Web Lecture
Mind and Nature
Psychology 3130
Department of Psychology
University of Utah
Tom Malloy
TAO Tool Description (Mind & Nature) |
The Recursive Application of TAO:
Higher order derivatives
I have said that what gets from territory to map is transforms of differences and that these (somehow selected) differences are elementary ideas. But there are differences between differences. Every effective difference denotes a demarcation, a line of classification, and all classifications are hierarchic. In other words, differences are themselves to be differentiated and classified. In this context I will only touch lightly on the matter of classes of difference, because to carry the matter further would land us in the problems of Principia Mathematica. Let me invite you to a psychological experience, if only to demonstrate the frailty of the human computer. First note that differences in texture are different (a) from differences in color. Now note that differences in size are different (b) from differences in shape. Similarly ratios are different (c) from subtractive differences. Now let me invite you... to define the differences between "different (a)," "different (b)," and "different (c)" in the above paragraph. The computer in the human head boggles at the task. --Gregory Bateson (1972), pp. 463, 464. |
TAO can be applied recursively
This means we find differences in differences in differences...
SEE: Table 5, below.
NOTE: Attractors are Circles and we represent Time as a line
so even though we represent a basin as a list of vectors from the "first" to the "last,"
the "first" vector is repeated immediately after the "last" and so on in an endless cycle.
Thus, the "last" state vector precedes the "first" state vector.
Table 5 |
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Recursive application of TAO to attracor cycle of Basin 1 |
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State Vector |
TAO-1 |
TAO-2 |
TAO-3 |
{1001} |
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{1101} |
2 vs1 {0100} |
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{1111} |
3 vs 2 {0010} |
(3 vs 2) vs (2 vs 1) {0110} |
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{1011} |
4 vs 3 {0100} |
(4 vs 3) vs (3 vs 2) {0110} |
[(4 vs 3) vs (3 vs 2)] vs[(3 vs 2) vs (2 vs 1)] {0000} |
1 vs 4 {0010} |
(1 vs 4) vs (4 vs 3) {0110} |
[(1 vs 4) vs (4 vs 3)] vs [(3 vs 2) vs (2 vs 1) {0000} |
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(2 vs 1) vs (1 vs 4) {0110} |
[(2 vs 1) vs (1 vs 4)] vs [(1 vs 4) vs (4 vs 3)] {0000} |
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[(3 vs 2) vs (2 vs 1)] vs [(2 vs 1) vs (1 vs 4)] {0000} |
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Attractor Cycle for Basin 1
STATE VECTOR COLUMN of Table 5
IS a list of state vectors for the attractor cycle of Basin 1
The pattern of differences in the first column is relatively rich.
TAO-1 COLUMN
compares each state vector
in the list
with the state vector that preceded it.NOTE: when we get to the bottom (the fourth state vector), since we are in an L = 4 basin, vector 4 precedes vector 1.
The pattern of differences (in the differences)
In the TAO-1 column is less rich than the first column.In fact there are only two patterns of difference:
({0100} and {0010})
whereas there were four patterns of difference in the first column.
TAO-2 COLUMN
Compares each TAO-1 vector with the vector that preceded it.
NOTE: Again, the last TAO-1 vector is assumed to precede the first TAO-1 vector.
Notice that for TAO-2 the pattern of differences is even less rich than it was in column two.
There is only one pattern of differences,
{0110}
TAO-3 COLUMN examines
[the Differences among the
{(TAO-2) Differences
{in the (TAO-1) Differences}
{in the state vectors Differences}
]
In this example, there are no differences in TAO-3.
All vectors = 0 = {0000}
This makes sense, since there was only one pattern of TAO-2 differences so they cannot differ from each other.
The epistemological implications of higher order derivatives are considered on the Differences in Differences web page.
Defining TAO matrices allows us to begin to create a procedure to categorize the form of basin patterns.
