## Sign Relations, Triadic Relations, Relations • 5

Re: Ontolog ForumRS

I chose those examples of triadic relations to be as simple as possible without being completely trivial but they already exemplify many features we need to keep in mind in all the more complex cases as we use relational models of realistic phenomena and objective domains.

The mathematical examples are typical of many in linguistic, logical, and mathematical contexts where we start out with compact, ready-made axioms, definitions, equations, expressions, formulas, predicates, or terms that denote the relations of interest.

For example, we might be discussing dyadic relative terms like “parent of —” or “square of —” and triadic relative terms like “giver of — to —” or “sum of — and —”.

If we spend the majority of our time in contexts like that we may form the impression that all the relational concepts we’ll ever need can be requisitioned off-the-shelf from pre-fab stock, no assembly required.

That’s a pretty picture of our mental equipment.  It may even be true if we cook the data long enough and fudge the meaning of pre-fab down to the level of amino acids or quarks or some other bosons on the bus.

As a practical matter, however, research pursued in experimental veins tends to push the envelope of pre-fab concepts into surprisingly novel realms of ideas.

I’ll discuss the examples of sign relations as I get more time …

## Sign Relations, Triadic Relations, Relations • 4

The middle ground between relations in general and the sign relations we need to do logic, inquiry, communication, and so on is occupied by triadic relations, also called ternary or 3-place relations.

Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic etc.

Here’s a primer with examples from mathematics and semiotics:

## Sign Relations, Triadic Relations, Relations • 3

At the other end of the funnel, here’s an intro to relations in general, focusing on the discrete mathematical variety we find most useful in applications, for example, as background for relational data bases and empirical data.

## Sign Relations, Triadic Relations, Relations • 2

I always have trouble deciding whether to start with the genus and drive down to the species or else to start with concrete examples and follow Sisyphus up Mt. Abstraction.

Soon after I made my 3rd try at grad school, this time in Systems Engineering, I was trying to explain sign relations to my advisor and he — being the very model of a modern systems engineer — asked me to give a single simple concrete example, as simple as possible without being trivial, and this is the example I came up with:

Here’s a more compact and self-contained article that starts from scratch and covers much of the same material:

Folks already registered with any Wikipedia system site may find it convenient to use the article talk page at Wikiversity for additional discussion.

## Theme One Discussion • 1

Warfield gets it right about the relationship between object languages and metalanguages.  Something about the prefix meta- has contributed to a not uncommon misconception that metalanguages are formalized to a higher degree than the languages they objectify whereas in fact the opposite is true.

As it happens, the relation of informal contexts to formal contexts and what I’ve elsewhere called the formalization arrows between them are themes of major importance in my study of Inquiry Driven Systems.  Being short on time at the moment, I’ll just give a pointer into one of many relevant discussions and hope to elaborate further at the next opportunity.

## Theme One Exposition • 3

### Coding Logical Graphs

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’s best to jump right in and proceed by way of generic examples.

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.

## Theme One Exposition • 2

The previous post described the elementary data structure used to represent nodes of graphs in the Theme One program.  This post describes the specific family of graphs employed by the program.

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
```

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 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 “$\texttt{@}$”.

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 indicated 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 can be encoded in the form of a character string called a painted and rooted cactus expression.  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.