Changes

==Note 1==

==Note 1==

I am going to excerpt some of my previous explorations on differential logic and dynamic systems and bring them to bear on the sorts of discrete dynamical themes that we find of interest in the NKS Forum.  This adaptation draws on the "Cactus Rules", "Propositional Equation Reasoning Systems", and "Reductions Among Relations" threads, and will in time be applied to the "Differential Analytic Turing Automata" thread:

I am going to excerpt some of my previous explorations

on differential logic and dynamic systems and bring them

to bear on the sorts of discrete dynamical themes that we

find of interest in the NKS Forum.  This adaptation draws on

the "Cactus Rules", "Propositional Equation Reasoning Systems",

and "Reductions Among Relations" threads, and will in time be

applied to the "Differential Analytic Turing Automata" thread:

CR.    http://forum.wolframscience.com/showthread.php?threadid=256

* CR.    http://forum.wolframscience.com/showthread.php?threadid=256

PERS.  http://forum.wolframscience.com/showthread.php?threadid=297

* PERS.  http://forum.wolframscience.com/showthread.php?threadid=297

RAR.  http://forum.wolframscience.com/showthread.php?threadid=400

* RAR.  http://forum.wolframscience.com/showthread.php?threadid=400

DATA.  http://forum.wolframscience.com/showthread.php?threadid=228

* DATA.  http://forum.wolframscience.com/showthread.php?threadid=228

One of the first things that you can do, once you have

One of the first things that you can do, once you have a moderately functional calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.

a moderately functional calculus for boolean functions

or propositional logic, whatever you choose to call it,

is to start thinking about, and even start computing,

the differentials of these functions or propositions.

Let us start with a proposition of the form "p and q",

Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:

that is graphed as two labels attached to a root node:

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |

|                    p and q                    |

|                    p and q                    |

o-------------------------------------------------o

o-------------------------------------------------o

Written as a string, this is just the concatenation "p q".

Written as a string, this is just the concatenation "<math>p~q</math>".

The proposition pq may be taken as a boolean function f<p, q>

The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>

having the abstract type f : B x B -> B, where B = {0, 1} is

read in such a way that 0 means "false" and 1 means "true".

In this style of graphical representation,

In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.

the value "true" looks like a blank label

and the value "false" looks like an edge.

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |

|                      true                      |

|                      true                      |

o-------------------------------------------------o

o-------------------------------------------------o

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |

|                      false                      |

|                      false                      |

o-------------------------------------------------o

o-------------------------------------------------o

Back to the proposition pq.  Imagine yourself standing

Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown here:

in a fixed cell of the corresponding venn diagram, say,

the cell where the proposition pq is true, as pictured:

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |