Monthly Archives: June 2025

It is fitting to wrap up the foregoing developments by summarizing the notion of a formal grammar which appeared to evolve in the analysis of cactus languages.  For the sake of future reference and further application it is useful to … Continue reading →

Grammar 6 Grammar 6 has the intermediate alphabet with the set of covering rules listed in the next display. Our exploration of the grammar space for the language shows how an initially effective and succinct definition of a formal language … Continue reading →

Grammar 5 With the foregoing array of considerations in mind, one is gradually led to a grammar for in which all of the covering productions have one of the following two forms. A grammar fitting that description is called a … Continue reading →

Grammar 4 (concl.) As we have seen, Grammar 4 partitions the intermediate type as in parallel fashion with the division of its overlying type as   That is an option we will close off for now but leave open to consider … Continue reading →

Grammar 4 If one imposes the distinction between empty and significant types on each non‑terminal symbol in Grammar 2 then the symbols and give rise to the expanded set of non‑terminal symbols leaving the last three to form a new intermediate … Continue reading →

Grammar 3 (concl.) Returning to the cactus language and fixing the parameter for the moment, we have a language of painted and rooted cactus expressions.  It serves the purpose of efficient accounting to divide the language into two sublanguages. The emptily … Continue reading →

Grammar 3 (cont.) In Grammar 3, the first three Rules say a sentence (a string of type ), is either a rune (a string of type ), a foil (a string of type ), or formed by concatenating strings of … Continue reading →

Grammar 3 It is possible to organize the materials of our developing grammar in a more easily graspable fashion by recognizing two recurring types of strings appearing in typical cactus expressions.  In doing so one arrives at the next two definitions. … Continue reading →

Grammar 2 One way to analyze the surcatenation of any number of sentences is to introduce an auxiliary type of string, not itself a sentence but a proper component of any sentence formed by surcatenation.  Doing that brings one to … Continue reading →

Grammar 1 (concl.) Returning to the case of the cactus language, the process of recognizing iterative or recursive types can be illustrated in the following way.  The operative phrases in the simplest form of recursive definition are its initial part … Continue reading →