Changes

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"Aha!" we say, and think we see the way of things, writing down the rule <math>\texttt{x' = (x)}</math> where <math>\texttt{x'}</math> is the state that comes next after <math>\texttt{x},</math> and <math>\texttt{(x)}</math> is the negation of <math>\texttt{x}</math> in boolean logic.

"Aha!" we say, and think we see the way of things, writing down the rule <math>\texttt{x}^\prime = \texttt{(x)}</math> where <math>\texttt{x}^\prime</math> is the next state after <math>\texttt{x},</math> and <math>\texttt{(x)}</math> is the negation of <math>\texttt{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:

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 <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>\texttt{x' = (x)}</math> can now be expressed by the rule <math>\texttt{dx = 1}.</math>

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>\texttt{x}^\prime = \texttt{(x)}</math> can now be expressed by the rule <math>\texttt{dx} = \texttt{1}.</math>

There is a more detailed account of differential logic in the following paper:

There is a more detailed account of differential logic in the following paper: