Boundary Conditions and Limitations for
The Principle of Dynamic Constancy

A new perceptual principle: Dynamic Constancy

Patterns whose dynamics have the same pattern of changes over time will be grouped perceptually together. Patterns whose dynamics have differences in their patterns over time will be perceptually distinct. The catch (ceteris paribus) is that the principle of dynamic constancy will hold when all things are equal, that is, when no other principles of perceptual grouping over-ride it. Table 1 focuses on examples of the failure of the principle of dynamic constancy.

On the Differences in Differences page we noted that in some ways TAO sorting corresponds to human sorting. Now we are going to look at counter examples and examine in detail how it comes about that the principle of dynamic constancy fails when indeed it fails.

Table 1. A TAO sorting that does not correspond well to human sorting.
Category 5 seems coherent enough to humans but how is it that Basin 89 isn't placed with basins 74 and 95 in Category 5? And how did basins 89 and 57 get put together when they look so different?
Basin 74	Basin 95	Basin 89	Basin 57
TAO-2 Category 5		TAO-2 Category 2
NOTE: The images shown are selected details from larger images representing basin dynamics.

Table 1 makes obvious that there are boundary conditions on the principle of dynamic constancy. How this happens is that other principles of perceptual organization can over-ride the perceptual effects of dynamic constancy. There are two broad categories of the failure of the principle that are obvious in Table1: First, sorting by the first derivative sometimes places two patterns (89 and 57) that look different in the same category; and, second, first derivative sorting sometimes places patterns that look similar (74, 95, 89) in different categories.

The following discussion will address how dynamic constancy works and how it doesn't work.

Ceteris Paribus

When everything else is equal. Principles of perceptual grouping have a broad boundary condition: Since they compete with other principles of perceptual organization, they apply most obviously when no such competition is present, i.e., when all other things are equal.

For example Table 1 shows how the the principle of proximity can be put into conflict with the principle of closure.

Table 2. Proximity in conflict with Closure
The top row shows how placing dots close together produces columns (left) versus rows (right). The bottom row shows how the tendency to close brackets [ ] into a square over-rides proximity. The ][ brackets are closer together than the [ ] brackets, yet we tend to see (or to see at least as easily) the "closed" squares [ ] rather than the ][ slits. The one exception is on the far left where there is no mate for the left-most ].

Limits and Boundary Conditions
For the correspondence between perceptual sorting and sorting by derivative

Case 1: When TAO places patterns that look different in the same category

Figure/ground. The representation of dynamic relations as equally-spaced black and white squares leads to figure ground issues. The first derivative (XOR) function does not know content; it does not know a 0 from 1 or a black square from the white square. XOR only knows if two tokens are the same or different on the next iteration; it treats 1,1 in the same way it treats 0,0. It returns a 0 in both cases because in both cases the content is the same. XOR only knows if the content is the same or different; it does not care if the this sameness is due to two consecutive 0's or two consecutive 1's.

Punctuating a sequence of alternating black and white squares. Suppose we consider for the moment only a single node, isolated from the behavior of other nodes in a system. Suppose that the behavior of that node is to alternate on and off. Suppose that we represent that alternating behavior as a series of black and white squares across a sequence of three iterations. To do so requires that we decide how to punctuate the flow of alternating states; do we start with an off and end with an off, or do we start with an on and end with an on? Table 3 shows that the visual representation of the behavior that node is perceived strikingly differently depending on how we punctuate the sequence of iterations.

Taking the first derivative of either of these two punctuation of the sequence will yield TAO-1 = {1,1,0}. That is, XOR compares iteration T=1 to T=2 and finds them different and so returns a 1; it compares iteration T=2 to T=3 and finds them different and returns a 1 and finally it compares iteration T=3 to T=1 and finds them the same and returns a 0. The first derivative of both of these two sequences is the same, {1,1,0}. On that basis, TAO-1 will sort these two basins into the same category. The first derivative (XOR) does not care what the content is, it only cares about the relation between content. Humans, in contrast, care very much about the shifts from black-white-black to white-black-white. Humans tend to see such shifts as shifts in figure/ground and as triggers for other organizing principles of perception.

Phase relations.

When perception does not correspond to derivative equality. What happens when we consider (see Table 4, below) more than one node changing dynamically over time rather than, as above, a single node? Suppose that there are two adjacent nodes that we want to focus on for observing perceptual phenomena but that these two nodes are part of a larger system that has three nodes. Suppose that, primarily due to that third node, the larger system falls into to two distinct basins, each of length L=4. Suppose that Nodes 1 and 2 both alternate on-off-on-off as shown visually in Table 4, below. [Technically, we would say that these two nodes have sub-basins of length L=2 embedded in a larger system that has basins of length L=4.]
Notice that, when these two adjacent nodes (Nodes 1 and 2 in Table 4) alternate out-of-phase (basin 1) versus in-phase (basin 2), the patterns formed are perceived differently. Yet their first derivatives are equal.

