On 3/1/2018 9:36 AM, Jon Awbrey wrote: > > 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/ >
Peircers, My earliest experiments with coding logical graphs as pointer data structures taught me that conceptual and computational efficiencies of a critical sort could be achieved by generalizing their abstract graphs from trees to the variety graph theorists call “cacti”. The genesis of that generalization is a tale worth telling another time, but for now it makes a quicker start to proceed by way of generic examples. I gave a rough sketch of a typical cactus graph in my last post. I'll make a better graphic eventually but for now see this page: Theme One • A Program Of Inquiry : 8 https://inquiryintoinquiry.com/2018/03/10/theme-one-%e2%80%a2-a-program-of-inquiry-8/ Figure 1 shows a typical example of a “painted and rooted cactus”. o a | d o---o o \ / b c | o----o----o b e \ / \ / \ / \ / @ a c e Figure 1. Painted And Rooted Cactus Figure 2 shows a way to visualize the correspondence between cactus graphs and cactus strings, demonstrated on the cactus from Figure 1. By way of convenient terminology, the polygons of a cactus graph are called its “lobes”. An edge not a part of a larger polygon is called a “2-gon” or a “bi-gon”. A terminal bi-gon is called a “spike”. o a (|) d o---o o (\ /) b c (|) o--,--o--,--o b e \ / \ / ( \ / ) \ / \ / @ a c e ( ( a , ( ) ) , b c , ( d ) b e ) a c e Figure 2. Cactus Graph and Cactus Expression The correspondence between a cactus graph and a cactus string is obtained by an operation called “traversing” the graph in question. * One traverses a cactus graph by beginning at the left hand side of the root node, reading off the list of paints one encounters at that point. Since the order of elements at any node is not significant, one may start the cactus string with that list of paints or save them for the end. We have done the latter in this case. * One continues by climbing up the left hand side of the leftmost lobe, marking the ascent by means of a left parenthesis, traversing whatever cactus one happens to reach at the first node above the root, that done, proceeding from left to right along the top side of the lobe, marking each interlobal span by means of a comma, traversing each cactus in turn one meets along the way, on completing the last of them climbing down the right hand side of the lobe, marking the descent by means of a right parenthesis, and then traversing each cactus in turn, in left to right order, that is incident with the root node. The string of letters, parentheses, and commas one obtains by this procedure is called the “traversal string” of the graph, in this case, a “cactus string”. Regards, Jon -- inquiry into inquiry: https://inquiryintoinquiry.com/ academia: https://independent.academia.edu/JonAwbrey oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey isw: http://intersci.ss.uci.edu/wiki/index.php/JLA facebook page: https://www.facebook.com/JonnyCache
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