Equivalent Context-free Grammar Research Articles

On the Complexity of Nonassociative Lambek Calculus with Unit

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). Moreover, we show that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time.

On Greibach normal form construction

We present an algorithm which, given an arbitrary ε-free and chain-free context-free grammar, produces an equivalent context-free grammar in Greibach normal form (GNF). This algorithm is a generalization of an algorithm recently presented by Ehrenfeucht and Rozenberg. Using this algorithm the well-known GNF construction of Rosenkrantz can be obtained. Furthermore it is possible to rewrite a given context-free grammar by a GNF grammar which has at most twice as much nonterminals as the original grammar. This result is shown to be optimal (with respect to the number of nonterminals).

An easy proof of Greibach normal form

We present an algorithm which given an arbitrary A -free context-free grammar produces an equivalent context-free grammar in 2 Greibach normal form. The upper bound on the size of the resulting grammar in terms of the size of the initially given grammar is given. Our algorithm consists of an elementary construction, while the upper bound on the size of the resulting grammar is not bigger than the bounds known for other algorithms for converting context-free grammars into equivalent context-free grammars in Greibach normal form.

Direct parsing of ID/LP grammars

The Immediate Dominance/Linear Precedence (ID/LP) formalism is a recent extension of Generalized Phrase Structure Grammar (GPSG) (Gazdar and Pullum, 1982; Gazdar and Pullum, 1981) designed to perform some of the tasks previously assigned to metarules - for example, modeling the word-order characteristics of so-called free-word-order languages. (See, e.g., Stucky, 1981; Pullum, 1982; Uszkoreit, 1982.) It allows a simple specification of classes of rules that differ only in constituent order. ID/LP grammars (as well as metarule grammars) have been proposed for use in parsing by expanding them into equivalent context-free grammars (Gazdar and Pullum, 1982; Thompson, 1982). We develop a parsing algorithm, based on the algorithm of Earley, for parsing ID/LP grammars directly, circumventing the initial expansion phase. A proof of correctness of the algorithm is supplied. We also discuss some aspects of the time complexity of the algorithm and some formal properties associated with 1D/LP grammars and their relationship to context-free grammars.

Operator Precedence Grammars and the Noncounting Property

The notion of noncounting language, initially introduced for regular languages recognized by counter-free finite machines, and recently extended to parenthesized context-free languages, is here further studied for general (i.e., nonparenthesized) context-free languages. While weakly equivalent context-free grammars do not, in general, fall in the same class with respect to the noncounting property, it is shown by a complex proof that weakly equivalent operator precedence grammars are all counting or all noncounting (a property which distinguishes the operator precedence languages from classical deterministically parsable families).