TAO-1 Matrix
TAO-1 Matrix for the attractor cycle of Basin 1. Notice in Table 5 that the application of the TAO function to the four successive state vectors in Basin 1 results in four TAO-1 vectors. These four TAO-1 vectors can be assembled into a TAO-1 matrix: [{0100}, {0010}, {0100}, {0110}]. HTML is not friendly to expressing matrices in their standard formats, so using a table format, Table 6 approximates a TAO-1 matrix for Basin 1 (T(1,1). Notice that the TAO-1 vectors alternate between two values (either {0100} or {0010}) in the T(1,1) matrix.
| Table 6. TAO-1 Matrix for Basin 1 | |
| T(1,1) = | 0100 |
0010 |
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0100 |
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0010 |
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TAO-2 Matrix
TAO-2 Matrix for attractor of Basin 1. The third column of Table 5 shows TAO-2: The recursive application of the XOR function; that is the application of XOR to TAO-1 (which is the application of XOR to the state vectors). Table 7 shows the four TAO-2 vectors arranged in matrix form. Notice that there is only one TAO-2 vector (0110) in the TAO-2 matrix for Basin 1 (T(2,1). That is there are no differences left.
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TAO-3 Matrix
TAO-3 Matrix for the attractor cycle of Basin 1. The fourth column of Table 5 shows TAO-3: The application of XOR to the output of TAO-2. Table 8 puts this in matrix form. Notice that all row vectors in T(3,1) = 0.
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TAO-4 Matrix for Basin 1. For completeness we include Table 9 which makes obvious the trivial outcome that seeking differences where there are no differences results in no differences.
| Table 9. TAO-4 Matrix for Basin 1 | |
| T(4,1) = | 0000 |
0000 |
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0000 |
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0000 |
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Differences in Differences
Putting it all together
Transposed and Visualized Vectors Vertical Axis = Nodes A, B, C, D Horistonal Axis = Time |
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State Vector |
TAO-1 |
TAO-2 |
TAO-3 |
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OR:
Basin 2 We have not derived TAO-2 and TAO-3 or TAO-4 for Basin 2 as we did (in Table 5) for Basin 1. But the procedure is straightforward and Table 10 shows the results of TAO-2 TAO-3 and TAO-4 derivations for Basin 2 without the detailed derivational support.
TRANSPOSE
VISUALIZE:
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OR:
Matrix Equality. Once we have the TAO matrices defined we can use the operation of matrix equality (see next section) to begin a qualitative analysis of the basin pattern. For the moment let us note that for two matrices to be equal, every corresponding element in each of them must be equal. For example the two TAO-4 matrices for Basins 1 and 2 are equal, that is, T(4,1) = T(4,2). Although we will make no use of the fact, technically we can say that T(2,1) = T(3,2).
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Toward Qualitative Analysis:
A Step toward Knowing the Form of Process
Going beyond finding Basins and Basin Lengths
There are epistemological Limits of ONLY Finding Basins and their Lengths Analytically.
The analytic procedure for finding a basin and its length tells nothing of the qualities of the dynamic pattern that constitute a deeper knowledge of that basin. Even static snapshots of basins will give some idea that there is more to the story than analytically detecting a basin and knowing its length. For example below are the visual representations of four basins (arranged in two categories). All four basins are L=8 (each image repeats each basin four times). All of these are basins of length L=8 from the same dynamic system; but saying that does not address the qualities of the patterns that characterize each basin. Finding basins is only the first step. Learning about pattern is a more challenging and interesting endeavor.
Using TAO Matrix Equality
To categorize Basin Patterns
As an enticement, I will point out here that in the image below the TAO-1 matrices are equal for the two basins on the left; likewise the two basins on the right have TAO-1 matrices that are equal. That is, when patterns are categorized by their TAO-1 (first derivatives) most humans find such categories to make perceptual sense. This is an interesting hint of possible ways of pattern hunting.
Here are two more examples of this correspondence between TAO based categories and human sorting:
Notice that two patterns on the left (Basins 32 and Basin 15) comprise Category 1 (based on the first derivative or TAO-1). Similarly the two patterns on the right (Basins 33 and 21) comprise a distinct TAO-1 category (Category 25--note that the category numbers are simply names arbitrarily assigned by the TAO tool). Human perceptual judgment tends to agree with these model generated categories.
