Web Lecture
Mind and Nature
Psychology 3130
Department of Psychology
University of Utah
Tom Malloy

TAO Tool Description

Contents:

TAO Overview
TAO-1: The First Derivative
Finding Basins using TAO-1
Recursive TAO: Higher Order Derivatives
TAO Matrices
Steps toward Knowing the Form of Process
Visualizing Higher Order TAO's
The Augmented Boolean Network Model
TAO Tool Interface in E42
TAO Tool Output

 

 

TAO Overview.

TAO is a suite of tools
for analyzing and for representing:

The Qualities Change over Time
in Dynamic Systems
(hence the name).

Imagine two state vectors (six nodes)

TOA uses the XOR logical operator

to compare each respective position in the two state vectors

Same returns 0

Different returns a 1

 

Detecting Basins

The basic idea is that

TAO examines the flow of state vectors

When it detects that a state vector is equal to some prior state vector

(which has already flowed past)

TAO "knows" that the system is in a basin.

Archiving

TAO extracts a copy of the system's state vectors for every basin it finds.
It then

indexes (names)
catalogues
and archives

This allows the dynamics of a specific basin to be studied on demand, without the tedious, sometimes impossible, wait for the system itself to repeat that basin.

Higher-order derivatives.

The TAO tool suite

has the capacity to examine the changes in the changes in a flow of process.

Described below there will TAO-1, TAO-2, TAO-3,...

functions akin to the first, second, third, ... derivatives in calculus.

Representing and modeling the flow of change.

TAO has advanced features that allow alternative methods of representing the flow of change in a system in ways that have implications for model building.

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Example

Consider the simple dynamic system, "4-Node Standard," used in the tutorial explicating the logic of N, K Boolean systems.That logic led to the development of a basin structure which is shown in Figure 2 from the tutorial.

On any iteration, T, we can fully describe the state of a system by its state vector, S(T). The state vector shows the four states corresponding to the four nodes in order.

Figure 1 shows the deterministic flow from one state vector to another resulting from the logical relations among the four nodes in the "4-Node Standard" system. For example, the state vector {0000} always returns to itself, while the state vector {0101} (shown on the bottom left of Figure 2 and used as the starting point of Table 1, below) flows deterministically to {1101} which then flows to {1111} and so on.

A series of state vectors is required in order to get to know something useful about the dynamics of a system.

TAO-1: The first Derivative.

(The phrase first derivative (or TAO-1) implies a second derivative (or TAO-2); these higher-order derivatives will be discussed below.)

The 4-Node Standard system had four nodes named A, B, C, D. The state vector show the states of those nodes in order. So S(1) = {0101}means that on iteration 1 A=0, B=1, C=0 and D=1.

It is relatively easy to compute the TAO function by hand in the following format which shows two state vectors and the TAO that would result from their comparison:

Simply compare the corresponding nodes in the two state vectors. If they are different the TAO will return a 1; if they are the same it returns a 0.

As an arbitrary starting point, suppose the 4-Node Standard system started running at T=1 with the state vector, S(1) = {0101}
(which is the state vector in the lower left-hand corner of Figure 2.).

Consider how the system changes from T=1 to T=2. S(1) is {0101} and changes to S(2) which is {1101}. Look at these two vectors in the T=2 row of the third, (right-mos)t column of Table 1. Notice that the first position (first node) changes from a 0 to a 1 but the last three nodes remain unchanged from T=1 to T=2. Therefore the TAO-1 vector of changes (row T=2, third column) is {1000}.

Similar comparisons are made between each iteration and the prior iteration in the third column of Table 1.

Notation. Consider the functional notation, e.g., "TAO-1 (4,3)," from the third row of the table. The "1" in TAO-1 means we are talking about the first derivative. The parentheses (4, 3) means we are comparing the state vectors for T=4 and T=3. Finally, of course, TAO is the symbol of the function that returns a 0 when there is no change in the elements of state vectors over time and a 1 where there is. The elaborate indexing (4,3) of iterations usually is not necessary, but in the discussion below of how TAO-1 is used to discover basin structure, it will be useful. So in this section indexing will be detailed and operators (e.g., TAO-1) will appear in bold while vectors (e.g., S(T)) will both be bold and underlined. In other contexts where the meaning is clear the notation will be simplified to TAO and S(T).

Using TAO-1 to find Basins Top

In our above description TAO described the changes in the dynamic flow from one iteration to the next iteration. But this same function can be applied across longer time spans to detect basins.

General idea: Backwards Comparisons.

The basic rational for finding basins analytically with TAO is simple--

At any moment, T,

Compare the current state vector backwards in time
with all previous state vectors

Until

you find one that is the same,
that is, until TAO returns all 0's (the 0 vector)

OR until
you go back all the way to T=1.

We are looking for any comparisons where there is no change from the current state vector to a prior state vector.

0 means a basin.

Because the dynamic systems we are considering are deterministic, any state vector that occurs more than once must be followed by the same sequence of state vectors each time it occurs. So if the same vector occurs twice, once at T and and again at T+L, that means that the identical sequence of state vectors that fell between time T and time T+L must repeat again between T+L and T+2L and yet again between T+2L and T+3L and so on forever or until the system is perturbed. The system is in a basin and L will be the basin length. In our four-node example, when we compare backwards from the current state vector to previous state vectors we are searching to find a TAO vector that has four zeros (0={0000}) indicating that all four nodes have not changed in a particular comparison.

