Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

Authors:
Artur Jeż;Alexander Okhotin
Affiliations:
University of Wrocław, Institute of Computer Science, Wrocław, Poland;University of Turku, Department of Mathematics, Turku, Finland and Academy of Finland, Helsinki, Finland
Venue:
Theory of Computing Systems - Special Issue: Symposium on Computer Science, Guest Editors: Sergei Artemov, Volker Diekert and Dima Grigoriev
Year:
2009

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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 non-regular 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 non-existence of a recursive function bounding the growth rate of the generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.