Edwin H. Spanier

Control sets on grammars

Given a setC of strings of rewriting rules of a phrase structure grammarG, we consider the setLc(G) of those words generated by leftmost derivations inG whose corresponding string of rewriting rules is an element ofC. The paper concerns the nature of the setLc(G) whenC andG are assumed to have special form. For example, forG an arbitrary phrase structure grammar,Lc(G) is an abstract family of languages ifC is an abstract family of languages, andLc(G) is bounded ifC is bounded.

BOUNDED REGULAR SETS

Abstract : Two characterizations of bounded regular sets are given. In addition, certain bounded regular sets are related to their commutative closure. (Author)

Derivation-bounded languages

A derivation in a phrase-structure grammar is said to be k-bounded if each word in the derivation contains at most k occurrences of nonterminals. A set L is said to be derivation bounded if there exists a phrase-structure grammar G and a positive integer k such that L is the set of words in the language generated by G which have some k-bounded derivation. The main result is that every derivation-bounded set is a context-free language. Various characterizations of the derivation-bounded languages are then given. For example, the derivation-bounded languages coincide with the standard matching-choice sets discussed by Yntema. They also coincide with the smallest family of sets containing the linear context-free languages and closed under arbitrary substitution, a family discussed by Nivat.

AFL with the semilinear property

A slip language is a language whose Parikh mapping is a semilinear set. A slip family is a family containing only slip languages. The purpose of this paper is to study slip AFL. A sufficiency condition is given on a slip family which ensures that the family generates a slip AFL. Using this condition, it is proved that (i) there exists a largest slip AFL and (ii) if @? is a slip family, then the smallest AFL containing the commutative closure of @? is a slip AFL. A new operation called homomorphic replication is then introduced. It is shown that the smallest AFL containing a homomorphic replication of a slip AFL is also a slip AFL. Furthermore, the resulting AFL is principal if the original AFL is principal. It is then proved that the smallest AFL containing all homomorphic replications of the regular sets is not principal. Finally, abstract families of acceptors are presented which, respectively, define the smallest AFL containing a particular homomorphic replication of the regular sets and all homomorphic replications of the regular sets.

Mappings of languages by two-tape devices

Several devices with two input lines and one output line are introduced. These devices are viewed as transformations which operate on pairs of (ALGOL-like) languages. Among the results proved are the following: (i) a pair consisting of a language and a regular set is transformed into a language; (ii) let (<italic>V, W</italic>) be a pair consisting of a language and a regular set. Then the set of those words <italic>w</italic><subscrpt>1</subscrpt>, for which there exists a word <italic>w</italic><subscrpt>2</subscrpt> in <italic>V</italic> so that (<italic>w</italic><subscrpt>1</subscrpt>, <italic>w</italic><subscrpt>2</subscrpt>) is mapped into <italic>W</italic>, is a language.

Quotients of Context-Free Languages

Abstract : The following results on the quotient of context free languages CFL are shown: (1) It is recursively unsolvable to determine for arbitrary CFL whether the quotient of one by another is a CFL. (2) If either set is regular and the other is a CFL, then the quotient is a CFL. (Author)

Substitution in families of languages

The effect of substitution in families of languages, especially an AFL (i.e., abstract family of languages), is considered. Among the main results shown are the following: The substitution of one AFL into another is an AFL. Under suitable hypotheses, the AFL generated by the family obtained from the substitution of one family into another is the family obtained from the substitution of the corresponding AFL. A condition is given for the AFL generated by the substitution closure of a family to be the substitution closure of the AFL generated by the family.

The structure of context-free grammatical families*

Let ℒ CI 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 ℒ CF 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 ℒ CF 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 ^ .

On incomparable abstract family of languages (AFL)

Given an abstract family of languages (AFL) @? the question is considered if there exists an AFL incomparable with @?. In case there is an AFL @? incomparable with @? the paper considers if there exists a largest AFL incomparable with @?, and if there is a maximal AFL containing @? incomparable with @?. The main results characterize those full AFL @? having a largest full incomparable AFL @? and relate properties of @? to properties of @?.

Substitution of grammar forms

SummaryGiven grammar forms F and F′, the grammar form Sûb (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 Sûb (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.