Backwards from the Start. To do a TAO basin search, we start the system and it generates a state vector, S(1), for T=1. We let the system iterate once, generating another state vector, S(2) for T=2. Now we compare S(2) backwards to S(1). If the comparisons returns a 0, then we have found a immediately found a basin of length L=1. If that comparison returns a non-zero vector (meaning at least one node has changed) then we have not found a basin and let the system iterated to T=3, generating S(3). We then compare S(3) backwards with S(2) and S(1) in that order. This continues until we get a 0={0000}for some comparison indicating that none of our four nodes have changed and a basin has been found.

In general, in a deterministic system the reoccurrence of the same state vector indicates the system is in a basin.

We can look at the same idea visually. Examination of Figure 2, above, will make evident that when the flow of state vectors repeats the exact same state vector the system is in a basin. So when a first derivative taken between any two vectors returns 0, it means that there is no difference between those two state vectors, and a basin has been found.

The idea is to start with S(T) and compare S(T) backwards in time to S(T-1) and then to S(T-2) and then to S(T-3) and so on until you either find a comparison [S(T) versus S(T-L)] that yields0 or until you arrive back at S(1).

If, starting at T and comparing backwards, you reach the first iteration (T=1) without TAO = 0, then go on to the next iteration, T+1, and compare backwards from there. Continue this procedure until you find a basin or until you reach a user-defined parameter that tells you to stop. Then perturb the system and begin searching for another basin.

Detailed example of finding a basin. Let us examine this rational in more detail by by showing all the computational details of the procedure in Table 2, below, which expands the example started in Table 1. Let the system begin running at T=1 in state vector S(1) = {0101}, which, from visual examination of Figure 2, we know will result in the system falling into basin Gamma.

Begin comparing backwards at T=2. The only prior time is T=1, so there is only one backward TAO comparison, which yields {1000} which not 0.

Therefore move to the state vector for the next iteration, T=3, and make the TAO comparisons between current, T=3, state vector and the two previous state vectors. Neither of these produces a TAO = 0.

Therefore, move to T=4 and repeat all the TAO comparisons back to T=1. None of these yields 0.

Move on to T=5 and then to T=6. At T=6, the TAO comparison with the state vector for T=2 will produce 0, that is, TAO-1 (6,2) = {0000}= 0. At this point TAO has found a basin. The difference in the time index (T=6 minus T=2) is 4 which is the basin length. L = 4.

Archiving. At this point the TAO tool will archive the basin. It will record the four state vectors that constitute the basin, rotate them so that the state vector with the lowest boolean value is placed first, and name the basin. Since it is the first basin found, it will be called Basin 1. The number (e.g., "1") in the basin name has no numerical or counting function; it is simply a name, that is, it functions like the numbers on athletic jerseys which are simply a way of identifying players. TAO will create a a record that looks something like:

Learning to Think

Another name for this course might be:
Pre-requisites to Personal Genius

Loops, Cycles, and Time Lines

Time Line
(Left to Right Convention)

Convert Linear Representation
to
Looped (Circular) Representation

Cycles: Mental Proccess falls into Attractor Cycles
and Loops (until Perturbed)

Wheel

A Wheel of States (around circumference)
Rolling across Time

Visualizing Attractor Cycle Matrices

1. First TRANSPOSE the Basin Matrix
Because... we like to have time running along
the horizontal axis

 

Un-Transposed
Attractor 1 Matrix
Nodes ==>

1	0	0	1
1	1	0	1
1	1	1	1
1	0	1	1

Transposed
Attractor 1 Matrix
Time ==>

1	1	1	1
0	1	1	0
0	0	1	1
1	1	1	1

 

2. Replace 1's with BLACK and 0's with White

Re-Visualize the Time Line

Visualize the Calculus of Difference

Search for the next basin.

Once TAO has found and archived a basin , it moves on to another search for other basins. To do so, TAO examines the latest state vector, (e.g., S(6) = {1101}) and perturbs the system by pseudo-randomly selecting some (user defined) percentage of nodes and changing their states (0 to 1 or 1 to 0).

It then begins the process again. Using the 4_Node Standard example of Figure 2, remember that S(6) was last state vector that occurred when Basin 1 was found. Suppose the pseudo-random change of S(6) = {1101} changed S(6) from {1101) to {1100}. TAO then starts the next basin search with S(1) = {1100}. Figure 2, above, indicates that this starting point is in basin Beta, so that is the basin that will be found. Refer to Figure 2 for information about where the state vectors in Table 3 come from in the example below. The process is the same, and you may want to skip it; it is provided to develop a complete example for those who want one.

Once again, in Table 3, above, TAO begins at T=2 and compares the state vectors for T=2 and T=1, which do not yield 0. To TAO moves on to T=3 and then to T=4 and finally T=5 where it encounters a comparison (S(5) versus S(1)) whose TAO = 0. It then archives the new basin it has found:

Visualize the Calculus of Difference

Finally

TAO will initiate another search which is summarize in Table 4.

4-Node Standard Example

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