Theme One is a program for constructing and transforming a particular species of graph‑theoretic data structures, forms designed to support a variety of fundamental learning and reasoning tasks.
The program evolved over the course of an exploration into the integration of contrasting types of activities involved in learning and reasoning, especially the types of algorithms and data structures capable of supporting all sorts of inquiry processes, from everyday problem solving to scientific investigation. In its current state, Theme One integrates over a common data structure fundamental algorithms for one type of inductive learning and one type of deductive reasoning.
We begin by describing the class of graph-theoretic data structures used by the program, as determined by their local and global features. It will be the usual practice to shift around and view these graphs at many different levels of detail, from their abstract definition to their concrete implementation, and many points in between.
The main work of the Theme One program is achieved by building and transforming a single species of graph-theoretic data structures. In their abstract form these structures are closely related to the graphs called cacti and conifers in graph theory, so we’ll generally refer to them under those names.
The Idea↑Form Flag
The graph-theoretic data structures used by the program are built up from a basic data structure called an idea-form flag. That structure is defined as a pair of Pascal data types by means of the following specifications.
- An idea is a pointer to a form.
- A form is a record consisting of:
- A sign of type char;
- Four pointers, as, up, on, by, of type idea;
- A code of type numb, that is, an integer in [0, max integer].
Represented in terms of digraphs, or directed graphs, the combination of an idea pointer and a form record is most easily pictured as an arc, or directed edge, leading to a node labeled with the other data, in this case, a letter and a number.
At the roughest but quickest level of detail, an idea of a form can be drawn as follows.
When it is necessary to fill in more detail, the following schematic pattern can be used.
The idea-form type definition determines the local structure of the whole host of graphs used by the program, including a motley array of ephemeral buffers, temporary scratch lists, and other graph-theoretic data structures used for their transient utilities at specific points in the program.
I will put off discussing the more incidental graph structures until the points where they actually arise, focusing here on the particular varieties of cactoid graphs which constitute the main formal media of the program’s operation.
Resources
cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Theme One Program • Structural Modeling • Systems Science
called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result. Operators with optional numbers of arguments are called 
in that form gives the following formula.
evaluates the proposition 
on each model of the proposition 
and combines the results according to the method indicated by the connective parameter 
In principle, the index 
may specify any logical connective on as many as 
arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums. By convention, each of the accessory indices 
is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition 
for the lower index 
for the upper index 
if and only if 
In propositional terms, this is tantamount to the assertion that 
or that 
if and only if 
The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.
where 
is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse. As usual, we won’t be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough.
![X^\bullet = [u, v].](https://s0.wp.com/latex.php?latex=X%5E%5Cbullet+%3D+%5Bu%2C+v%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
A bridge between propositions and quantifications is afforded by a set of measures or qualifiers 
defined by the following equations.
namely, which elements in the space of type 
are assigned a positive value by 
Taken together, the 
operators give us a way to express many useful observations about the propositions in 
![\text{Figure 16. Higher Order Universe of Discourse}~ [ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} ] \subseteq [[ u, v ]]](https://s0.wp.com/latex.php?latex=%5Ctext%7BFigure+16.+Higher+Order+Universe+of+Discourse%7D%7E+%5B+%5Cell_%7B00%7D%2C+%5Cell_%7B01%7D%2C+%5Cell_%7B10%7D%2C+%5Cell_%7B11%7D+%5D+%5Csubseteq+%5B%5B+u%2C+v+%5D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\begin{matrix} \alpha_i f & = & \Upsilon (f_i, f) & = & \Upsilon (f_i \Rightarrow f), \\[6pt] \beta_i f & = & \Upsilon (f, f_i) & = & \Upsilon (f \Rightarrow f_i). \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5Calpha_i+f+%26+%3D+%26+%5CUpsilon+%28f_i%2C+f%29+%26+%3D+%26+%5CUpsilon+%28f_i+%5CRightarrow+f%29%2C++%5C%5C%5B6pt%5D++%5Cbeta_i+f+%26+%3D+%26+%5CUpsilon+%28f%2C+f_i%29+%26+%3D+%26+%5CUpsilon+%28f+%5CRightarrow+f_i%29.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
on each of the 16 boolean functions 
In terms of the implication ordering on the 16 functions, 
says that 
is above or identical to 
in the implication lattice, that is, 
in the implication ordering.
on each of the 16 boolean functions 
says that 
in the implication ordering.
