One-Nonterminal Conjunctive Grammars over a Unary Alphabet

: Jez A, Okhotin A

Publisher: SPRINGER

: 2011

: Theory of Computing Systems

: THEORY OF COMPUTING SYSTEMS

: THEOR COMPUT SYST

: 2

: 49

: 2

: 319

: 342

: 24

: 1432-4350

DOI: https://doi.org/10.1007/s00224-011-9319-6

Conjunctive grammars over an alphabet I ={a} pound are studied, with the focus on the special case with a unique nonterminal symbol. Such a grammar is equivalent to an equation X=I center dot(X) over sets of natural numbers, using union, intersection and addition. It is shown that every grammar with multiple nonterminals can be encoded into a grammar with a single nonterminal, with a slight modification of the language. Based on this construction, the compressed membership problem for one-nonterminal conjunctive grammars over {a} is proved to be EXPTIME-complete; the same problem for the context-free grammars is decidable in NLOGSPACE, but becomes NP-complete if the grammar is compressed as well. The equivalence problem for these grammars is shown to be co-r.e.-complete, both finiteness and co-finiteness are r.e.-complete, while equivalence to a fixed unary language with a regular positional notation is decidable.