Grammatical Families Research Articles

On bounded interpretations of grammar forms

An interpretation of a grammar form is called ( k, i)-bounded iff all its nonterminals are substituted by at most k symbols and all its terminals are replaced by at most i words. The ( k, i)-bounded grammar family of a grammar form is the collection of its ( k, i)-bounded interpretations, and its ( k, i)-bounded grammatical family is the corresponding family of languages. The paper gives basic properties of these families. Especially, we show the decidability of the equivalence problem for bounded grammar families, the undecidability of the membership problem for bounded language families and give some hierarchy, closure and descriptional complexity results. Finally, some consequences in normal form theory of context-free languages are presented.

Denseness, maximality, and decidability of grammatical families

Denseness, maximality, and decidability of grammatical families

Density of grammar forms. part I

We give two characterizations for dense intervals consisting of grammatical families. These results are general enough to cover the eases of context-free grammar forms, general grammar forms and synchronized EOL forms. By means of the characterizations it is proved that denseness is undecidable for context-free forms. The characterization theorems also enable us to investigate the maximality of dense intervals. The paper is divided into two parts. The results mentioned above are presented in Part II. This first part contains definitions and preliminary results. Also in Part I, we investigate language families defined by subregular forms and clarify the role of the empty word in form theory.

On contextual grammar forms

The grammatical families of all languages generated by simple contextual grammars which are obtained by (strict) interpretations of a simple contextual grammar are investigated. Their hierarchies and their closure properties with respect to the AFL-operations are studied.

On the equality of grammatical families

A decision procedure involving the testing for containment is presented for determining whether two grammatical families (families generated by context-free grammar forms) are equal. It is also shown that every nontrivial grammatical family which is a proper subset of the family of context-free languages can be constructed in a unique canonical way from the family of regular sets by certain operations. Thus, two such grammatical families are equal iff their canonical representations are identical.

A prime decomposition theorem for grammatical families

Two basic operations on families of languages, in general, and grammatical families, in particular, are sum (formed by taking unions of member languages) and product (formed by taking unions of pairwise products of member languages). In this paper, a grammatical family is defined to be prime if it is contained in one of two grammatical families whenever it is contained in their product, and the following Prime Decomposition Theorem is then established: Every grammatical family can be represented as a minimal sum of products of primes in a unique way. This theorem leads to a general method for decomposing a grammatical family into simpler ones. A subsequent paper uses this method to obtain a decision procedure for determining whether two grammar forms generate the same grammatical family, as well as a canonical representation for grammatical families.

Decidability and density in two-symbol grammar forms

It is proved that form equivalence is decidable for context-free grammar forms with only one nonterminal and one terminal symbol. However it also proved that there are “many” grammatical families generated by such forms: with some trivial exceptions, the families are dense in the sense that between any two families one can “squeeze” in a third one. The results obtained are applied also to L forms.

On predecessors of finite languages

We continue the study of finite grammatical families by characterizing when a family has a predecessor. This characterization implies that (i) the existence of predecessors for finite families is decidable and (ii) the collection of context-free grammatical families, each family of which contains all regular languages, is not maximal dense.

The structure of context-free grammatical families

Let ℒCI be the family of context-free languages. Two characterization theorems are given for (context-free) grammatical families. The first says that the class of grammatical families not ℒCF is the smallest collection of sets of languages which contains the trivial ones and is closed under a union operator ∨, a concatenation operator ⊙, and ℐ, where ℐ is a ternary operator involving substitution into linear languages. The second asserts that the collection of all nontrivial grammatical families not ℒCF is the smallest collection of sets of languages which contains the family of regular sets, and is closed under ∨, ⊙, ℐ, and the full AFL operator ℱ^.

Substitution of grammar forms

Given grammar forms F and F?, the grammar form Sub (F, Ft') is defined as that obtained by substituting the start variable of F? for every occurrence of a terminal in F. The main result is that if F is a nontrivial grammar form, then the grammatical family defined by Sub (F, F?) is the set of languages obtained by substituting languages in the family defined by F? into the family defined by F. Thus the substitution of one grammatical family into another is a grammatical family. It follows as a corollary that the full AFL generated by a grammatical family is a grammatical family.