The Structure of Boolean Derivatives in TAO and Meta-TAO

Following the ecological epistemology of Gregory Bateson, we have modeled mental process as the flow of differences in a richly connected network. Bateson also suggested that taking differences in this flow of differences would produce higher order knowledge. Finally, Bateson considered symmetry to be the basis for the pattern which connects living forms and, while he never connected taking differences in differences with symmetry, we have done so here. We operationalize the taking of differences in the flow of differences through the XOR operator of symbolic logic. This makes the XOR operator key to our model of mental process. The Sierpinski gasket describes the pattern of the recursive application of the XOR operator in a Boolean flow of differences. Moreover, the recursive application of XOR interacts with the period (length) of attractor cycles in a way that under certain conditions it is self-canceling and under other circumstances wraps upon itself to create complex patterns. Thus Bateson's call to take differences in differences implies, within the dynamic system's perspective of Boolean simulations, an elegantly symmetrical Boolean process whose very symmetry is an ideal tool for discovering symmetry in a world of broken symmetries because the Sierpinski gasket has symmetry in terms of rotation, reflection, translation and magnification. Therefore processing any flow of differences through the gasket will detect and highlight how aspects of the flow of differences diverge from or conform to symmetry. This creates a perfect detection method for broken symmetry. The fact that taking multiple iterations of the difference taking process (TAO) can wrap upon itself now connects Bateson’s ecological epistemology to topological insights of chaos theory. Part 3: Sierpinski Gaskets: The Structure of Boolean Derivatives 2 Hidden Order We are using an NK Boolean System (Kauffman, 1993) to formalize Bateson's (2000, 2002) ecological epistemology within a modern nonlinear dynamic systems approach. This formalization has three fundamental premises. First, the map (what humans and other sentient beings know) is not the territory (that which is known); moreover, what gets onto maps from territories are differences in the territories. Second, mental process can be abstracted as a flow of differences in a richly connected network; in Boolean terms, then, James' stream of consciousness becomes a stream of 0's and 1's. This stream of differences in dynamic systems terms can be construed to flow within the constraints of an adaptive landscape consisting of many basins of attraction, each with an attractor and, usually, many tributaries. Third, higher order knowledge emerges in the process of taking differences in the flow of differences (Bateson, 2000, p. 454ff). In the formal Boolean model this process of finding differences in the flow of differences begins with finding discrete derivatives (TAO's) and continues with finding differences among the derivatives themselves (Meta-TAO's). Considered meta-theoretically, this three premise perspective is extremely simple compared to the complexity of many cognitive approaches. The third premise of the model, that mental phenomena of great interest will emerge from taking differences in the flow of differences (Bateson's thought experiment 2000, p. 463, 464) has led to the definition of the derivative taken on an attractor cycle in a discrete system in terms of the XOR operator and has produced hierarchical perceptual categories (Malloy, Bostic St Clair, & Grinder, 2005). Given our operationalization of Bateson’s difference-taking process as XOR, and given Kauffman's arguments that the order which emerges is a function of the relations that define a system, we now turn our attention to the nature of the XOR operator. XOR is a key function in our simple model and as such we want to describe its operating characteristics and how those characteristics contribute the flow of mental process as we have modeled it. What is the hidden order implied by Bateson's call to take differences in the flow of differences? And what does this hidden order have to do with knowing in a world of symmetry and broken symmetry? Methodological comments. The TAO-0 (attractor cycle) matrices used in Part 3 are arbitrary. By arbitrary we mean that they were not simulated Boolean systems generated by E42. We are concerned here in Part 3 with the properties of the XOR operator within the context of matrix algebra. Thus we start simply with a matrix that has convenient properties for our analyses and then use a spread sheet to produce the various transforms (TAO, Meta-TAO). When Boolean systems are simulated in E42 the emergent attractor cycles are described by NxL matrices, where N is the number of nodes and L is the length of the attractor cycle. Nodes are the rows (vertical axis) of the matrix and iterations (time) are the columns. In Boolean math the cells of these matrices contain either a 0 or a 1; in our figures we typically have converted a 0 cell to a white cell and a 1 cell to a black cell so that the pattern of the attractor cycle is apparent for qualitative analysis. Once we have a NxL attractor matrix, we use the XOR operator (see Part 1) to generate derivatives of the dynamics of the attractor cycles