Enumeration of Context-Free Languages and Related Structures

In this paper, we consider the enumeration of context-free languages. In particular, for any reasonable descriptional complexity measure for context-free grammars, we demonstrate that the exact number of context-free languages of size n is uncomputable. Nevertheless, we are able to give upper and lower bounds on the number of such languages. We also generalize our uncomputability results to a general theorem applicable to enumeration of equivalence classes or yes-instances of predicates.

[1]  Jozef Gruska,et al.  Complexity and Unambiguity of Context-Free Grammars and Languages , 1971, Inf. Control..

[2]  Jeffrey Shallit,et al.  Simulating finite automata with context-free grammars , 2002, Inf. Process. Lett..

[3]  Sampath Kannan,et al.  A Quasi-Polynomial-Time Algorithm for Sampling Words from a Context-Free Language , 1997, Inf. Comput..

[4]  Jeffrey Shallit,et al.  On the Number of Distinct Languages Accepted by Finite Automata with n States , 2002, DCFS.

[5]  Dennis F. Cudia.  The degree hierarchy of undecidable problems of formal grammars , 1970, STOC.

[6]  Alberto Bertoni,et al.  The Complexity of Computing the Number of Strings of Given Length in Context-Free Languages , 1991, Theor. Comput. Sci..

[7]  Jeffrey Shallit,et al.  Enumerating Regular Expressions and Their Languages , 2004, CIAA.

[8]  Pál Dömösi.  Unusual Algorithms for Lexicographical Enumeration , 2000, Acta Cybern..

[9]  Jim Gray,et al.  The Theory of Sequential Relations , 1966, Inf. Control..

[10]  Timothy V. Griffiths.  The unsolvability of the Equivalence Problem for Λ-Free nondeterministic generalized machines , 1968, JACM.

[11]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[12]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[13]  Gheorghe Paun,et al.  Grammar Systems , 1997, Handbook of Formal Languages.

[14]  H. Gordon Rice,et al.  On completely recursively enumerable classes and their key arrays , 1956, Journal of Symbolic Logic.

[15]  Michael Domaratzki,et al.  Enumeration of Formal Languages , 2006, Bull. EATCS.

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