DETAILS: How is it that the first derivatives are equal?

Node 3. First notice in Table 4 that Node 3 in the system is what is extending the two basins to length L=4. That is, it is Node 3 that has a pattern of length 4; in both basins it has 3 iterations that are in the same state followed by one that is in a different state. (I've separated Node 3 from the two we are talking about to avoid other perceptual effects that I will talk about later.) In basin 1 Node 3 is white-white-white-black; in basin 2 Node 3 is black-black-black-white. In either case the first derivative is {0010}. So for Node 3 alone, basins 1 and 2 would be put in the same category.

Nodes 1 and 2 have the same derivative in both basins. Note that both Node 1 and Node 2 always change from iteration to iteration, both in basin 1 and in basin 2. So the XOR function comparing each node at T and T+1 will always return a 1. In other words the first derivative for each node in each basin is {1,1,1,1}. Therefore first derivative is the same in basin 1 and basin for both nodes. Therefore TAO-1 on the basis of the derivatives of these two nodes, place basins 1 and 2 in the same category.

Basin 1 and Basin 2 have the same derivative. As we've noted the derivatives for every node in both basins is the same. On this basis Basin 1 and Basin 2 would be placed in the same category. In more detail the ordered set of vectors for the first derivative across all three nodes in Basin 1 is [{1,1,1,1},{1,1,1,1},{0010}]. The same is true of Basin 2, the ordered set of TAO-1 vectors is [{1,1,1,1},{1,1,1,1},{0010}]. The two ordered sets are identical so the two basins (for the three-node system as a whole) are placed in the same category.

Perceptual Grouping over-rides Dynamic constancy. Perceptually the patterns for the two adjacent nodes (1 and 2) organize into a checkerboard in basin 1 whereas in basin 2 they organize into columns. Perceptually they are very different even though they share the same dynamic invariances. This does not mean that dynamic constancy is not a valid principle of perceptual organization but rather that perceptual grouping principles can conflict with it and set limits for it.

Putting the 3-Node system together. Up to this point, I wanted to focus on phase relationships between Nodes 1 and 2 and so in Table 4, above, I kept the third node perceptually separated. Making all three nodes adjacent (as they would normally be) makes the perceptual grouping effect even stronger. Basin 1 forms a nondescript checkerboard-with-a-tail pattern while basin 2 forms a block U pattern. Again, as noted above, both basins have the same first derivative.

Returning to the main example. Table 6 below (essentially the same as Table 1, above) shows details from the principle example of TAO sorting used on the Differences in Differences web page to generate at hierarchy of levels of categories.

TAO-1, TAO-2, TAO-3, ... On the Differences in Differences page that is the primary page referring to this page, Tables 1 and 2 document a hierarchy of categories based on successively sorting basins by applying TAO recursively. Therefore there are categories for TAO-1, TAO-2, ... and so forth. The discussion above regarding phase shifts and principles of perceptual grouping that over-ride the principle of dynamic constancy used exclusively the first derivative (TAO-1). The reason for this focus is to keep the discussion simple; it's detailed enough to track the logic of TAO-1 let alone following through the recursive application of XOR to TAO-2 and beyond.

The limits and boundaries that apply to TAO sorting are perceptual and refer to the organization of the patterns in the original basins, so they apply to the success or failure of the sorting whether that sorting be by TAO-1 or TAO-2 or higher-order derivatives. Consequently, for this discussion, I have selected examples from whatever level of TAO returns interesting sorting of patterns. The overall hierarchy of patterns shown Table 2 of the Differences in Differences page integrates all these examples into a coherent whole. That said, Table 6 is based on TAO-2 sorting.

Table 6. A TAO sorting of fragments of static visual representations of four basins into two categories.
Time (iterations) runs horizontally in each basin. The vertical dimension corresponds to several adjacent nodes selected for detailed study from the system. The red arrows point to the behavior of one particular node. The arrows point to the same, corresponding node in basin 89 and in basin 57.
Basin 74	Basin 95	Basin 89	Basin 57
TAO-2 Category 5		TAO-2 Category 2

White Spaces affect perceptions. Let's use Table 6 to look at one more aspect of how TAO sorts in ways that DON'T correspond to human sorting. In Table 6, Category 2, there is a red arrow in both basin 89 and basin 57. These arrows point to our visualization of the behavior of a particular, target, node. That target node behaves rather differently in the two basins. It shows alternating black and white squares in basin 89 and a solid line of white in basin 57. In other words is alternating on-off-on-off in basin 89 but is frozen off in basin 57. How does TAO-2 calculate the second derivative to be the same for that node in those two categories? The alternating black and white squares of basin 89 essentially come out to be a {0,1} vector repeated over and over. TAO-1 applied to this vector returns {1,1}. Applying XOR to the the {1,1} vector returns TAO-2 = {0,0}. Obviously the white line in basin 57 is just a vector of 0's. Since the white line is a vector of 0's, TAO-1 is a zero vector as is TAO-2. Therefore the alternating black and white line and the solid white line have the same TAO-2 = 0.

