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User:Jon Awbrey/SANDBOX (view source)
Revision as of 20:02, 19 January 2009
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==Grammar Stuff==
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).</math> One way to do this proceeds as follows:
<pre>
1. The parse of the concatenation Conc^k of the k sentences S_j,
for j = 1 to k, is defined recursively as follows:
a. Parse(Conc^0) = Node^0.
b. For k > 0,
Parse(Conc^k_j S_j) = Node^k_j Parse(S_j).
2. The parse of the surcatenation Surc^k of the k sentences S_j,
for j = 1 to k, is defined recursively as follows:
a. Parse(Surc^0) = Lobe^0.
b. For k > 0,
Parse(Surc^k_j S_j) = Lobe^k_j Parse(S_j).
</pre>
---
<ol style="list-style-type:decimal">
<li>The parse of the concatenation <math>\operatorname{Conc}_{j=1}^k</math> of the sequence of <math>k\!</math> sentences <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
<ol style="list-style-type:lower-alpha">
<li><math>\operatorname{Parse} (\operatorname{Conc}^0) ~=~ \operatorname{Node}^0.</math>
<li>
<p>For <math>\ell > 1,\!</math></p>
<p><math>\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Conc}_{j=1}^{\ell - 1} s_j \, \cdot \, s_\ell.</math></p></li>
</ol>
<li>The ''surcatenation'' <math>\operatorname{Surc}_{j=1}^k</math> of the sequence of <math>k\!</math> strings <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
<ol style="list-style-type:lower-alpha">
<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
<li>
<p>For <math>\ell > 1,\!</math></p>
<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
</ol></ol>
==Table Stuff==
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