[Logic In Graphs] - A New Kind of Science: The NKS Forum

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# Logic In Graphs

Posted by: Jon Awbrey

LIG. Note 1

Alpha Graphs -> Laws Of Form -> Cactus Language -> Differential Logic

A program that I begain writing in Lisp and then ultimately developed
in Pascal all through the 80's implements a variant of C.S. Peirce's
logical graphs (at the "alpha" or prop calc level), that being the
progenitor of Spencer Brown's Laws Of Form calculus. This is now
called the "cactus language", because of the underlying species
of graphs that it involves. Re-entrant forms come into play
a slightly different way, with the "differential extension"
of this logical calculus, precisely analogous to the way
that differential calculus extends algebraic geometry.

I'm still getting around to a full documentation of the program.
This is being done at a function by function "commentary" level
and a bird's eye view "exposition" level at the following sites:

TOP. Theme One Program -- Source Code

TOP-COM. Theme One Program -- Commentary

TOP-EXP. Theme One Program -- Exposition

General introductions to the cactus language and its differential extension
can be found at the following sites:

PERS. Propositional Equation Reasoning Systems

CIL. Change In Logic

DAL. Dynamics And Logic

DIF. Differential Logic and Dynamic Systems

Jon Awbrey

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Posted by: Jon Awbrey

LIG. Note 2

People who pursue complexity via cellular automata
may be interested in the cactus forms for Wolfram's
elementary cellular automata rules, which are really
just a particular use of the 256 boolean functions on
three boolean variables, that is, functions of the type
B^3 -> B, where B = {0, 1}. More or less minimal cactus
expressions for these forms are listed and discussed here:

CR. Cactus Rules
CR. http://suo.ieee.org/ontology/thrd1.html#05486

Jon Awbrey

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Posted by: Jon Awbrey

LIG. Note 3

I probably ought to explain my remark that re-entrant forms are
related to the "differential extension" of propositional calculi,
as this may not be immediately clear from the links that I listed.

The easiest way to see this is to consider the re-entrant form "x = (x)".
As long as we stick to classical logic this will remain a false statement,
but we can understand its relation to the assignment statement "x := (x)",
that is, a command that sets the next value of x equal to the negation of
the current value of x, where x is a boolean variable, x in B = {0, 1}.

NB. In this text, I use parentheses for logical negation and
angle brackets for argument lists in functional notation.

A more explicit model of what is happening here may be formulated
by considering the space of functions {f : N -> B}, where N is the
set of natural numbers or non-negative integers N = {0, 1, 2, ...}.
In a typical scenario we may consider N to be a discrete dimension
of time, and x<n> to be the value of the variable x at the time n.
The assignment statement x := (x) is tantamount to a differential
equation that has two solutions in the function space {f : N -> B}.
With initial condition x<0> = 0, we get the sequence 0, 1, 0, 1, ...
With initial condition x<0> = 1, we get the sequence 1, 0, 1, 0, ...

Another way to write this differential equation, strictly speaking,
a boolean finite difference equation, is via the statement "dx = 1",
that is, an assertion that the first difference of x is a constant 1.
Moreover, since we are working in a logical domain, it is sufficient
simply to write "dx", reading "dx" as a "differential proposition"
that says that the first difference of x is true for all n in N.

We should note, however, that dx is not the same thing
as an imaginary truth value, since it does not satisfy
the initial equation, as in the expression "dx = (dx)".
The differential proposition dx asserts that x changes
at every step, while the differential proposition (dx)
asserts that dx = 0, in other words, that the value of
x<n> is the same for all values of n.

Jon Awbrey

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Posted by: Jon Awbrey

A new introduction to logical graphs — animated but still 2D — can be found here.

Don't it always seem to go — no sooner do you post a link to a page than the server decides to blank it for reasons unknown. Anyway, there's another copy here.