TAO-2 may return the same zero vector for an alternating black and white pattern as it does for a constant white pattern, but human perception of such patterns in the context of the patterns of adjacent nodes is very different. The detail (left) shows how the white line in basin 57 produces a coherent perceptual grouping (indicated by a red box) that is perceptually strong in basin 57 (Table 6, above). In contrast, the same area in basin 89 organizes very differently.

Table 7

Perhaps a deeper integration of this discussion might come from making the same points from the perspective of a different way of representing the material. Table 7 shows basins 89 and 57 again, but this time the original basin (noted as "Or" along the bottom of the image) is shown in the leftmost column and TAO-1 (noted as "1") in the next column and then TAO-2, TAO-3, and TAO-4. The red arrow points at the target node in our discussion. Notice that in basin 89, on the left, that node starts out alternating, then is the TAO-1 column becomes a solid black line (all 1's) since TAO-1 = 1, a vector of 1's. In the TAO-2 column that node becomes a white line since TAO-2 = 0. So that node's constant change from one state to another in the original basin is coded by TAO-1 as "constant difference" across all four iterations of the L=4 basin. And that constant change becomes "no change" in the "eyes" of TAO-2.

Conversely, the target node in basin 57 is frozen in the off state and so all derivatives are 0.

Looking at the TAO-2 column, you can see that the pattern of TAO-2's is identical for the two basins, and so they are sorting into the same category. That sort works poorly for some nodes by as we've seen before and will review below it works brilliantly for other nodes.

The Power of TAO to sort by depth of complexity. Table 7 also allows an examination of how TAO sorts basins.

XXXXXXXXXXXXXXXXXXXXXXX Under Construction xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Notice in the image below, successive applications of TAO reduce node patterns to 0 vectors (white space); only the most complex patterns survive. So sorting by high-level TAO's is sorting by the most complex parts of patterns. The cost is that perceptual over-rides like the one shown above will become more evident when this sort of sorting happens.

Case 2: When TAO places perceptually similar patterns in different categories

For Basins 74 and 95, Category 5 above notice carefully how the first column of each pattern. Notice, starting from the bottom solid black row that the checker board pattern starts with black white black white black etc. This is true of both basin 74 and basin 95. In contrast, basins 89 and 57, Category 2, start with the pattern shifted over one square. That is to say, if you look closely, Categories 5 and 2 are out of phase by one iteration. Otherwise the patterns in the two categories are rather similar. Humans typically ignore such subtle differences (although they can perceive them since we are talking about them). TAO, on the other hand, is very sensitive to phase shifts, and places them in different categories.

[ASIDE: Is out of phase (below) like size--irrelevant to the pattern of formal relations? Or is basin phase a product of counting iterations and therefore a different sort of thing than size (a quantity)--but possibly still irrelevant. Right now, I think that the out of phase relations (below) are an essential characteristic of the recursive application of XOR and that that these phase shifts may in fact, when combined with apparent motion as a pattern detector, the basis of all pattern perception. But to motivate that last rather grand statement will take some effort and a long journey.]

Other things being equal. There is no pretense being made that sorting by derivative (by changes in dynamics) is the the only or even the most important perceptual sorting principle. Figure/ground and various perceptual grouping effects are powerful perceptual principles that over-ride the perceptual correspondence we "see" with sorting by derivatives. What we are proposing is that sorting by changes in dynamics is a powerful new perceptual principle that corresponds to human sorting when all other things are equal (Ceteris Paribus). This limitation is held in common with most other perceptual grouping principles. The pretense we are asserting is that sorting by characteristics of dynamics (changes in differences in differences over time) is a powerful new perceptual principle.

Where Perception and Dynamic constancy DO Correspond. The

Table 7. Details TAO-3 Categories that DO Correspond to Perception
Basin 38	Basin 76	Basin 60	Basin 89	Basin 74	Basin 95	Basin 57	Basin 67	Basin 78
Category 1			Category 2				Category 3

This phase shift adjustment will produce TAO categories that fit better with human sorting. Might consider this adjustment useful or not. Certainly it can be argued that humans routinely adjust their phase relations with the environment and with each other. Drummers may want to play on the beat with each other or, within some constraints, play out of phase each other to create a richer pattern. That the TAO model has a similar option to shift different basins into phase or leave them out of phase is simply and interesting characteristic of the model.

Multiple Descriptions. In full list of length L=4 categories below, we can see the advantage of having multiple sorting principles. Circled in red are parts of the basin representations where perception corresponds well to dynamic invariances (see sorting by TAO-3, below). In responding to a complex and dynamic universe, it seems useful to have many ways of sorting the many aspects of our representations of that universe.

Perceptual grouping principles are very powerful. Elements that are close, particularly if they are touching, tend to be organized together while elements that are separated by space are perceived separately.