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→‎Inquiry Driven Systems : The Cactus Patch: split into sections
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{{DISPLAYTITLE:Cactus Language}}
 
{{DISPLAYTITLE:Cactus Language}}
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==Inquiry Driven Systems : The Cactus Patch==
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==The Cactus Patch==
    
<pre>
 
<pre>
−
1.3.10.8  The Cactus Patch
  −
   
| Thus, what looks to us like a sphere of scientific knowledge more accurately
 
| Thus, what looks to us like a sphere of scientific knowledge more accurately
 
| should be represented as the inside of a highly irregular and spiky object,
 
| should be represented as the inside of a highly irregular and spiky object,
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Tackling this task requires me to introduce a number of new definitions
 
Tackling this task requires me to introduce a number of new definitions
 
and a collection of additional notational devices, to which I now turn.
 
and a collection of additional notational devices, to which I now turn.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.8  The Cactus Patch (cont.)
      
Depending on whether a formal language is called by the type of sign
 
Depending on whether a formal language is called by the type of sign
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for it to denote.  A semantic interpretation of the "cactus language" is
 
for it to denote.  A semantic interpretation of the "cactus language" is
 
carried out in Subsection 1.3.10.12.
 
carried out in Subsection 1.3.10.12.
 +
</pre>
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==The Cactus Language : Syntax==
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  −
1.3.10.9  The Cactus Language: Syntax
      +
<pre>
 
| Picture two different configurations of such an irregular shape, superimposed
 
| Picture two different configurations of such an irregular shape, superimposed
 
| on each other in space, like a double exposure photograph.  Of the two images,
 
| on each other in space, like a double exposure photograph.  Of the two images,
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|
 
|
 
|  "blank "  =  "blank"·" "  =  "blank"·blank
 
|  "blank "  =  "blank"·" "  =  "blank"·blank
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
A few definitions from formal language theory are required at this point.
 
A few definitions from formal language theory are required at this point.
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A "token" is a particular appearance of a sign.
 
A "token" is a particular appearance of a sign.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
The informal mechanisms that have been illustrated in the immediately preceding
 
The informal mechanisms that have been illustrated in the immediately preceding
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in which they denote propositions, considered as indicator functions over
 
in which they denote propositions, considered as indicator functions over
 
some universe, then these operations amount to "propositional connectives".
 
some universe, then these operations amount to "propositional connectives".
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
Rather than presenting the most concise description of these languages
 
Rather than presenting the most concise description of these languages
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constants, whatever sequences of strings z_j may be
 
constants, whatever sequences of strings z_j may be
 
listed as their ostensible arguments.
 
listed as their ostensible arguments.
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
Having defined the basic operations of concatenation and surcatenation
 
Having defined the basic operations of concatenation and surcatenation
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these complementary forms of analysis and synthesis to articulate the
 
these complementary forms of analysis and synthesis to articulate the
 
structures of strings and sentences in two directions.
 
structures of strings and sentences in two directions.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
The "painted cactus language" with paints in the
 
The "painted cactus language" with paints in the
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the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
 
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
 
as the "sentential forms" of the associated grammar.
 
as the "sentential forms" of the associated grammar.
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
In forming a grammar for a language, statements of the form W :> W',
 
In forming a grammar for a language, statements of the form W :> W',
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
A notation in the style of "S :> T" is now introduced,
 
A notation in the style of "S :> T" is now introduced,
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
Employing the notion of a covering relation it becomes possible to
 
Employing the notion of a covering relation it becomes possible to
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how the two features interact with one another, and then I return to address
 
how the two features interact with one another, and then I return to address
 
in further detail the questions that they engender on their individual bases.
 
in further detail the questions that they engender on their individual bases.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
In the process of developing a grammar for a language, it is possible
 
In the process of developing a grammar for a language, it is possible
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expression merely to account for and to recount every increment in the parameter
 
expression merely to account for and to recount every increment in the parameter
 
of iteration.
 
of iteration.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
Returning to the case of the cactus language, the process of recognizing an
 
Returning to the case of the cactus language, the process of recognizing an
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T = !e! and S = !e! on the covered side of the rule,
 
T = !e! and S = !e! on the covered side of the rule,
 
bears the germinal implication that T :> ",".
 
bears the germinal implication that T :> ",".
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
Grammar 2 achieves a portion of its success through a higher degree of
 
Grammar 2 achieves a portion of its success through a higher degree of
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significant expressions are taken up for discussion next time.
 
significant expressions are taken up for discussion next time.
   −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
With the distinction between empty and significant expressions in mind,
−
 
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
  −
 
  −
With the distinction between empty and significant expressions in mind,
   
I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
 
I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
 
that is afforded by Grammar 2, and, taking that as a point of departure,
 
that is afforded by Grammar 2, and, taking that as a point of departure,
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and also to enumerate a larger number of the smallest cases that can
 
and also to enumerate a larger number of the smallest cases that can
 
be observed to fall under each significant type.
 
be observed to fall under each significant type.
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  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
With the array of foregoing considerations in mind,
 
With the array of foregoing considerations in mind,
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|
 
|
 
|      is a sentence.
 
|      is a sentence.
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.9  The Cactus Language:  Syntax (cont.)
      