Visualizing Higher Order TAO's.
Some Prerequisites. Large matrices of 0's and 1's are difficult to think about. As a result we have developed several strategies for visualizing the flow of process in a Boolean network. We assume the reader of this page has read the discussion of representing, in ways that are perceptually obvious to humans, the processes of Boolean systems. We particularly assume that the reader is familiar with the form of representation we call the Historical Trace (Smilie 3). See the relevant sections of the documents found on the following links. In short, a historical trace simply represents a basin or TAO matrix with a white square replacing a 0 and a black square replacing a 1.
Mapping Knowledge to Boolean Dynamic Systems in Bateson's Epistemology PDF (more formal)
Discrete Dynamic Systems & Epistemology HTML (more chatty)
Genius: A more Complex Example of Historical Trace (Applet) (interactive)
Briefly, a historical trace takes the state vectors (transposed so that they are column vectors) and replaces each 0 with a white square and each 1 with a black square. As the system iterates over time this trace moves right to left across the screen leaving a black and white pattern (a historical trace of the states of the system) behind.
Visualizing TAO Matrices. This (historical trace) representational process can be extended to TAO matrices: What the TAO images (below) do is replace every "1" in a TAO matrix with a black square and every "0 " with a white square. So, the images are simply visualizations of TAO matrices which make it perceptually simple to determine if two TAO matrices are equal. An identical perceptual pattern means two Tao Matrices are equal.
Below are four images, one each, for basins 15 and 32 (Category 1) and for basins 21 and 33 (Category 25).
In columns starting on the left and moving right
Each image shows:
a historical trace of the original basin matrix,
a historical trace of TAO-1 matrix,
a historical trace of TAO-2 matrix,
a historical trace of TAO-3 matrix,
a historical trace of TAO-4 matrix,
and so on.
If you look below and examine the first two TAO images you'll notice that, while the original basins have a different pattern, the TAO-1 patterns are identical. (All subsequent TAO's are also identical.) Basins 15 and 32 are placed in the same category precisely because their TAO-1 matrices are equal.) [To see the original basins in a pop up window click here: (Category 1, Basins 15 and 32)]
[NOTE: An advanced issue is that we have "rotated" the TAO-1 matrices so that each begins with the lowest Boolean valued vector. This has nuanced implications for how the TAO models map to epistemology which will be considered at a later time. Such rotation essentially makes the epistemological points we are making easier to understand.]
Likewise, if you examine the next two images you'll notice that the original basins again have a different pattern but the TAO-1 patterns are identical. Basins 21 and 33 are placed in the same category precisely because their TAO-1 matrices are equal. [To see the original basins in a pop up window click here: Category 25, Basins 21 and 33)
TAO-1 CATEGORY 1: BASINS 15 AND 32:
CATEGORY 25: BASINS 21 AND 33:
Recall that the images, above, represent static snapshots of what is a dynamic process of taking differences in differences. The flow of transformation of differences extant in the original basins (by finding the differences in differences over time, i.e., TAO-1, TAO-2, TAO-3, ...) is a process. To get a dynamic representation of this process you can examine the following Applets. Simply press PLAY (the Green Arrow) and the Applet will run:
NOTE: On each Applet you may want to press the Delay radio button and drag the Delay Slider down to a short delay to get the speed of the dynamics to a perceptual level you find pleasing. You can also adjust the Window Size slider which affects the Apparent Motion effects (see discussion of Apparent Motion elsewhere on this site). For an Applet that has good instructions and a discussion of the flow of differences in differences similar to the four applets above click here. |
Computationally Speaking: The main computational point is that we can easily see the basis of E42's perceptual categories when we look at the TAO matrices represented as black and white squares (Historical Trace). Basin's 15 and 32 have identical TAO-1 patterns and so are placed in the same category. Similarly, Basins 21 and 33 have identical TAO-1 patterns and are placed in the same category.
