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Directory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 01:14, 24 December 2008
, 01:14, 24 December 2008→The Cactus Language : Syntax: set formula display as TeX array
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<math>\begin{array}{lccc}
<math>\begin{array}{lccc}
1. & s & \rightarrow & o. \\
1. & s & \rightarrow & o \\
\\
\\
2. & o & \leftarrow & s. \\
2. & o & \leftarrow & s \\
\end{array}</math>
\end{array}</math>
|}
|}
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:
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:
{| align="center" cellpadding="8" width="90%"
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<math>\begin{array}{lll}
^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\
</pre>
\end{array}</math>
|}
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:
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:
{| align="center" cellpadding="8" width="90%"
| " " = " "·" " = blank·blank
|
|
<math>\begin{array}{lclcl}
^{\backprime\backprime}\operatorname{~~}^{\prime\prime}
& = &
</pre>
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
\cdot\,
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
& = &
\operatorname{blank}
\cdot\,
\operatorname{blank} \\
\\
^{\backprime\backprime}\operatorname{~blank}^{\prime\prime}
& = &
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
\cdot\,
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
& = &
\operatorname{blank}
\cdot\,
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\
\\
^{\backprime\backprime}\operatorname{blank~}^{\prime\prime}
& = &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
\cdot\,
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
& = &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
\cdot\,
\operatorname{blank}
\end{array}</math>
|}
A few definitions from formal language theory are required at this point.
A few definitions from formal language theory are required at this point.
An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
An ''alphabet'' is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
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. 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:
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:
s_1 s_2 ... s_(k-1) s_k,
s_1 s_2 ... s_(k-1) s_k,