Changes

There are several abuses of notation that commonly tolerated in the use of covering relations.  The worst offense is that of allowing symbols to stand equivocally either for individual strings or else for their types.  There is a measure of consistency to this practice, considering the fact that perfectly individual entities are rarely if ever grasped by means of signs and finite expressions, which entails that every appearance of an apparent token is only a type of more particular tokens, and meaning in the end that there is never any recourse but to the sort of discerning interpretation that can decide just how each sign is intended.  In view of all this, I continue to permit expressions like <math>t <: T\!</math> and <math>T <: S,\!</math> where any of the symbols <math>t, T, S\!</math> can be taken to signify either the tokens or the subtypes of their covering types.

There are several abuses of notation that commonly tolerated in the use of covering relations.  The worst offense is that of allowing symbols to stand equivocally either for individual strings or else for their types.  There is a measure of consistency to this practice, considering the fact that perfectly individual entities are rarely if ever grasped by means of signs and finite expressions, which entails that every appearance of an apparent token is only a type of more particular tokens, and meaning in the end that there is never any recourse but to the sort of discerning interpretation that can decide just how each sign is intended.  In view of all this, I continue to permit expressions like <math>t <: T\!</math> and <math>T <: S,\!</math> where any of the symbols <math>t, T, S\!</math> can be taken to signify either the tokens or the subtypes of their covering types.

'''Note.'''  For some time to come in the discussion that follows, although I will continue to focus on the cactus language as my principal object example, my more general purpose will be to develop the subject matter of the formal languages and grammars.  I will do this by taking up a particular method of ''stepwise refinement" and using it to extract a rigorous formal grammar for the cactus language, starting with little more than a rough description of the target language and applying a systematic analysis to develop a sequence of increasingly

'''Note.'''  For some time to come in the discussion that follows, although I will continue to focus on the cactus language as my principal object example, my more general purpose will be to develop the subject matter of the formal languages and grammars.  I will do this by taking up a particular method of ''stepwise refinement" and using it to extract a rigorous formal grammar for the cactus language, starting with little more than a rough description of the target language and applying a systematic analysis to develop a sequence of increasingly more effective and more exact approximations to the desired grammar.

more effective and more exact approximations to the desired grammar.

Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.

Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.