The decidability of the equivalence of context-free grammar forms

Abstract

In 1975 Cremers and Cinsburg published a paper that described the notion of a "grammar form" or a master grammar [CG]. A grammar form enables one to group grammars into grammatical families with similar production rules. All grammars dnd grammar forms discussed both in this paper and in the work of Cremers and Ginsburg are restricted to the context free. In the publication mentioned above an open problem was posed: is the weak equivalence of grammar forms deciable'? In the 1977 Colloquium on Automata and Formal Languages in Szeged, Hungary,it was announced that Ginsburg, Goldstine and Spanier believe they have a solution to the decidability of the equivalence of grammar forms [Gi]. For the past two years they have been working on a complete manuscript of that proof. A number of years ago the author of this abstract began to work with certain constructs called "matched languages" which seemed to lend themselves to the solution of the equivalence problem. In the abs trac t "Inheren t Amb ig ui ties in the solution to the equivalence problem is implicit in the results. The current abstract sununarizes the techniques and develops the specific results needed to probe the equivalence problem based on the material in "Inherent Ambiguities." These ideas are summarized informally in the next paragraph. A context-free grammar form G is a context-free grammar tOgetller with a set of substitutions into ~he produc~ion rules of G or llinterpretations" that replace terminal symbols by strings of terminals and nonterminals by nonterminals with the condition that distinct nonterminals replace distinct non-terminals. The results of such substitutions are grammars. The grammatical family of the grammar form is simply the family of languages generated by these interpretations. A grammar G = (V, ,P, S) is expansive if there is a nonterminal X in G such thatX;> llXVXW, where u, v, ware in V* and Xt>w for some WEL+. If a grammar is expansive then its grammatical family is the entire set of context-free grammars while if the grammar is not expansive its grammatical family must lie within the derivation bounded grammars. The ~erivation bounded grammars are structurally between the nonterminal bounded and the expansive grammars. Many problems which are un-decidable for arbitrary context-free grammars are decidable for the nonterminal bounded due to their underlying structure which can be characterized by scmilinear sets formed by the languages they generate [CS]. A similar type of characterization …