Epistemologically Speaking. The main epistemological point regards your perceptual judgments. When you look at the categories generated by E42 (which is designed to explore Bateson's difference-based epistemology) do the basin patterns within an E42 category look more similar to each other than they do to the basins patterns in a different category. If so, then the computational model has utility in its mapping to perceptual processes.
The Augented Boolean Network Model
As we mentioned at the beginning of this web page, the addition of a function like TAO to a simple Boolean model adds interesting features that enable a mapping of the augmented model to descriptions of epistemology and knowledge. Some of these implications are detailed elsewhere in:
Mapping Knowledge to Boolean Dynamic Systems in Bateson's Epistemology
and in
Steps to an Ecology of Emergence.
We enumerate briefly some of the interesting consequences of augmenting a Boolean model with TAO:
1. TAO is one example of how a system may perturb itself. While one can think of many different ways a system can self-perturb, at the very least, TAO provides E42 with one such mechanism.
2. TAO allows a way for a system to explore its own attractor cycles, moving from one to another.
3. By archiving attractor cycles, TAO allows the model to perform meta-functions on its basins. The example above of categorizing basins in terms of their equality of TAO matrices is one example of that meta operation. As we have seen the model's categorizing of its basins corresponds in interesting ways to human perception and hints at one way a knowing system might perceive and create categories.
These three functional additions to the basic Boolean model augment the model in ways that open up interesting and provocative correspondences to a knowing system.
Below is a quick summary of the parameters a user can set in using the TAO tool to search for basins in a system. After that summary, an example of the the output of a TAO search is shown.
Tao Frame. The Windows Menu allows you to show the Tao Frame. As can be seen by the figure below, an investigator can type in integers in two fields. First we will consider the lower field, "Iterations per Perturbation," which in the figure is set to its default value of 1000. This integer in essence determines the maximum basin length that a searched-for basin can have. Tao will track the system for as many iterations as are specified in the "Iterations per Perturbation" field. It searches for an exact repeat of the vector of states as described above. If it reaches the maximum number of iterations specified without detecting a basin it decides there is no basin and perturbs the system (i.e., randomly changes the state of each node with a probability of .5). You can also specify how many times Tao will perturb the system in its search for basins. This allows relatively cursory or relatively thorough searches of the basin structure of a system. There is no theoretical limit to the integers you enter in the two fields, but computing time sets pragmatic limits. As a point of reference, perturbing the system 5000 times while searching for all basins up to length 1000 takes a few minutes on a 1 gigabyte clock speed computer. Leaving Tao running over a weekend allows analyses with much larger numbers. When Tao completes its basin search, you can click the "Print" button and it will post a printable web page that contains a summary of basic information about basins found (their lengths and the frequency of basins of the same length). It also prints a full list of all basins detected. At the user's option it will also post all the vectors (see below) for all the basins detected. The latter case leads to lengthy printouts, but there are times when such complete information is of deep theoretical interest.
Example printout pages. Below is an example of a Tao printout for system that has three basin. At the top, a SUMMARY give the number of basins, and frequency of different basin lengths. Below the SUMMARY is a list of the details of all three basins. The printout shows only the first two of those basins. If you select the "bits" option in the Tao Options Menu, all the state vectors for each basin will be printed out. In that case, each horizontal vector of 0's and 1's shows the states of all the nodes on a single iteration.
