[Inquiry] Re: Cactus Rules

[Jon Awbrey]

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CR.  Note 5

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So long as we're seeing the sights at Cactus Junction,
we might as well take a gander at a computational way
to assay the import of any ole cactus expression that
comes down the pike.  Way out here, and elsewhere, too,
the computational clarification of a formal expression
is claimed to yield its canonical or its "normal" form.
Finer distinctions can be weighed, of course, and there
is always the problem of just how, exactly, and, indeed,
even whether such forms will be forthcoming from a given
cut of syntax for a given objective domain, or any other
wide open space.  But the notion of a "normal form" is
cast in the right direction, and so it'll do for now.

By way of example, let's examine the subtype of cactoid expression
that is typified by q_97 and its complement q_158, and that hardly
got its just deserts in the way of attention the last time around.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o o ` ` ` |
| ` ` ` ` q r ` ` ` | ` ` ` ` | ` ` ` p | | ` ` ` |
| ` ` ` ` o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q),(r)) ` | ` ` ` ` | `(( p ,(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_97` ` ` ` | ` ` ` ` | ` ` ` q_158 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

Cactus forms of the generic shape (g, (s_1), ..., (s_k))
are those that arise when we have a "genus and species"
or a "pie chart" arrangement of logical features, where
g is the genus and the k species are s_1 through s_k,
or g is the whole pie and the slices are the s_j.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `s_1` `s_k` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` g ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-----o-...-o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

We can reason out the meaning of all such expressions
by using the case analysis tactic that we used before.
If g is true, then it's just like "g" wasn't there at
all, and the expression comes down to the case below:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `s_1` ` `s_k` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o--...--o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

But this expresses the "just one true" condition that partitions
the remaining space, that is to say, the space where g is true,
into k sectors where each of the s_j in its own turn is true.

On the other hand, in the case where g is false,
we are left with a k-lobe with one bare spike:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `s_1` `s_k` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` | ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-----o-...-o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

If that expression as a whole is going to turn out to be true,
then there can be only one expression that evaluates to false
on its argument list, and since we already have it in custody,
we know that the remaining arguments, (s_1), ..., (s_k), will
all have to be true.  In effect, the spike collapses the lobe
to a node, leaving a conjunction of the negations of the s_j.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `s_1` ` `s_k` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o `...` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

In summation, we have the following interpretation:
If g is true, then exactly one of the s_j is true;
if g is false, then all of the s_j are false, too.

That is not yet a method that would be amenable to
computational routine, but it does get us part way.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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