One-Nonterminal Conjunctive Grammars over a Unary Alphabet

Authors:
Artur Jeż;Alexander Okhotin
Affiliations:
University of Wrocław, Institute of Computer Science, Wrocław, Poland;University of Turku, Department of Mathematics, Turku, Finland
Venue:
Theory of Computing Systems
Year:
2011

Citing 0
Cited 2

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Abstract

Conjunctive grammars over an alphabet Σ={a} are studied, with the focus on the special case with a unique nonterminal symbol. Such a grammar is equivalent to an equation X=ϕ(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.