[Inquiry] Dynamics And Logic

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 1

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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
PERS.  http://forum.wolframscience.com/showthread.php?threadid=297
RAR.   http://forum.wolframscience.com/showthread.php?threadid=400
DATA.  http://forum.wolframscience.com/showthread.php?threadid=228

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.

Let us start with a proposition of the form "p and q",
that is graphed as two labels attached to a root node:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` p and q ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

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

The proposition pq may be taken as a boolean function f<p, q>
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,
the value "true" looks like a blank label
and the value "false" looks like an edge.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` `true ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` `false` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Back to the proposition pq.  Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say,
the cell where the proposition pq is true, as pictured:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` `o` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` P ` ` ` |%%%%%| ` ` ` Q ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `o` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / \ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Now ask yourself:  What is the value of the
proposition pq at a distance of dp and dq
from the cell pq where you are standing?

Don't think about it -- just compute:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` dp o` `o dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / \ / \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o--- at ---o q` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

To make future graphs easier to draw in Asciiland,
I'll use devices like @=@=@ and o=o=o to identify
several nodes into one, as in this next redrawing:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

However you draw it, these expressions follow because the
expression p + dp, where the plus sign indicates addition
in B and thus corresponds to the exclusive-or in logic,
parses to a graph of the following form:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` p ` `dp ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o---o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` (p, dp) ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Next question:  What is the difference between the value of the
proposition pq "over there", at a remove of dp dq, and the value
of the proposition pq where you are, all expressed in the form of
a general formula, of course?  Here is the appropriate formulation:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `p q` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` `((p, dp)(q, dq), p q)` ` ` ` ` ` ` |
o-------------------------------------------------o

There is one thing that I ought to mention at this point:
Computed over B, plus and minus are identical operations.
This will make the relation between the differential and
the integral parts of the appropriate calculus slightly
stranger than usual, but we will get into that later.

Last question, for now:  What is the value of this expression
from your current standpoint, that is, evaluated at the point
where pq is true?  Well, substituting 1 for p and 1 for q in
the graph amounts to erasing the labels "p" and "q", like so:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `dp ` `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` `(( , dp)( , dq), ` `)` ` ` ` ` ` ` |
o-------------------------------------------------o

And this is equivalent to the following graph:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp` `dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ((dp) (dq)) ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o