Associate Languages and Derivational Complexity of Formal Grammars and Languages

A criterion to measure derivational complexity of formal grammars and languages is proposed and discussed. That is, the associate language and the L-associate language are defined for a grammar such that the former represents all the valid derivations and the latter represents all the valid leftmost derivations. It is shown that for any phrase\3-structure grammar, the associate language is a contex\3-sensitive language and the L\3-associate language is a context\3-free language. Necessary and sufficient conditions for an associate language to be a regular set and to be a context\3-free language are found. The idea in the above necessary and sufficient conditions is extended to the notion of “rank≓ for a measure of derivational complexity of context\3-free grammars and languages. It is shown that for each nonnegative integer k, there exists a context\3-free language whose rank is k. The paper also includes a few solvable decision problems concerning derivational complexity of grammars.

[1]  John E. Hopcroft,et al.  Simple Deterministic Languages , 1966, SWAT.

[2]  Peter S. Landweber,et al.  Decision Problems of Phrase-Structure Grammars , 1964, IEEE Trans. Electron. Comput..

[3]  Jozef Gruska,et al.  Some Classifications of Context-Free Languages , 1969, Inf. Control..

[4]  Seymour Ginsburg,et al.  The mathematical theory of context free languages , 1966 .

[5]  Otto Mayer,et al.  Some Restrictive Devices for Context-Free Grammars , 1972, Inf. Control..

[6]  Jeffrey D. Ullman,et al.  Formal languages and their relation to automata , 1969, Addison-Wesley series in computer science and information processing.

[7]  Seymour Ginsburg,et al.  Mappings which Preserve Context Sensitive Languages , 1966, Inf. Control..

[8]  Ronald V. Book.  Time-Bounded Grammars and Their Languages , 1971, J. Comput. Syst. Sci..

[9]  Ranan B. Banerji,et al.  Some Problems of Finite Representability , 1965, Inf. Control..

[10]  A. Nerode,et al.  Linear automaton transformations , 1958 .

[11]  Arto Salomaa,et al.  Periodically Time-Variant Context-Free Grammars , 1970, Inf. Control..

[12]  Jozef Gruska.  On a classification of context-free languages , 1967, Kybernetika.

[13]  Noam Chomsky,et al.  On Certain Formal Properties of Grammars , 1959, Inf. Control..

[14]  Daniel J. Rosenkrantz,et al.  Programmed Grammars and Classes of Formal Languages , 1969, JACM.

[15]  S. ABRAHAM,et al.  SOME QUESTIONS OF PHRASE STRUCTURE GRAMMARS, I , 1967 .

[16]  G. H. Matthews.  A Note on Assymmetry in Phrase Structure Grammars , 1964, Inf. Control..

[17]  Arthur C. Fleck,et al.  On the Combinatorial Complexity of Context-free Grammars , 1971, IFIP Congress.

[18]  John E. Hopcroft,et al.  Scattered Context Grammars , 1969, J. Comput. Syst. Sci..

[19]  S. Ginsburg,et al.  Finite-Turn Pushdown Automata , 1966 .

[20]  Barron Brainerd.  An Analog of a Theorem about Contex-Free Languages , 1967, Inf. Control..

[21]  Arto Salomaa,et al.  On the Index of a Context-Free Grammar and Language , 1969, Inf. Control..

[22]  Takumi Kasai,et al.  An Hierarchy Between Context-Free and Context-Sensitive Languages , 1970, J. Comput. Syst. Sci..

[23]  Ranan B. Banerji,et al.  Phrase Structure Languages, Finite Machines, and Channel Capacity , 1963, Inf. Control..