Difference between revisions of "User:Jon Awbrey/SANDBOX"
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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== | ||
| + | |||
| + | <br> | ||
| + | |||
{| border="1" | {| border="1" | ||
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''› | | rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''› | ||
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|} | |} | ||
|} | |} | ||
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<br> | <br> | ||
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|} | |} | ||
|} | |} | ||
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<br> | <br> | ||
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<br> | <br> | ||
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<br> | <br> | ||
Revision as of 20:02, 19 January 2009
Grammar Stuff
Working from a structural description of the cactus language, or any suitable formal grammar for \(\mathfrak{C} (\mathfrak{P}),\) it is possible to give a recursive definition of the function called \(\operatorname{Parse}\) that maps each sentence in \(\operatorname{PARCE} (\mathfrak{P})\) to the corresponding graph in \(\operatorname{PARC} (\mathfrak{P}).\) One way to do this proceeds as follows:
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).
---
- The parse of the concatenation \(\operatorname{Conc}_{j=1}^k\) of the sequence of \(k\!\) sentences \((s_j)_{j=1}^k\) is defined recursively as follows:
- \(\operatorname{Parse} (\operatorname{Conc}^0) ~=~ \operatorname{Node}^0.\)
-
For \(\ell > 1,\!\)
\(\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Conc}_{j=1}^{\ell - 1} s_j \, \cdot \, s_\ell.\)
- The surcatenation \(\operatorname{Surc}_{j=1}^k\) of the sequence of \(k\!\) strings \((s_j)_{j=1}^k\) is defined recursively as follows:
- \(\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
-
For \(\ell > 1,\!\)
\(\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
Table Stuff
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