Not guaranteed to preserve the order of Boolean operators.
Thursday, December 13, 2012
Thursday, December 6, 2012
Dynamical Systems, Posets, and Graphs
Suppose we have a set of nodes \(X = \{ x_1, \ldots, x_n \}\), with entries in \(\mathbb{Z}/2\mathbb{Z}\). If we also have a set of unknown "change-of-state" vectors \(V = \{v^*_1, \ldots, v^*_t\}\), we say that \(S = (X, V)\) is an information system.
Whilst our "change-of-state" vectors are initially unknown, we understand that they also have entries in \(\mathbb{Z}/2\mathbb{Z}\), with composition defined as the boolean \( \vee \).
The stabilization of the system can be determined by applying the "change-of-state" vectors, iteratively to \(X\):
\begin{equation*}
X_0 \overset{v^*_1}{\to} X_1 \overset{v^*_2}{\to} \cdots \overset{v^*_n}{\to} X_n = C
\end{equation*}
We say that every state \(X_i\) is an observation of \(S\) and that a system has stabilized when all nodes are equal to \( \mathbf{1} \). This state is denoted as \( C \), as seen above. Of course, we may examine other systems in which there may not exist a finite stabilization state. Also, the "change-of-state" vectors may be composed by some other boolean operator other than \(\vee\).
Let \(A_i = X_i \setminus X_{i-1}\). We say that \(A_i\) is the set of nodes activated in the \(i\)th observation of the system, and denote the set of all such \(A_i\) as simply \(A\). We define a function \(\tau: A_i \to \mathbb{Z}\) that measures the future increase in activation.
\begin{equation*}
\tau(A_i) = |A_{i + 1}| - |A_i|
\end{equation*}
Following stabilization, we may examine the fibers of \(\tau(A)\). We say that the fibers of \(\tau(A)\) are components of \(S\), whilst \(\tau^{-1}(\{\text{sup}\,\tau(A)\})\) is called the core of \(S\).
Let \(\pi\) be a functor, where \(\pi : \text{Set} \to \text{Grph}\), such that \(\pi(x)\) forms a complete graph with nodes elements of \(x\). If we let \(\pi\) act on the fibers of \(\tau(A)\), we will of course get a set of graphs. Join these graphs according to their partial ordering from \(\tau(A)\), and we will have formed a graph whose connectivity represents the discrete activation of the system \(S\).
Undeniably, this may be a little rough around the edges. The last few steps will surely be clarified with the application of some categorical language. However, the more important issue at hand is, what can we do with all of this?
EDIT: This seems quite related to Network Theory as discussed here: http://johncarlosbaez.wordpress.com/2011/03/04/network-theory-part-1/ . Although, it seems I have some reading to get done, before coming to a conclusion.
Whilst our "change-of-state" vectors are initially unknown, we understand that they also have entries in \(\mathbb{Z}/2\mathbb{Z}\), with composition defined as the boolean \( \vee \).
The stabilization of the system can be determined by applying the "change-of-state" vectors, iteratively to \(X\):
\begin{equation*}
X_0 \overset{v^*_1}{\to} X_1 \overset{v^*_2}{\to} \cdots \overset{v^*_n}{\to} X_n = C
\end{equation*}
We say that every state \(X_i\) is an observation of \(S\) and that a system has stabilized when all nodes are equal to \( \mathbf{1} \). This state is denoted as \( C \), as seen above. Of course, we may examine other systems in which there may not exist a finite stabilization state. Also, the "change-of-state" vectors may be composed by some other boolean operator other than \(\vee\).
Let \(A_i = X_i \setminus X_{i-1}\). We say that \(A_i\) is the set of nodes activated in the \(i\)th observation of the system, and denote the set of all such \(A_i\) as simply \(A\). We define a function \(\tau: A_i \to \mathbb{Z}\) that measures the future increase in activation.
\begin{equation*}
\tau(A_i) = |A_{i + 1}| - |A_i|
\end{equation*}
Following stabilization, we may examine the fibers of \(\tau(A)\). We say that the fibers of \(\tau(A)\) are components of \(S\), whilst \(\tau^{-1}(\{\text{sup}\,\tau(A)\})\) is called the core of \(S\).
Let \(\pi\) be a functor, where \(\pi : \text{Set} \to \text{Grph}\), such that \(\pi(x)\) forms a complete graph with nodes elements of \(x\). If we let \(\pi\) act on the fibers of \(\tau(A)\), we will of course get a set of graphs. Join these graphs according to their partial ordering from \(\tau(A)\), and we will have formed a graph whose connectivity represents the discrete activation of the system \(S\).
Undeniably, this may be a little rough around the edges. The last few steps will surely be clarified with the application of some categorical language. However, the more important issue at hand is, what can we do with all of this?
EDIT: This seems quite related to Network Theory as discussed here: http://johncarlosbaez.wordpress.com/2011/03/04/network-theory-part-1/ . Although, it seems I have some reading to get done, before coming to a conclusion.
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