Conjunctive grammars over a unary alphabet: undecidability and unbounded growth

Authors:
Artur Jeż;Alexander Okhotin
Affiliations:
Institute of Computer Science, University of Wrocław, Poland;Academy of Finland and Department of Mathematics, University of Turku, Finland
Venue:
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
Year:
2007

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Abstract

It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some nonregular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as nonexistence of an r.e. bound on the growth rate of generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.