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Directory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 22:28, 3 January 2009
, 22:28, 3 January 2009→The Cactus Language : Syntax: markup & segment
There are several abuses of notation that commonly tolerated in the use of covering relations. The worst offense is that of allowing symbols to stand equivocally either for individual strings or else for their types. There is a measure of consistency to this practice, considering the fact that perfectly individual entities are rarely if ever grasped by means of signs and finite expressions, which entails that every appearance of an apparent token is only a type of more particular tokens, and meaning in the end that there is never any recourse but to the sort of discerning interpretation that can decide just how each sign is intended. In view of all this, I continue to permit expressions like <math>t <: T\!</math> and <math>T <: S,\!</math> where any of the symbols <math>t, T, S\!</math> can be taken to signify either the tokens or the subtypes of their covering types.
There are several abuses of notation that commonly tolerated in the use of covering relations. The worst offense is that of allowing symbols to stand equivocally either for individual strings or else for their types. There is a measure of consistency to this practice, considering the fact that perfectly individual entities are rarely if ever grasped by means of signs and finite expressions, which entails that every appearance of an apparent token is only a type of more particular tokens, and meaning in the end that there is never any recourse but to the sort of discerning interpretation that can decide just how each sign is intended. In view of all this, I continue to permit expressions like <math>t <: T\!</math> and <math>T <: S,\!</math> where any of the symbols <math>t, T, S\!</math> can be taken to signify either the tokens or the subtypes of their covering types.
'''Note.''' For some time to come in the discussion that follows, although I will continue to focus on the cactus language as my principal object example, my more general purpose will be to develop the subject matter of the formal languages and grammars. I will do this by taking up a particular method of ''stepwise refinement" and using it to extract a rigorous formal grammar for the cactus language, starting with little more than a rough description of the target language and applying a systematic analysis to develop a sequence of increasingly
For some time to come in the discussion that follows,
although I will continue to focus on the cactus language
as my principal object example, my more general purpose will
be to develop and to demonstrate the subject materials and the
I will do this by taking up a particular method of "stepwise refinement"
and using it to extract a rigorous formal grammar for the cactus language,
starting with little more than a rough description of the target language
and applying a systematic analysis to develop a sequence of increasingly
more effective and more exact approximations to the desired grammar.
more effective and more exact approximations to the desired grammar.
Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.
===Grammar 1===
<pre>
Grammar 1 is something of a misnomer. It is nowhere near exemplifying
Grammar 1 is something of a misnomer. It is nowhere near exemplifying
any kind of a standard form and it is only intended as a starting point
any kind of a standard form and it is only intended as a starting point
surcatenation of sentences, and so to obtain the following
surcatenation of sentences, and so to obtain the following
form of a grammar:
form of a grammar:
</pre>
===Grammar 2===
<pre>
| !C!(!P!). Grammar 2
| !C!(!P!). Grammar 2
|
|
intermediate symbols, as string variables and as type names, apply
intermediate symbols, as string variables and as type names, apply
again to the letter "F".
again to the letter "F".
</pre>
===Grammar 3===
<pre>
| !C!(!P!). Grammar 3
| !C!(!P!). Grammar 3
|
|
Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
the set !K! of covering production rules as listed in the next display.
the set !K! of covering production rules as listed in the next display.
</pre>
===Grammar 4===
<pre>
| !C!(!P!). Grammar 4
| !C!(!P!). Grammar 4
|
|
Grammar 5 is a context-free grammar for the painted cactus language
Grammar 5 is a context-free grammar for the painted cactus language
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
</pre>
===Grammar 5===
<pre>
| !C!(!P!). Grammar 5
| !C!(!P!). Grammar 5
|
|
accumulating all the benefits of the context-free format in Grammar 5.
accumulating all the benefits of the context-free format in Grammar 5.
A plausible synthesis of most of these features is given in Grammar 6.
A plausible synthesis of most of these features is given in Grammar 6.
</pre>
===Grammar 6===
<pre>
| !C!(!P!). Grammar 6
| !C!(!P!). Grammar 6
|
|