It is fitting to wrap up the foregoing developments by summarizing the
 
It is fitting to wrap up the foregoing developments by summarizing the
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|'Machines, Languages, and Computation',
 
|'Machines, Languages, and Computation',
 
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
 
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
 +
</pre>
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==The Cactus Language : Stylistics==
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  −
1.3.10.10  The Cactus Language: Stylistics
      +
<pre>
 
| As a result, we can hardly conceive of how many possibilities there are
 
| As a result, we can hardly conceive of how many possibilities there are
 
| for what we call objective reality.  Our sharp quills of knowledge are so
 
| for what we call objective reality.  Our sharp quills of knowledge are so
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and whether it can be compounded to cover the complexity that actually exists
 
and whether it can be compounded to cover the complexity that actually exists
 
in the realm of meaning -- these are questions for an extended future inquiry.
 
in the realm of meaning -- these are questions for an extended future inquiry.
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
Perhaps I ought to comment on the differences between the present and
 
Perhaps I ought to comment on the differences between the present and
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to permit the initial symbol to characterize, to cover, to regenerate,
 
to permit the initial symbol to characterize, to cover, to regenerate,
 
or to produce itself, if that turns out to be the best way in the end.
 
or to produce itself, if that turns out to be the best way in the end.
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
Before I leave this Subsection, I need to say a few things about
 
Before I leave this Subsection, I need to say a few things about
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are demanded by the application and intended by the designer, and not so much the
 
are demanded by the application and intended by the designer, and not so much the
 
peculiar characters of the signs that represent these classes of practical meaning.
 
peculiar characters of the signs that represent these classes of practical meaning.
−
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
Any description of a language is bound to have alternative descriptions.
 
Any description of a language is bound to have alternative descriptions.
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this piece not only fits the type q but even comes to be generated under
 
this piece not only fits the type q but even comes to be generated under
 
the auspices of the non-terminal symbol "q".
 
the auspices of the non-terminal symbol "q".
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
A moment's reflection on the issue of style, giving due consideration to the
 
A moment's reflection on the issue of style, giving due consideration to the
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as the simplest and the most primitive in principle, even if they are
 
as the simplest and the most primitive in principle, even if they are
 
scarcely recognized as lying among the more familiar elements of logic.
 
scarcely recognized as lying among the more familiar elements of logic.
−
  −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
I am throwing together a wide variety of different operations into each
 
I am throwing together a wide variety of different operations into each
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by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
 
by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
 
albeit through the posting of editorial remarks from an external "mark-up" language.
 
albeit through the posting of editorial remarks from an external "mark-up" language.
−
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.10  The Cactus Language:  Stylistics (cont.)
      
In order to systematize the relations that strictures and straits placed
 
In order to systematize the relations that strictures and straits placed
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There is a measure of ambiguity that remains in this formulation,
 
There is a measure of ambiguity that remains in this formulation,
 
but it is the best that I can do in the present informal context.
 
but it is the best that I can do in the present informal context.
 +
</pre>
   −
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+
==The Cactus Language : Mechanics==
−
 
  −
1.3.10.11  The Cactus Language: Mechanics
      +
<pre>
 
| We are only now beginning to see how this works.  Clearly one of the
 
| We are only now beginning to see how this works.  Clearly one of the
 
| mechanisms for picking a reality is the sociohistorical sense of what
 
| mechanisms for picking a reality is the sociohistorical sense of what
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                   =  a lobe with the accoutrements C_1, ..., C_k.
 
                   =  a lobe with the accoutrements C_1, ..., C_k.
−
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.11  The Cactus Language:  Mechanics (cont.)
      
Working from a structural description of the cactus language,
 
Working from a structural description of the cactus language,
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     In particular, this generally makes it easier to observe the
 
     In particular, this generally makes it easier to observe the
 
     various properties of its underlying graphical structure.
 
     various properties of its underlying graphical structure.
 +
</pre>
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+
==The Cactus Language : Semantics==
−
 
  −
1.3.10.12  The Cactus Language: Semantics
      +
<pre>
 
| Alas, and yet what 'are' you, my written and painted thoughts!
 
| Alas, and yet what 'are' you, my written and painted thoughts!
 
| It is not long ago that you were still so many-coloured,
 
| It is not long ago that you were still so many-coloured,
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     For the generic case of k > 0, the "bound connective" takes the
 
     For the generic case of k > 0, the "bound connective" takes the
 
     form "-(S_1, ..., S_k)-".
 
     form "-(S_1, ..., S_k)-".
−
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.12  The Cactus Language:  Semantics (cont.)
      
At this point, there are actually two different "dialects", "scripts",
 
At this point, there are actually two different "dialects", "scripts",
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Indeed, the "roughly" can be made "exactly" as soon as the domains of
 
Indeed, the "roughly" can be made "exactly" as soon as the domains of
 
a suitable sign relation are specified precisely.
 
a suitable sign relation are specified precisely.
−
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  −
  −
1.3.10.12  The Cactus Language:  Semantics (cont.)
      
A good way to illustrate the action of the conjunction and surjunction
 
A good way to illustrate the action of the conjunction and surjunction
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070.  http://suo.ieee.org/ontology/msg03473.html
 
070.  http://suo.ieee.org/ontology/msg03473.html
 
071.  http://suo.ieee.org/ontology/msg03479.html
 
071.  http://suo.ieee.org/ontology/msg03479.html
 +
</pre>
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+
==Stretching Exercises==
−
 
  −
1.3.10.13  Stretching Exercises
      +
<pre>
 
Taking up the preceding arrays of particular connections, namely,
 
Taking up the preceding arrays of particular connections, namely,
 
the boolean functions on two or less variables, it possible to
 
the boolean functions on two or less variables, it possible to
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Which was to be shown.
 
Which was to be shown.
 +
</pre>
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==Document History==
−
<pre>
  −
 
  −
===Document History===
      
<pre>
 
<pre>
Jon Awbrey
12,080

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