the qualifiers 
respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.![\begin{array}{*{8}{r}} \alpha_{0} f = 1 & \mathrm{iff} & f_{0} \Rightarrow f & \mathrm{iff} & 0 \Rightarrow f, & \mathrm{hence} & \alpha_{0} f = 1 & \mathrm{for~all} ~ f. \\[4pt] \alpha_{15} f = 1 & \mathrm{iff} & f_{15} \Rightarrow f & \mathrm{iff} & 1 \Rightarrow f, & \mathrm{hence} & \alpha_{15} f = 1 & \mathrm{iff} ~ f = 1. \\[4pt] \beta_{0} f = 1 & \mathrm{iff} & f \Rightarrow f_{0} & \mathrm{iff} & f \Rightarrow 0, & \mathrm{hence} & \beta_{0} f = 1 & \mathrm{iff} ~ f = 0. \\[4pt] \beta_{15} f = 1 & \mathrm{iff} & f \Rightarrow f_{15} & \mathrm{iff} & f \Rightarrow 1, & \mathrm{hence} & \beta_{15} f = 1 & \mathrm{for~all} ~ f. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B8%7D%7Br%7D%7D++%5Calpha_%7B0%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D++%26+f_%7B0%7D+%5CRightarrow+f++%26+%5Cmathrm%7Biff%7D++%26+0+%5CRightarrow+f%2C++%26+%5Cmathrm%7Bhence%7D++%26+%5Calpha_%7B0%7D+f+%3D+1++%26+%5Cmathrm%7Bfor%7Eall%7D+%7E+f.++%5C%5C%5B4pt%5D++%5Calpha_%7B15%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D++%26+f_%7B15%7D+%5CRightarrow+f++%26+%5Cmathrm%7Biff%7D++%26+1+%5CRightarrow+f%2C++%26+%5Cmathrm%7Bhence%7D++%26+%5Calpha_%7B15%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D+%7E+f+%3D+1.++%5C%5C%5B4pt%5D++%5Cbeta_%7B0%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D++%26+f+%5CRightarrow+f_%7B0%7D++%26+%5Cmathrm%7Biff%7D++%26+f+%5CRightarrow+0%2C++%26+%5Cmathrm%7Bhence%7D++%26+%5Cbeta_%7B0%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D+%7E+f+%3D+0.++%5C%5C%5B4pt%5D++%5Cbeta_%7B15%7D+f+%3D+1++%26+%5Cmathrm%7Biff%7D++%26+f+%5CRightarrow+f_%7B15%7D++%26+%5Cmathrm%7Biff%7D++%26+f+%5CRightarrow+1%2C++%26+%5Cmathrm%7Bhence%7D++%26+%5Cbeta_%7B15%7D+f+%3D+1++%26+%5Cmathrm%7Bfor%7Eall%7D+%7E+f.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is a wholly indifferent or indiscriminate measure, accepting every proposition 
whereas the measures 
and 
value the constant propositions 
and 
respectively, above all others.![\begin{matrix} [| \alpha_i |] & = & \alpha_i^{-1}(1), \\[6pt] [| \beta_i |] & = & \beta_i^{-1}(1), \\[6pt] [| \Upsilon_p |] & = & \Upsilon_p^{-1}(1). \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5B%7C+%5Calpha_i+%7C%5D+%26+%3D+%26+%5Calpha_i%5E%7B-1%7D%281%29%2C++%5C%5C%5B6pt%5D++%5B%7C+%5Cbeta_i+%7C%5D+%26+%3D+%26+%5Cbeta_i%5E%7B-1%7D%281%29%2C++%5C%5C%5B6pt%5D++%5B%7C+%5CUpsilon_p+%7C%5D+%26+%3D+%26+%5CUpsilon_p%5E%7B-1%7D%281%29.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
measures of type 
present a formidable array of propositions about propositions about 2‑dimensional universes of discourse. The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.
be a two‑dimensional boolean space, 
generated by two boolean variables or logical features 
and 
as arguments, the relative umpire operator reports the value 
if the first implies the second, otherwise it reports the value 
it is possible to express this more succinctly as follows.
defined by the equation 
In this case the subscript 
In most settings where 
is applied to arguments it is safe to omit the subscript 
![[u, v],](https://s0.wp.com/latex.php?latex=%5Bu%2C+v%5D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
giving a score of 
to everything else, counting all contingent statements as no better than falsehoods.
Inquiry Into Inquiry 