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Revision as of 22:47, 27 February 2009
, 22:47, 27 February 2009→Note 3: markup
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
<math>\begin{array}{ll}
<math>\begin{array}{cc}
t & x \\
t & x \\
0 & 0 \\
0 & 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.
"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.
Another way to detect patterns is to write out a table of finite differences. For this example, we would get:
Another way to detect patterns is to write out a table
of finite differences. `For this example, we would get:
x dx d2x
{| align="center" cellpadding="8" width="90%"
|
0 1 0 ...
<math>\begin{array}{ccccc}
1 1 0
t & x & dx & d^2 x & \ldots \\
0 1 0
0 & 0 & 1 & 0 & \ldots \\
1 1 0
1 & 1 & 1 & 0 & \\
0 1 0
2 & 0 & 1 & 0 & \\
1 1 0
3 & 1 & 1 & 0 & \\
0 1
4 & 0 & 1 & 0 & \\
5 & 1 & 1 & 0 & \\
6 & 0 & 1 & 0 & \\
7 & 1 & 1 & 0 & \\
8 & 0 & 1 & \ldots & \\
9 & \ldots & \ldots & \ldots & \\
\end{array}</math>
|}
And of course, all the higher order differences are zero.
And of course, all the higher order differences are zero.
This leads to thinking of X as having an extended state
This leads to thinking of <math>X\!</math> as having an extended state <math>(x, dx, d^2 x, \ldots, d^k x),</math> and this additional language gives us the facility of describing state transitions in terms of the various orders of differences. For example, the rule <math>x' = (x)\!</math> can now be expressed by the rule <math>dx = 1.\!</math>
<x, dx, d^2.x, ..., d^k.x>, and this additional language
gives us the facility of describing state transitions in
terms of the various orders of differences. `For example,
the rule x' = (x) can now be expressed by the rule dx = 1.
I'll leave you to muse on
I'll leave you to muse on the possibilities of that.
the possibilities of that.
==Note 4==
==Note 4==
