MyWikiBiz, Author Your Legacy — Friday October 04, 2024
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, 19:48, 27 February 2009
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| ==Note 3== | | ==Note 3== |
| | | |
− | <pre>
| + | I will draw on those previously advertised resources of notation and theory as needed, but right now I sense the need for some concrete examples. |
− | I will draw on those previously advertized resources | |
− | of notation and theory as needed, but right now | |
− | I sense the need for some concrete examples. | |
| | | |
− | Let's say we have a system that is known by the name of | + | Let's say we have a system that is known by the name of its state space <math>X\!</math> and we have a boolean state variable <math>x : X \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}.</math> |
− | its state space X and we have a boolean state variable | |
− | x : X -> B, where B = {0, 1}. | |
| | | |
− | We observe X for a while, relative to a discrete time frame, | + | We observe <math>X\!</math> for a while, relative to a discrete time frame, and we write down the following sequence of values for <math>x.\!</math> |
− | and we write down the following sequence of values for x. | |
| | | |
− | x | + | {| align="center" cellpadding="8" width="90%" |
| + | | |
| + | <math>\begin{array}{ll} |
| + | t & x \\ |
| + | 0 & 0 \\ |
| + | 1 & 1 \\ |
| + | 2 & 0 \\ |
| + | 3 & 1 \\ |
| + | 4 & 0 \\ |
| + | 5 & 1 \\ |
| + | 6 & 0 \\ |
| + | 7 & 1 \\ |
| + | 8 & 0 \\ |
| + | 9 & \ldots |
| + | \end{array}</math> |
| + | |} |
| | | |
− | 0
| + | "Aha!" we say, and think we see the way of things, writing down the rule <math>x' = (x),\!</math> where <math>x'\!</math> is the state that comes next after <math>x,\!</math> and <math>(x)\!</math> is the negation of <math>x\!</math> in boolean logic. |
− | 1
| |
− | 0
| |
− | 1
| |
− | 0
| |
− | 1
| |
− | 0
| |
− | 1
| |
− | ...
| |
− | | |
− | "Aha!" we say, and think we see the way of things, | |
− | writing down the rule x' = (x), where x' is the | |
− | state that comes next after x, and (x) is the | |
− | negation of x in boolean logic. | |
| | | |
| + | <pre> |
| Another way to detect patterns is to write out a table | | Another way to detect patterns is to write out a table |
| of finite differences. `For this example, we would get: | | of finite differences. `For this example, we would get: |