way to make predictive analysis. The only mathematical treatments in the past were thus mostly statistical. This parallels the developments of quatum physics by the way. The precise functionality of the interacting parts may or may not be known for systems, but in general, simple deterministic systems can produce behaviour which is computationally irreducible. Individual trajectories of a system may be followed in time as orbits about attractors. These orbits may described as convergent to a fixed point, periodic, quasi-periodic, or aperiodic as the system moves between ordered and chaotic.

Attractor basins are regions of state space where the system behaviour is convergent. Convergence means that future states become limited to certain more probable states as time passes. Attractors thus filter out certain possibilites and introduce the notion of probability. An attractor basin with a certain density of state points has a, in some sense, measureable degree of probability. The problem, though, is that there can be an innumerable number of starting states which would have to be evolved from to get accurate estimates. This forward-in-time analysis is a brute force approach and very computationally expensive. There is a better way. To do it requires back-in-time analysis!

Probability and Entropy of Attractor Basins

I discovered a new method of analyzing attractor basins at:

Discrete Dynamical Networks and their Attractor Basins

it was written by:

Andrew Wuensche
Santa Fe Institute, 1399 Hyde Park Road,
Santa Fe, New Mexico 87501 USA,
Email: wuensch@santafe.edu
WWW: www.santafe.edu/~wuensch/

The method described by Wuensche is easy to visualize even though the actual computational methods are difficult. The material below was clipped from the web site. Apparently the Sante Fe Institute is using (developed?) a program called DDLab. It implements a backward-chaining algorithm.

Given a system in state space we have:

For a network size n, an example of one of its states B might be ... State-space is made up of all such states, the space of all possible bitstrings or patterns

Part of a trajectory in state-space, where C is a successor of B, and A is a pre-image of B, according to the dynamics of the network.

The state B may have other pre-images besides A, the total is the in-degree. The pre-image states may have their own pre-images or none. States without pre-images are known as garden-of-Eden states.

Any trajectory must sooner or later encounter a state that occurred previously - it has entered an attractor cycle. The trajectory leading to the attractor a transient. The period of the attractor is the number of states in its cycle, which may be only just one -- a point attractor.

Take a state on the attractor, find its pre-images (excluding the pre-image on the attractor). Now find the pre-images of each pre-image, and so on, until all garden-of-Eden states are reached. The graph of linked states is a transient tree rooted on the attractor state. Part of the transient tree is a subtree defined by its root.

Construct each transient tree (if any) from each attractor state. The complete graph is the basin of attraction. Some basins of attraction have no transient trees, just the bare ``attractor''.

Now find every attractor cycle in state-space and construct its basin of attraction. This is the basin of attraction field containing all 2^n states in state-space, but now linked according to the dynamics of the network. Each discrete dynamical network imposes a particular basin of attraction field on state-space.

Very cool! Now there is a way to catalog all the stages and states leading up to a particular state. This is like the inverse of superposition. The superposition of all the pre-images of B gives birth to B.

..."The number of pre-images is the state's ``in-degree''. In-degrees greater than one require that transient states exist outside the attractor. Tracing connections backwards to successive pre-images of transient states will reveals a tree-like topology where the ``leaves'' are states without pre-images, known as garden-of-Eden  states. Conversely, the flow in state-space is convergent. Measures of convergence are G-density, the fraction of states that are garden-of-Eden, and the distribution of in-degrees,..."

By counting the particular input configurations that occur during sliding time windows, one can get a Shannon measure of the input entropy. A histogram can then be made of all the attractors in a system. It turns out that the variance of entropy is an excellent quantitative way to classify behaviours of the system. Ordered regions of state space have low entropy and low entropy variance. Chatotic regions have high entropy and low variance of entropy. Complex regions have medium entropy, but the statistical variance is high.

Another quantification is the Garden of Eden density. Ordered regions are highly convergent and have a large number of 'garden of eden' points. Chaotic regions have fewer G-points because they have less convergence on certain more probable states. Chaotic regions have sparse pre-image trees.

Attractor basins and their sub-trees are precisely identified with 'network memory' when talking about neural type systems.

In addition we can now visualize a computational method that can quantify the probability of landing on a given attractor basin given random starting states. It is a function of the number of pre-images belonging to each basin. The reverse algorithm also gives the probability for convergence or divergence at any step in time also. We now have a computational way to measure state probabilities.

State probabilities are thus an inverse function of superposition given the existence of attractors. Looking backward in time one sees that present order is the convergent result of furcitive possibilities. Only one time line into the past was actualized but the process still has a probalistic measure. The future is unpredictable and this probalistic measure is the best we can do. Seen from this perspective, wave collapse is a philosophical 'red herring' .

postnote:

I believe the next wave of research will focus on how different systems interact. The ideas above are for mostly closed systems. What happens when we deal with open systems exposed to extraneous and unpredictable outside influences? What happens as systems based on different functionalities and geometries interact?