Ontologgers,
Discussion arose in the Laws Of Form Group about computational
explorations of George Spencer Brown's calculus of indications.
Readers of Peirce are generally aware that Spencer Brown revived
certain aspects of Peirce's logical graphs, focusing on the Alpha
level interpretable for Boolean Algebra and Propositional Calculus
but adding some hints of potential extension and generalization.
He used what amounts to Peirce's “entitative” interpretation of the
graphical forms in his exposition but was clear about the abstract
character of the forms themselves, as evidenced by their admitting
dual interpretations, dubbed “entitative” and “existential” by Peirce.
In computational context the question naturally arises how to code
the abstract formal structures used by the calculi of CSP and GSB
into the relatively concrete forms that a computer can process.
I began my response to that question as follows ...
Theme One • A Program Of Inquiry : 6
https://inquiryintoinquiry.com/2018/02/28/theme-one-%e2%80%a2-a-program-of-inquiry-6/
{A plaintext transcript follows, but see my blog copy
if the ASCII figure below gets mushed in this email.)
Programs are algorithms that operate on data structures (Wirth).
How do we turn abstract graphs like those used by C.S. Peirce
and Spencer Brown into concrete data structures that algorithms
can manipulate? There are many ways to do this, but one very
efficient way is through the use of “pointer data structures”.
The full documentation of my Theme One Program is still in progress,
but here's a link to a page of exposition, describing the family of
graphs used in the program, how to code the graphs as strings of
parentheses, commas, and letters, and how the program parses the
strings into pointer structures that live in computer memory.
Theme One Program • Exposition
http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program_%E2%80%A2_Exposition
Here's a link to a suitable point of entry for our present purpose:
Theme One Program • Expository Note 2
http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program_%E2%80%A2_Exposition#Expository_Note_2
2. Painted And Rooted Cacti And Conifers
=========================================
Figure 1 depicts a typical example of a painted and rooted cactus (PARCA).
o
a | d
o---o o
\ / b c |
o----o----o b e
\ /
\ /
\ /
\ /
@ a c e
Figure 1. Painted And Rooted Cactus
The graph itself is a mathematical object and does not inhabit
the page or other medium before our eyes, and it must not be
confused with any picture or other representation of it, anymore
than we'd want someone to confuse us with a picture of ourselves,
but it's a fair enough picture, once we understand the conventions
of representation involved.
Let V(G) be the set of nodes in a graph G and let L be a set of
identifiers. We very often find ourselves in situations where we
have to consider many different ways of associating the nodes of G
with the identifiers in L. Various manners of associating nodes with
identifiers have been given conventional names by different schools of
graph theorists. I will give one way of describing a few of the most
common patterns of association.
• A graph is “painted” if there is a relation between its node set
and a set of identifiers, in which case the relation is called a
painting and the identifiers are called paints.
• A graph is “colored” if there is a function from its node set
to a set of identifiers, in which case the function is called
a coloring and the identifiers are called colors.
• A graph is “labeled” if there is a one-to-one mapping between
its node set and a set of identifiers, in which case the mapping
is called a labeling and the identifiers are called labels.
• A graph is said to be “rooted” if it has a unique distinguished node,
in which case the distinguished node is called the root of the graph.
The graph in Figure 1 has a root node marked by the “at” sign or
amphora symbol “@”.
The graph in Figure 1 has eight nodes plus the five paints in the set
{ a, b, c, d, e }. The painting of nodes is depicted by drawing the
paints of each node next to the node they paint. Observe that some
nodes may be painted with an empty set of paints.
The structure of a painted and rooted cactus (PARC) can be encoded
in the form of character string called a painted and rooted cactus
expression (PARCE). For the remainder of this discussion the terms
cactus and cactus expression will be used to mean the painted and
rooted varieties. A cactus expression is formed on an alphabet
consisting of the relevant set of identifiers, the paints, together
with three punctuation marks: the left parenthesis, the comma, and
the right parenthesis.
To be continued ...
Jon