which we call the TAO functions; thus the derivatives are labeled as TAO-1 (first derivative), TAO-2 (second derivative), and so on. At times we refer to the original attractor cycle as the zero order derivative (TAO-0). All the TAO's are also NxL matrices filled with 0's and 1's that for visual inspection we often convert to white and black cells. In short Boolean simulations produce NxL matrices that describe the dynamic behavior of the system’s attractor cycles. But the behavior and thus the matrices of a particular randomly generated or even engineered system is emergent and it is impossible to predict the details of the emergent landscape Part 3: Sierpinski Gaskets: The Structure of Boolean Derivatives 3 of simulated Boolean systems. Thus, in Part 3 because we want to examine matrices with certain properties and these properties may or may not show up in a simulation, we simply use arbitrary matrices that are convenient for exploring the nature of the XOR operator acting on matrices in various ways. Matrix algebra assures that the analyses transfer back to matrices generated by E42. In Part 3 we will be particularly interested in row vectors selected from the NxL matrices. For example, look at Figure 3.1 and at the L=4 set of 5x4 matrices running from the TAO-0 matrix to the TAO-6 matrix along the top of the figure. The vertical dimension of these matrices is individual nodes and the horizontal dimension is time (iterations). Thus each node has its own row vector, n, which expresses the change in its states (0 vs 1) across time as the system cycles through its attractor. For the Node 1 in the TAO-0 matrix of the L=4 system, n(0) = {1011}. The same node has a corresponding vector in the TAO-1 matrix, which is n(1) = {1100}, and so on across TAO levels up to TAO-6. In this paper we will focus on how a particular node's vectors change across TAO levels. [Technically, we could specify these node vectors to indicate n(1,3) = {1111} means the vector for Node 1 in the TAO-3 matrix. We won't normally use this degree of specificity because we will focus on a single node in our discussions so that it will be obvious what n refers to.] If it is not clear to the reader how, in Figure 3.1, the node vectors change from the TAO-0 matrix to the TAO-1 matrix to the TAO-2 matrix and so on, please review Part 1. In short, the TAO function in E42 can be used to find the difference in each node of a Boolean system generated in E42 as that node changes across iterations in an attractor cycle; it does so by applying the XOR function to that node's state across moments in time. Recall that, as defined in Part 2, Meta-TAO is the XOR comparison, cell by cell, of any two matrices. Typically we use the Meta-TAO analysis to compare the original attractor cycle matrix (TAO-0) with one of its derivatives. So, when we compare an attractor matrix with one of its derivative matrices, Meta-TAO-1 is the cell by cell XOR comparison of TAO-0 and TAO-1 (that is, of an attractor and its first derivative), and Meta-TAO-2 is the XOR of an attractor matrix with its second derivative matrix. Meta-TAO in these cases indicates, for every cell, if the attractor matrix is the the same (0) or different (1) than a given derivative matrix. At the end of Part 3 we will use the Meta-TAO tool to compare matrices in a more general way. Finally, rather than solely using black and white squares to indicate node states, examples will use 1’s and 0’s or BLACK and WHITE interchangeably to represent the states of nodes. Patterns in the Recursive Application of TAO. As previously demonstrated (Figure 1.6, Part 1), attractor matrices that are a length, L, that is a power of 2 resolve to a 0 matrix after repeated applications of the TAO function. However attractor cycles whose length is not a power of 2 fail to resolve through recursive Tao application. Figure 3.1 demonstrates the effect of recursive TAO on both a power-of-two and a non-power-of-two attractor cycle. The bottom row of Figure 3.1 shows that, while not resolving to zero, the recursive application of TAO a to non-power-2 attractor cycle results in a cyclic repetition of TAO patterns. In this case, the attractor cycle of basin length 5 enters a TAO-loop at TAO-1 which is out of phase with but otherwise identical to TAO-4. Since the matrices in this figure have not been rotated, we recommend that the reader rotate the matrices mentally; this is most easily done by noticing, for example, that if you slide the top row of TAO-1 over one cell (iteration) to the right it will be identical to the top row of TAO-4, and so on down all the rows of TAO-1. This procedure will indicate that TAO-2 is identical (when shifted in phase to the right one iteration) to TAO-5 and that TAO-3 is identical to TAO-6. Note that TAO-7 (not shown) would be a repetition of TAO-4, and so on, ad infinitum. Part 3: Sierpinski Gaskets: The Structure of Boolean Derivatives 4 Figure 3.1. In the matrices show in this paper, the white cells represent 0 in Boolean algebra and black cells represent 1. In this figure TAO matrices have not been rotated to begin with the lowest Boolean value. Top Row: Recursive application of the TAO function (discrete derivative) to an attractor cycle whose length (period) is a power of 2 diminishes to the 0 matrix. Bottom Row: Recursive application of the TAO function does not