******************************* NEW TAO FILE *******************************
Date: Mon Jun 02 14:21:16 MDT 2003
File: 03-06-02_L6-8-32_TAO-Description-Example
Number of Nodes: 66
Number of Connections: 3
************************************ SUMMARY ************************************
Number of Basins Found: 3
Number of Basin Lengths Found: 3
Number of Perturbations: 100
Number of Iterations per Perturbation: 1000
Number of Searches Past 1000 Iterations: 0
---------------------------------------------------------------------------------
For Basin Length 6: Number of Basin of Length 6 Found = 1
For Basin Length 8: Number of Basin of Length 8 Found = 1
For Basin Length 32: Number of Basin of Length 32 Found = 1
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*********************************************************************************
******************************** SEPARATE BASINS ********************************
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Basin 3
Jun 2, 2003 2:21:16 PM
Basin Length = 6
Frequency = 1
Relative Frequency = 0.01
Tributary Length = 6
Total Sum = 7244
SqRoot of Sum of the Squares = 2963.1780236766067
{31, 32, 35, 36, 38, 39, 41, 43, 44, 45, 46, 47, 48, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65}
000000000000000000000000000000011001101101011111100011011111111111
000000000000000000000000000000111111101110011110011110111101110010
000000000000000000000000000000011101111111010011111111100101010101
000000000000000000000000000000011001111011011111100011011111111111
000000000000000000000000000000111111111000011110011110111101110010
000000000000000000000000000000011101111101010011111111100101010101
---------------------------------------------------------------------------------
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Basin 2
Jun 2, 2003 2:21:16 PM
Basin Length = 8
Frequency = 57
Relative Frequency = 0.57
Tributary Length = 18
Total Sum = 9306
SqRoot of Sum of the Squares = 3295.05326208849
{33, 35, 36, 37, 38, 41, 43, 44, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 60, 61, 62, 65}
000000000000000000000000000000000101111001011010111110101111111001
000000000000000000000000000000111011111000110111100110111100101010
000000000000000000000000000000000101111111111011000111111111110101
000000000000000000000000000000111011111000111111001010101101011110
000000000000000000000000000000000101111111011010111110101111111001
000000000000000000000000000000111011111110110111100110111100101010
000000000000000000000000000000000101111101111011000111111111110101
000000000000000000000000000000111011101110111111001010101101011110
---------------------------------------------------------------------------------
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Basin 1
Jun 2, 2003 2:21:16 PM
Basin Length = 32
Frequency = 42
Relative Frequency = 0.42
Tributary Length = 21
Total Sum = 36782
SqRoot of Sum of the Squares = 6515.342508264627
{33, 35, 36, 38, 39, 40, 41, 43, 44, 46, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 63, 65}
000000000000000000000000000000000101101111011010001110111101110101
000000000000000000000000000000111011111001110110001111101101011110
000000000000000000000000000000000101111111010011011011111111111001
000000000000000000000000000000111011111110111110001100101101101010
000000000000000000000000000000000101111101010011001110101101110101
000000000000000000000000000000111011101111111111001110101101011110
000000000000000000000000000000000101111001011011011110101111111001
000000000000000000000000000000111011111000111111101110101100101010
000000000000000000000000000000000101111111111011000110111111110101
000000000000000000000000000000111011111000111111001111101101011110
000000000000000000000000000000000101111111011011011011101111111001
000000000000000000000000000000111011111110111110001101101100101010
000000000000000000000000000000000101111101110011001111101111110101
000000000000000000000000000000111011101110111111001011101101001110
000000000000000000000000000000000101111001011010101111101111111001
000000000000000000000000000000111011111010110111000011111101101010
000000000000000000000000000000000101111101011010001110111101110101
000000000000000000000000000000111011101111110111001111101101011110
000000000000000000000000000000000101111001011011011011111111111001
000000000000000000000000000000111011111000111110001100101101101010
000000000000000000000000000000000101111111010011001110101101110101
000000000000000000000000000000111011111001111111001110101101011110
000000000000000000000000000000000101111111011011011110101111111001
000000000000000000000000000000111011111110111111101110101100101010
000000000000000000000000000000000101111101111011000110111111110101
000000000000000000000000000000111011101110111111001111101101011110
000000000000000000000000000000000101111001011011011011101111111001
000000000000000000000000000000111011111000111110001101101100101010
000000000000000000000000000000000101111111110011001111101111110101
000000000000000000000000000000111011111000111111001011101101001110
000000000000000000000000000000000101111111011010101111101111111001
000000000000000000000000000000111011111100110111000011111101101010
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********************************** END OF FILE **********************************
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