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).
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- 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 \(\ell > 1,\!\)
\(\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
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