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Directory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 17:06, 6 January 2009
, 17:06, 6 January 2009→Grammar 4: format grammar box
If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>^{\backprime\backprime} S ^{\prime\prime}</math> and <math>^{\backprime\backprime} T ^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>^{\backprime\backprime} S ^{\prime\prime}, \, ^{\backprime\backprime} \, S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T ^{\prime\prime}, \, ^{\backprime\backprime} \, T' \, ^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} \, S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T ^{\prime\prime}, \, ^{\backprime\backprime} \, T' \, ^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.
If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>^{\backprime\backprime} S ^{\prime\prime}</math> and <math>^{\backprime\backprime} T ^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>^{\backprime\backprime} S ^{\prime\prime}, \, ^{\backprime\backprime} \, S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T ^{\prime\prime}, \, ^{\backprime\backprime} \, T' \, ^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} \, S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T ^{\prime\prime}, \, ^{\backprime\backprime} \, T' \, ^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.
<br>
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
| align="left" style="border-left:1px solid black;" width="50%" |
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} \, S' \, ^{\prime\prime}, \, ^{\backprime\backprime} \, T \, ^{\prime\prime}, \, ^{\backprime\backprime} \, T' \, ^{\prime\prime} \, \}</math>
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
<math>\begin{array}{rcll}
1.
& S
& :>
& \varepsilon
\\
2.
& S
& :>
& S'
\\
3.
& S'
& :>
& m_1
\\
4.
& S'
& :>
& p_j, \, \text{for each} \, j \in J
\\
5.
& S'
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
\\
6.
& S'
& :>
& S' \, \cdot \, S'
\\
7.
& T
& :>
& \varepsilon
\\
8.
& T
& :>
& T'
\\
9.
& T'
& :>
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
\\
\end{array}</math>
|}
<br>
<pre>
<pre>
In this version of a grammar for !L! = !C!(!P!), the intermediate type T
In this version of a grammar for !L! = !C!(!P!), the intermediate type T
is partitioned as T = %e% + T', thereby parsing the intermediate symbol T
is partitioned as T = %e% + T', thereby parsing the intermediate symbol T