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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
Revision as of 13:42, 9 March 2009
, 13:42, 9 March 2009→Note 5: \texttt
For a slightly more interesting example, let's suppose that we have a dynamic system that is known by its state space <math>X,\!</math> and we have a boolean state variable <math>x : X \to \mathbb{B}.</math> In addition, we are given an initial condition <math>\texttt{x~=~dx}</math> and a law <math>\begin{matrix}\texttt{d}^\texttt{2}\texttt{x~=~(x)}.\end{matrix}</math>
For a slightly more interesting example, let's suppose that we have a dynamic system that is known by its state space <math>X,\!</math> and we have a boolean state variable <math>x : X \to \mathbb{B}.</math> In addition, we are given an initial condition <math>\texttt{x~=~dx}</math> and a law <math>\begin{matrix}\texttt{d}^\texttt{2}\texttt{x~=~(x)}.\end{matrix}</math>
The initial condition has two cases: either <math>x = dx = 0\!</math> or <math>x = dx = 1.\!</math>
The initial condition has two cases:
Here is a table of the two trajectories or ''orbits'' that we get by starting from each of the two permissible initial states and staying within the constraints of the dynamic law <math>d^2 x = (x).\!</math>
{| align="center" cellpadding="8" width="95%"
|
<math>\begin{array}{ll}
1. & \texttt{x~=~dx~=~0}
\\
2. & \texttt{x~=~dx~=~1}
\end{array}</math>
|}
Here is a table of the two trajectories or ''orbits'' that we get by starting from each of the two permissible initial states and staying within the constraints of the dynamic law <math>\begin{matrix}\texttt{d}^\texttt{2}\texttt{x~=~(x)}.\end{matrix}</math>
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"