Changes

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> that is afforded by Grammar&nbsp;2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of these expressions.  In order to observe the effects of this alteration as clearly as possible, in isolation from any other potential factors, it is useful to strip away the higher levels intermediate organization that are present in Grammar&nbsp;3, and start again with a single intermediate symbol, as used in Grammar&nbsp;2.  One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> that is afforded by Grammar&nbsp;2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of these expressions.  In order to observe the effects of this alteration as clearly as possible, in isolation from any other potential factors, it is useful to strip away the higher levels intermediate organization that are present in Grammar&nbsp;3, and start again with a single intermediate symbol, as used in Grammar&nbsp;2.  One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar&nbsp;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 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&nbsp;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 production rules as listed in the next display.

===Grammar 4===

===Grammar 4===

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar&nbsp;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&nbsp;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.

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