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

Revision as of 13:05, 22 January 2009

790 bytes added ,  13:05, 22 January 2009
→‎The Cactus Language : Semantics: set table
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| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
 
| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
−
| width="20%" | <math>\operatorname{Node}^k_j c_j</math>
+
| width="20%" | <math>\operatorname{Node}^k_j C_j</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
 
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
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| width="20%" | <math>\operatorname{Surc}^k_j s_j</math>
 
| width="20%" | <math>\operatorname{Surc}^k_j s_j</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
−
| width="20%" | <math>\operatorname{Lobe}^k_j c_j</math>
+
| width="20%" | <math>\operatorname{Lobe}^k_j C_j</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
 
| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
 
| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
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<br>
 
<br>
   −
<pre>
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
Table 14.2  Semantic Translations : Equational Form
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|+ '''Table 14.2  Semantic Translation : Equational Form'''
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o-------------------o-----o-------------------o-----o-------------------o
+
|- style="background:whitesmoke"
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|                   | Par |                   | Den |                   |
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|
−
| -[Sentence]-      | = | -[Graph]-        | = | Proposition       |
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:whitesmoke; width:100%"
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o-------------------o-----o-------------------o-----o-------------------o
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| width="20%" | <math>\downharpoonleft \operatorname{Sentence} \downharpoonright</math>
−
|                   |     |                   |     |                   |
+
| width="20%" | <math>\stackrel{\operatorname{Parse}}{=}</math>
−
| -[S_j]-          |  = | -[C_j]-          | = | Q_j              |
+
| width="20%" | <math>\downharpoonleft \operatorname{Graph} \downharpoonright</math>
−
|                   |     |                   |     |                  |
+
| width="20%" | <math>\stackrel{\operatorname{Denotation}}{=}</math>
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o-------------------o-----o-------------------o-----o-------------------o
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| width="20%" | <math>\operatorname{Proposition}</math>
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|                   |     |                   |     |                   |
+
|}
−
| -[Conc^0]-        |  = | -[Node^0]-        | = | %1%              |
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|-
−
|                   |     |                   |     |                   |
+
|
−
| -[Conc^k_j S_j]- | = | -[Node^k_j C_j]- | = | Conj^k_j Q_j    |
+
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
|                   |    |                  |    |                  |
+
| width="20%" | <math>\downharpoonleft s_j \downharpoonright</math>
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o-------------------o-----o-------------------o-----o-------------------o
+
| width="20%" | <math>=\!</math>
−
|                   |     |                   |     |                   |
+
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
−
| -[Surc^0]-        |  = | -[Lobe^0]-        | = | %0%              |
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| width="20%" | <math>=\!</math>
−
|                   |     |                   |     |                   |
+
| width="20%" | <math>q_j\!</math>
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| -[Surc^k_j S_j]- | = | -[Lobe^k_j C_j]- | = | Surj^k_j Q_j    |
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|}
−
|                   |    |                  |    |                  |
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|-
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o-------------------o-----o-------------------o-----o-------------------o
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|
−
</pre>
+
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^0 \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \operatorname{Node}^0 \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\underline{1}</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \operatorname{Surc}^0 \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \operatorname{Lobe}^0 \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\underline{0}</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
 +
|}
 +
|}
 +
 
 +
<br>
    
Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.  Table&nbsp;14.1 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j\!</math> as basic data, and fixing the rest by recursion on these.  Table&nbsp;14.2 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes.  It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation.  Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.
 
Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.  Table&nbsp;14.1 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j\!</math> as basic data, and fixing the rest by recursion on these.  Table&nbsp;14.2 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes.  It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation.  Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.
Jon Awbrey
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