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Directory:Jon Awbrey/Papers/Cactus Language (view source)

Revision as of 20:30, 23 December 2008

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→‎The Cactus Language : Syntax: wikify
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\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
 +
 +
When I say that the sign "blank" denotes the sign "&nbsp;", it means that the string of characters inside the first pair of quotation marks can be used as another name for the string of characters inside the second pair of quotes.  In other words, "blank" is a higher order sign whose object is "&nbsp;", and the string of five characters inside the first pair of quotation marks is a sign at a higher level of signification than the string of one character inside the second pair of quotation marks.  This relationship can be abbreviated in either one of the following ways:
    
<pre>
 
<pre>
−
When I say that the sign "blank" denotes the sign " ",
  −
it means that the string of characters inside the first
  −
pair of quotation marks can be used as another name for
  −
the string of characters inside the second pair of quotes.
  −
In other words, "blank" is a HO sign whose object is " ",
  −
and the string of five characters inside the first pair of
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quotation marks is a sign at a higher level of signification
  −
than the string of one character inside the second pair of
  −
quotation marks.  This relationship can be abbreviated in
  −
either one of the following ways:
  −
   
|  " "      <-<  "blank"
 
|  " "      <-<  "blank"
 
|
 
|
 
|  "blank"  >->  " "
 
|  "blank"  >->  " "
 +
</pre>
   −
Using the raised dot "·" as a sign to mark the articulation of a
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Using the raised dot "<math>\cdot</math>" as a sign to mark the articulation of a quoted string into a sequence of possibly shorter quoted strings, and thus to mark the concatenation of a sequence of quoted strings into a possibly larger quoted string, one can write:
−
quoted string into a sequence of possibly shorter quoted strings,
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and thus to mark the concatenation of a sequence of quoted strings
  −
into a possibly larger quoted string, one can write:
      +
<pre>
 
|
 
|
 
|  " "  <-<  "blank"  =  "b"·"l"·"a"·"n"·"k"
 
|  " "  <-<  "blank"  =  "b"·"l"·"a"·"n"·"k"
 
|
 
|
 +
</pre>
   −
This usage allows us to refer to the blank as a type of character, and
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This usage allows us to refer to the blank as a type of character, and also to refer any blank we choose as a token of this type, referring to either of them in a marked way, but without the use of quotation marks, as I just did.  Now, since a blank is just what the name "blank" names, it is possible to represent the denotation of the sign " " by the name "blank" in the form of an identity between the named objects, thus:
−
also to refer any blank we choose as a token of this type, referring to
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either of them in a marked way, but without the use of quotation marks,
  −
as I just did.  Now, since a blank is just what the name "blank" names,
  −
it is possible to represent the denotation of the sign " " by the name
  −
"blank" in the form of an identity between the named objects, thus:
      +
<pre>
 
|
 
|
 
|  " "  =  blank
 
|  " "  =  blank
 
|
 
|
 +
</pre>
   −
With these kinds of identity in mind, it is possible to extend the use of
+
With these kinds of identity in mind, it is possible to extend the use of the "<math>\cdot</math>" sign to mark the articulation of either named or quoted strings into both named and quoted strings.  For example:
−
the "·" sign to mark the articulation of either named or quoted strings
  −
into both named and quoted strings.  For example:
      +
<pre>
 
|  "  "      =  " "·" "      =  blank·blank
 
|  "  "      =  " "·" "      =  blank·blank
 
|
 
|
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|
 
|
 
|  "blank "  =  "blank"·" "  =  "blank"·blank
 
|  "blank "  =  "blank"·" "  =  "blank"·blank
 +
</pre>
    
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 "string" over an alphabet !A! is a finite sequence of signs from !A!.
 
A "string" over an alphabet !A! is a finite sequence of signs from !A!.
   −
The "length" of a string is just its length as a sequence of signs.
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The "length" of a string is just its length as a sequence of signs. A sequence of length 0 yields the "empty string", here presented as "". A sequence of length k > 0 is typically presented in the concatenated forms:
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A sequence of length 0 yields the "empty string", here presented as "".
  −
A sequence of length k > 0 is typically presented in the concatenated forms:
      
s_1 s_2 ... s_(k-1) s_k,
 
s_1 s_2 ... s_(k-1) s_k,
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Two alternative notations are often useful:
 
Two alternative notations are often useful:
   −
1.  !e!  =  @e@  =  ""  =  the empty string.
+
# !e!  =  @e@  =  ""  =  the empty string.
−
 
+
# %e%  =  {!e!}  =  {""}  =  the language consisting of a single empty string.
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2.  %e%  =  {!e!}  =  {""}  =  the language consisting of a single empty string.
     −
The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
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The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!. In particular, !A!* includes among its elements the empty string !e!.
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In particular, !A!* includes among its elements the empty string !e!.
     −
The "surplus" !A!^+ of an alphabet !A! is the set of all positive length
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The "surplus" !A!^+ of an alphabet !A! is the set of all positive length strings over !A!, in other words, everything in !A!* but the empty string.
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strings over !A!, in other words, everything in !A!* but the empty string.
     −
A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.
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A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*. If z is a string over !A! and if z is an element of !L!, then it is customary to call z a "sentence" of !L!.  Thus, a formal language !L! is defined by specifying its elements, which amounts to saying what it means to be a sentence of !L!.
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If z is a string over !A! and if z is an element of !L!, then it is
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customary to call z a "sentence" of !L!.  Thus, a formal language !L!
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is defined by specifying its elements, which amounts to saying what it
  −
means to be a sentence of !L!.
     −
One last device turns out to be useful in this connection.
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One last device turns out to be useful in this connection. If z is a string that ends with a sign t, then z · t^-1 is the string that results by "deleting" from z the terminal t.
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If z is a string that ends with a sign t, then z · t^-1 is
  −
the string that results by "deleting" from z the terminal t.
      
In this context, I make the following distinction:
 
In this context, I make the following distinction:
   −
1.  By "deleting" an appearance of a sign,
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# By "deleting" an appearance of a sign, I mean replacing it with an appearance of the empty string "".
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    I mean replacing it with an appearance
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# By "erasing" an appearance of a sign, I mean replacing it with an appearance of the blank symbol " ".
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    of the empty string "".
  −
 
  −
2.  By "erasing" an appearance of a sign,  
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    I mean replacing it with an appearance
  −
    of the blank symbol " ".
      
A "token" is a particular appearance of a sign.
 
A "token" is a particular appearance of a sign.
    +
<pre>
 
The informal mechanisms that have been illustrated in the immediately preceding
 
The informal mechanisms that have been illustrated in the immediately preceding
 
discussion are enough to equip the rest of this discussion with a moderately
 
discussion are enough to equip the rest of this discussion with a moderately
Jon Awbrey
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