diminish to a 0 matrix but rather settles into a repeating set of TAO matrices (or TAO patterns). In this case the TAO patterns loop every third application of TAO (assuming you adjust them for phase). We can start with the dynamics of an attractor cycle and take the first, second, third, fourth, etc. derivatives in a sequence for as long as we want. This is a recursive process in that the output of one derivative is the input for the next derivative. If the cycle length of the attractor is a power of 2 then the derivatives will diminish to 0. But for attractor cycle lengths that are not powers of 2, the patterns of 0's and 1's found in the derivative matrices will not diminish to 0. Indeed these patterns will repeat previous derivative patterns in a looping sequence (except that they are out of phase with the previous derivative). We refer to these descriptively as derivative loops and examine how they come about later in this paper. In Figure 3.1 (bottom row) the derivatives loop every third derivative (TAO-1 = TAO-4, TAO-2 = TAO-5, and so on). Figure 3.2. TAO’s and Meta-TAO’s for an arbitrary attractor cycle (TAO-0) of length L=5. Since Meta-TAO, as we are using it here, is a joint function of TAO-0 (original attractor cycle) and some other—higher order—derivative, we expect that there will be some repetitive Part 3: Sierpinski Gaskets: The Structure of Boolean Derivatives 5 pattern in the sequence of Meta-TAO's since there is a sequence in the TAO's. Notice in the bottom row of Figure 3.2 that Meta-TAO-1 is identical to TAO-0 but rotated one iteration to the right. Meta-TAO-2 is identical to TAO-0 but rotated two iterations to the right. (We skip Meta-TAO-3 for a moment.) Meta-TAO-4 is identical to TAO-0 except it is rotated four iterations to the right (or two to the left). In contrast, no matter how the matrices are rotated, Meta-TAO's 3 and 5 are not identical to the original attractor. This is a similar result to the one found with the pure XOR Ring in Part 2. For convenience we repeat a point made in Part 2 since it applies here. In our current example as well as in the pure XOR Ring used in Part 2, Meta-TAO's 1, 2, and 4 are identical to the original basin when they are rotated. In terms of matrix algebra these rotated Meta-TAO's can be generated by acting on the original attractor with the Identity operator. For example: [TAO-0] x I = [Meta-TAO-1]. Thus we can call them identity Meta-TAO's. But Meta-TAO's 3 and 5 are different than the original attractor; a simple descriptive term for these is non-identity Meta-TAO's because whatever transformation matrix T is applied to TAO0 is certainly not the identity matrix. In terms of matrix algebra and using Meta-TAO-3 as an example, we can express this non-identity as: [TAO-0] x T = [Meta-TAO-3]. A primary focus of this symposium is on the nature of T, on the relation of T to symmetry, and on how T underlies how knowledge begets knowledge. As a summary up to this point, we have found a hidden order in the dynamics of attractor cycles. First, derivatives of attractors whose cycle length is a power of 2 diminish to 0 but not so for attractors whose length is not a power of 2. Second, both TAO's and Meta-TAO's have repeating sequences. But the Meta-TAO sequence is not the same as the TAO sequence. In our example in Figure 3.1 the TAO sequence is that 1 =4, 2=5, 3=6, and so on (assuming we rotate the matrices). But in Figure 3.2 for the identity Meta-TAO's in the lower row (when rotated) the sequence of equivalent Meta-TAO's is 1=2 =4. Moreover, for non-identity Meta-TAO's, there is a repeating sequence which is not shown in Figure 3.2 because the figure does not show enough higher order Meta-TAO's. Both the sequence of TAO's and the sequence of Meta-TAO's have interesting patterns which are puzzles calling for solutions. Our focus here is on “knowing begetting knowing” in symmetry groups so we will not fully solve these puzzles. But we note them and we turn our focus to the nature of the patterns generated by the XOR operator because it is crucial to our definition of symmetry groups (since XOR is the basis of both the TAO's and Meta-TAO's that define symmetry groups) and because the functional properties of the XOR operator is related to the interesting sequences of TAO's and Meta-TAO's we've just described. Recall that XOR is not an arbitrary operator from an epistemological viewpoint; it describes formally Bateson's principle that knowledge is generated by finding differences in differences (2002, p. 454ff, particularly pp. 463, 464). So XOR is a theoretically driven operator; moreover, as we have examined recursive and repeated applications of XOR, we have found that it has interesting properties in and of itself. Indeed, work by Wolfram (2002) in cellular automata suggests that simple logical operators can result in complex and surprising patterns. Specifically Guy (1990), Schroeder (1991), and Wolfram (2002) all point out that recursive use of the XOR operator can be used to create the Sierpinski gasket/Pascal’s triangle; this suggests that perhaps the key to the strange and perplexing behavior from both the TAO and Meta-TAO functions may be found in understanding the emergent structure of the XOR Boolean operator as it relates to the Sierpinski gasket. Part 3: Sierpinski Gaskets: The Structure of Boolean Derivatives