The following definitions round out the concepts we need to begin applying formal grammar theory to the efficient description of formal languages, in particular, the family of cactus languages.

It is convenient to refer to the full set of symbols in as the augmented alphabet of the candidate formal grammar for and thus to refer to the strings in as the augmented strings of the grammar for in effect, articulating the forms superimposed on a language by one of its conceivable grammars.

In certain settings it becomes desirable to separate the augmented strings containing the symbol from all other cases of augmented strings.  In those situations the strings in the disjoint union are known as the sentential forms of the associated grammar.

In forming a grammar for a language, statements of the form where and are augmented strings or sentential forms of specified types which depend on the style of the grammar being sought, are variously known as characterizations, covering rules, productions, rewrite rules, subsumptions, transformations, or typing rules.  Statements of that form are collected together in a set which serves to complete the definition of the formal grammar in question.

The relation has the converse form which may be read as is covered by and understood in the sense that is of the type   Depending on the context, can be taken in one of the following two ways.

• Treating as a string variable, it means the individual string is an instance of the type
• Treating as a type name, it means every string of the type is an instance of the type

In light of the above conventions, an expression of the form can be read in all the ways one typically reads an expression of the form

Resources

• Cactus Language • Preliminaries
• Survey of Animated Logical Graphs
• Survey of Theme One Program

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